5 Geometry for DraftingDrafting Career Architect Zaha Hadid’s designs for the Cincinnati Contemporary Art Center were “like a rollercoaster, a little scary, but exhilarating,”

Section 5.1 Applied Geometry for Board Drafting

Section 5.2 Applied Geometry forCAD Systems

5 Geometry for Drafting

Chapter ObjectivesIdentify geometric shapes and construc-tions used by drafters.Construct various geometric shapes.Solve technical and mathematical prob-lems through geomet-ric constructions using drafting instruments.Solve technical and mathematical prob-lems through geomet-ric constructions using a CAD system.Use geometry to reduce or enlarge a drawing or to change its proportions.

•

•

•

•

•

Defying Convention It has been said that Zaha Hadid has built a career on defying convention—conventional ideas of architectural space, and of construction. What do you see in the building shown here that defi es convention?

132

Drafting Career

Architect Zaha Hadid’s designs for the Cincinnati Contemporary Art Center were “like a rollercoaster, a little scary, but exhilarating,” says Center direc-tor Charles Desmarais. Critics said “she was a paper architect, someone who had great respect as a theo-rist and as a thinker about architecture but who hadn't had the opportunity to build.”

“She totally got what we were trying to do,” said

Desmarais, “which was to try and bridge that sort of gap between the inside and the outside, between the world and the museum.” She certainly did. Zaha Hadid is the fi rst woman in the world to design a museum and to win the prestigious Pritzker Architec-ture Prize.

Academic Skills and AbilitiesMath Computer sciencesBusiness management skillsVerbal and written communication skillsOrganizing and planning skills

Career PathwaysThere is a wealth of opportunities outside the

classroom for expanding your drafting knowledge. Learn about annual drafting contests. Even if you do not intend to apply, read about the projects. Find groups such as the Solar-Powered Car Chal-lenge; their ideas will inspire you.

•••••

Go to glencoe.com for this book’s OLC to learn more about Zaha Hadid.

Zaha Hadid, Architect

133Zaha Hadid

http://www.glencoe.com

Applied Geometry for Board Drafting

5.1

Preview In this chapter, you will learn to construct geometric shapes using board drafting techniques. Have you learned geometric terms and formulas in other courses?

Content Vocabulary• geometry• geometric

construction

• vertex• bisect• perpendicular

• parallel• polygon• inscribe

• circumscribe• regular

polygon

• ellipse

Academic VocabularyLearning these words while you read this section will also help you in your other subjects and tests.• accurate • methods

Graphic Organizer

Use a table like the one below to organize the major concepts about the types of geometric constructions.

Academic Standards

English Language Arts

Students read a wide range of print and nonprint texts to build an understanding of texts (NCTE)

Mathematics

Students recognize and use connections among mathematical ideas (NCTM)

NCTE National Council of Teachers of English

NCTM National Council of Teachers of Mathematics

Go to glencoe.com for this book’s OLC for a downloadable version of this graphic organizer.

Bisect Construct Lines Construct

1. Arc 1. Triangle 1.

2. 2. 2.

3. 3.

4.

5.6.

134 Chapter 5 Geometry for Drafting


4

3

5

A

BC

90°

4 = 162

3 = 92

5 = 25

2

A + B = C2 2 2

3 + 4 = C2 2 2

9 + 16 = 25

Geometry and Geometric ConstructionsWhat do you need to be able to understand geometric constructions?

Geometry is the study of the size and shape of objects, and of as the relationship between straight and curved lines in draw-ing shapes. In ancient times, geometrywas used for measuring land and makingaccurate right-angle, or 90-degree, corners for constructing buildings and other proj-ects. When building the great pyramids, Ancient Egyptians formed right-angle cor-ners by using rope with marks or knots at 3-, 4-, and 5-space sections, and stretching the rope around carefully placed pegs driven into the ground. (See Figure 5-1.)

In the sixth century BCE, the math-ematician Pythagoras studied this method of forming right angles and proved the theory that the 3-4-5 triangle makes a right angle. This theorem (a2 + b2 = c2) or proof, called the Pythagorean theorem, is shown in Figure 5-2.

This method also works well for triangles that have the same proportions, such as 6, 8, and 10 units:

62 + 82 = 10036 + 64 = 100100 = 100

Figure 5-1

Egyptian rope-stretchers used knots divided into 3-4-5 triangles to lay out square corners for buildings.

Figure 5-2

The Pythagorean theorem shown graphically and mathematically

Section 5.1 Applied Geometry for Board Drafting 135

STRAIGHT LINE(SHORTEST DISTANCE

BETWEEN TWO POINTS)

PARALLEL LINES

POINT OF INTERSECTION

INTERSECTING LINESB

RIGHT ANGLE (90°)A

COMPLEMENTARYANGLES

SUPPLEMENTARY ANGLES

180°

A

B

60°

60° 60°

EQUILATERAL TRIANGLEALL SIDES EQUAL LENGTH

BASEISOSCELES TRIANGLE

TWO SIDES EQUAL LENGTH

SID

E

SID

ESYMBOL FORRIGHT ANGLE (90°)

90°ALT

ITU

DE

HYPOTEN

USE

SIDE

SIDE

BASE

SCALENE TRIANGLE

SEMI-CIRCLE

CHORDDIAMETER

RADIUS

SEGMENT

QUADRANT (ONE-QUARTER OF A CIRCLE)

SECTOR

ANGLE

TANGENT ARC

POINT OFTANGENCY

TANGENT LINE

RIGHT ANGLESIN A SEMI-CIRCLE

CONCENTRIC CIRCLES ECCENTRIC CIRCLES

EQUALSIDES

90° ANGLES

SQUARE

OPPOSITE SIDESARE EQUAL

90° ANGLES

RECTANGLE

EQUAL SIDES

OPPOSITEANGLES

ARE EQUAL

RHOMBUS

OPPOSITE SIDESARE EQUAL

OPPOSITE ANGLESARE EQUAL

RHOMBOID

TWO SIDESARE PARALLEL

TRAPEZOID

NO TWO SIDESARE PARALLEL

TRAPEZIUM

5 SIDES

PENTAGON

6 SIDES

HEXAGON

8 SIDES

OCTAGON

12 SIDES

DODECAGON

7 SIDES

HEPTAGON

9 SIDES

NONAGON

10 SIDES

DECAGON

Figure 5-3

Dictionary of drafting geometry


Illustrations made of individual lines and points drawn in proper relationship to one another are known as geometric con-structions. Drafters, surveyors, engineers, architects, scientists, mathematicians, and designers use geometric constructions.

To understand geometric constructions, you must understand how to describe various lines, arcs, and other shapes. This chapter fol-lows the identifi cation rules used in geometry.

The units may be millimeters, meters, inches, fractions of an inch, or any other unit of measure. Geometric fi gures used in drafting include circles, squares, triangles, hexagons, and octagons. Many other shapes are shown in Figure 5-3.

A2 � B2 � C2

32 � 42 � C2

9 � 16 � 25

Geometry Formulas In addition to solving drafting problems using geometric constructions, drafters often need to be able to calculate various aspects of geometric constructions. While hundreds of these formulas exist, a few are given here as examples.

What is the area of triangle A where the base is 10″ and the height is 7″?

To fi nd the area of any tri-angle, multiply the base (b) times the height (h) and divide by two.

Example:Area = bh/2Area = 2 × 6/2Area = 6 square inches

The diameter of a circle is 15″. What is the circumference?

To fi nd the circumference of a circle, multiply pi (π) times the diam-eter of the circle. The approximate decimal equivalent of pi is 3.1416.

Example:Circumference = πdCircumference = 3.1416 × 2.50Circumference = 7.85

For help with this math activity, go to the Math Appendix at the back of this book.

6"

2"

ALTITUDE (h)

BASE (b)

DIAMETER (d) = 2.50

Academic Standards

Mathematics

Measurement Apply appropriate techniques, tools,

and formulas to determine measurements (NCTM)


A

B

A

B

A

B

A

B

C

R

D

R

R

DR

C

A

B

A

B

C

R

D

R

R

DR

C

F

E

E

F

A B C

A BLINE AB

B

A

C

ANGLE ABC

A

B

ARC AB

A

CIRCLE A

Figure 5-5

Bisecting a straight line, an arc, and an angle

Figure 5-4

Identifi cation of lines, angles, arcs, and circles

Lines and arcs are described using their end-points. Therefore, line AB is a line segment that extends from point A to point B. Arc AB is an arc whose endpoints are A and B. Angles are described using three points: both end-points and the vertex, or the point at which the two arms of the angle meet. Angle ABC is an angle whose endpoints are A and C and whose vertex is at point B. Circles are usually specifi ed using their center points, so circle A is a circle whose center is at point A. See Figure 5-4.

Explain How is the Pythagorean theorem used in geometry?

Bisect a Line, an Arc, or an AngleBisect means to divide into two equal parts.

Bisect a Line or an ArcFollow these steps to bisect a straight line

or an arc.

1. Draw a line AB and arc AB as shown in Figure 5-5A.

2. With points A and B as centers and any radius R greater than one-half of AB, draw arcs to intersect, or cross, line AB as in Figure 5-5B. The radius is the dis-tance from the center of an arc or circle to any point on the arc or circle. The two places where the arcs intersect create points C and D.

3. Draw line EF through points C and D (Figure 5-5C).


A

C

B A

C

BA

C

B

1

0

2

3

4

5

6

7

8

A B C

A

O

B

GIVEN ANGLE

A

A

O

B

R1

C

D

B

A

O

B

R1

C

D

R2

R2

E

AOE = EOB

C

Figure 5-6

Bisecting an angle

Figure 5-7

Dividing a straight line into any number of equal parts

Bisect an AngleThis construction demonstrates a method

for bisecting a given angle. Refer to Figure 5-6.

1. Draw given angle AOB (Figure 5-6A). 2. With point O as the center and any con-

venient radius R1, draw an arc to inter-sect AO and OB to locate points C and D (Figure 5-6B).

Divide a Line into Any Number of Equal Parts

Two methods of dividing a line into equal parts are described next. Try both methods. Can you think of situations in which you would need to use one method instead of the other?

Divide a Line into Equal PartsThis method can be applied to create any

number of equal divisions. In this construc-tion, you will divide a straight line into eight equal parts. Refer to Figure 5-7 and follow these steps:

3. With C and D as centers and any radius R2 more than one- half the radius of arc CD, draw two arcs to intersect, locating point E. (Figure 5-6C).

4. Draw a line through points O and E to bisect angle AOB (Figure 5-6C).

1. Construct a line of any length at A per-pendicular to line AB, as in Figure 5-7A. Lines are perpendicular when they cross at 90° angles.

2. Position the scale, placing zero on line AC at such an angle that the scale touches point B, as in Figure 5-7B. Keeping zero on line AC, adjust the angle of the scale until any eight equal divisions are included between line AC and point B (in this case, at 8″). Mark the divisions.

3. Draw lines parallel to AC through the division marks to intersect line AB Figure 5-7C. Two lines are parallel when they are always the same distance apart.


A

O

B

A

O

O

B

C

D

R1

A

B

C

D

R1

R2

R2

A B C

A B

C

A B

C 54

32

1

A B C

A B

C5

43

21

Divide a Line into Five

Equal PartsFollow these instructions to divide a line

into fi ve equal parts.

1. Draw line BC from point B at any conve-nient angle and length (Figure 5-8A).

2. Use dividers or a scale to step off fi ve equal spaces on line BC beginning at point B (Figure 5-8B).

3. Draw a line connecting point A and the last point on line BC (Figure 5-8C.) Draw lines through each point on BC parallel to this line as shown.

Summarize Explain how to bisect a line using board drafting techniques.

Figure 5-9

Constructing a line perpendicular to a given line through a given point on the line (Method 1)

Figure 5-8

Dividing a straight line into fi ve equal parts

Construct a Perpendicular Line

Each one of the many procedures to con-struct a line perpendicular to another line is useful in certain drafting situations. Four methods are discussed below.

Method 1Figure 5-9A shows the given line AB and

point O that lies on line AB. Follow these steps to draw a line at point O on line AB so that the two lines are perpendicular.

1. Draw line AB and point O (Figure 5-9A).2. With O as the center and any conve-

nient radius R1, construct an arc inter-secting line AB, locating points C and D (Figure 5-9B).

3. With C and D as centers and any radius R2 larger than OC, draw arcs intersecting at point E (Figure 5-9C).

4. Draw a line connecting points E and O toform the perpendicular line (Figure 5-9C).


BO

A

BO

A

C

D

R

BO

A

C

D

E

A B C

A

O

B

A

O

B

A

O

B

C

D

A B C

Method 2Use this method when the given point

through which a perpendicular line is drawn lies near one end of the line.

1. Construct given line AB and point O (Figure 5-10A).

2. From any point C above line AB, con-struct an arc using CO as the radius and

passing through line AB to locate point D (Figure 5-10B).

3. Construct a line through points D and C, extending it through the arc to locate point E (Figure 5-10C).

4. Connect points E and O to form the per-pendicular line (Figure 5-10C).

Figure 5-10

Constructing a line perpendicular to a given line through a given point on the line

Figure 5-11

Constructing a line perpendicular to a given line through a given point on the line

Method 3This construction demonstrates another

way to draw a line perpendicular to a given line through a given point on the line. Follow the steps to create a line at O that is perpen-dicular to line AB. Refer to Figure 5-11.


2. Place the T-square and triangle (Figure 5-11B).

3. Slide the triangle along the T-square until the edge aligns with point O on line AB (Figure 5-11C).

4. Draw a perpendicular line through point O (Figure 5-11C).


P

A

BP

A

BR1R1

D

C

P

A

BR1R1

D

C

R2

E

A B C

Method 4Figure 5-12A shows another given line AB

and point O that is not on the line. Follow the steps to practice another way to draw a line perpendicular to a given line through a point that is not on the line.


2. Construct lines from point O to any two points on line AB, locating points C and D (Figure 5-12B).

3. With C and D as centers and CO and DO as radii, draw arcs to intersect, locating point E (Figure 5-12C).

4. Connect points O and E to form the per-pendicular line (Figure 5-12C).

Identify What are perpendicular lines?

Figure 5-13

Using a compass to construct a line parallel to a given line through a given point

O

B

A

O

B

AC

D

O

B

AC

D

EA B C

Figure 5-12

Constructing a line perpendicular to a given line through a point that is not one the given line

Draw a Parallel LineThe following construction methods create

a line that is parallel to another line. Recall that lines are parallel when they are always the same distance apart.

Method 1This construction allows you to place a line

parallel to a given line. Refer to Figure 5-13.

1. Draw given line AB and point P (Figure 5-13A).

2. With point P as the center and any convenient radius R1, draw an arc

intersecting line AB to locate point C (Figure 5-13B).

3. With point C as the center and the same radius R1, draw an arc through point P and line AB to locate point D (Figure 5-13B).

4. With C as the center and radius R2 equal to chord PD, draw an arc to locate point E. A chord is a straight line between two points on a circle (Figure 5-13C).

5. Draw a line through points P and E. Line PE is parallel to line AB (Figure 5-13C).


A

BA

A

B

R

R

B

A

B

C

DR

R

C

Method 2The following steps demonstrate another

way to construct a line parallel to another line through a given point. Refer to Figure 5-14.

1. Draw given line AB and point P (Figure 5-14A).

Method 3Use this method to construct a line parallel to

a given line at a specifi ed distance from the given line. Refer to Figure 5-15. Note: See “Construct a Tangent Line to a Circle” later in this chapter for instructions on creating a tangent line.

1. Draw given line AB (Figure 5-15A).2. Draw two arcs with centers anywhere along

line AB. The arcs should have a radius R equal to the specifi ed distance between the two parallel lines (Figure 5-15B).

Figure 5-15

Constructing a line parallel to a given line at a specifi ed distance from the given line

2. Place the T-square and triangle (Figure 5-14B).

3. Slide the triangle until the edge aligns with point P (Figure 5-14C).

4. Draw a parallel line through point P. See (Figure 5-14C).

3. Draw a parallel line CD tangent to the arcs. Recall that a line is tangent to an arc or circle when it touches the arc or circle at one point only (Figure 5-15C).

Explain What is a chord?

A

B

P

A

B

P

A

B

P

C

D

A B C

Figure 5-14

Using a triangle and T-square to construct a line parallel to a given line through a given point


BASEBA BA

R R

VERTEX

BA

EQUAL SIDES OFDESIRED LENGTH

A B

C

Copy an AngleThis construction demonstrates a method

of copying a given angle to a new location and orientation. Refer to Figure 5-16.

1. Draw given angle AOB (Figure 5-16A).2. Draw one side O1A1 in the new position

(Figure 5-16B).3. With O and O1 as centers and any conve-

nient radius R1, construct arcs to intersect

BO and AO at C and D and A1O1 at D1. Refer again to Figure 5-16B.

4. With D1 as the center and radius R2 equal to chord DC, draw an arc to locate point C1 at the intersection of the two arcs (Figure 5-16C).

5. Draw a line through points O1 and C1

to complete the angle. Refer again toFigure 5-16C.

Figure 5-17

Constructing an isosceles triangle

Construct a TriangleA triangle is a polygon, or closed fi gure,

with three sides. The following constructions show methods for drawing various types of triangles.

Method 1This method constructs an isosceles triangle,

which has two sides that are of equal length. Refer to Figure 5-17.

1. Draw base line AB (Figure 5-17A).2. With points A and B as centers and a

radius R equal to the length of the two sides you want, draw intersecting arcs to locate the third vertex of the triangle (Figure 5-17B). The other two vertices (plural of vertex) are at the endpoints of the base line.

3. Draw lines through point A and the ver-tex and through point B and the vertex to complete the triangle (Figure 5-17C).

AO

BGIVEN ANGLE

GIVENPOSITION

AO

BGIVENPOSITION

D

CR1

R1 NEWPOSITION

A1

O1D1

AO

B

GIVENPOSITION

D

C

C1

NEWPOSITION

A1

O1D1

CHORD

R2

R2

A

B C

Figure 5-16

Copying an angle


Method 2This method constructs an equilateral trian-

gle. An equilateral triangle is one in which all three sides are of equal length and all three angles are equal. Refer to Figure 5-18.

1. Draw base line AB as in Figure 5-18A.2. With points A and B as centers and a

radius R equal to the length of line AB,

AB

BC

A B

C

A B

C

A B C

Figure 5-19

Constructing a right triangle given the lengths of two sides

draw intersecting arcs to locate the third vertex Refer again to Figure 5-18B.

3. Draw lines through point A and the vertex and through point B and the ver-tex to complete the triangle (Figure 5-18C).

Method 3Construct a right triangle using this method

when you know the length of two sides of the triangle. A right triangle is one that has a right (90°) angle at one of its vertices. Given sides AB and BC are shown in Figure 5-19A.

1. Draw side AB in the desired position (Figure 5-19B).

2. Draw a line perpendicular to AB at B equal to BC. Note: Construct the perpen-dicular line using the method in Figure 5-11 or Figure 5-12.

3. Draw a line connecting points A and C to complete the right triangle (Figure 5-19C).

A BBASE

A B

VERTEX

R

A B

60°

60°

60°

A B C

Figure 5-18

Constructing an equilateral triangle


Method 4Use this method to construct a right triangle

when you know the length of one side and the length of the hypotenuse. See Figure 5-20A for the given side AB and hypotenuse AC.

1. Draw the hypotenuse AC in the desired location (Figure 5-20B).

2. Draw a semicircle on AC using ½AC as the radius. Refer again to Figure 5-20B.

3. With point A as the center and a radius equal to side AB, draw an arc to intersect the semicircle to locate point B (Figure 5-20C).

4. Draw line AB and then draw a line to connect B and C to complete the triangle (Figure 5-20C).

Method 5You can use this method to construct a tri-

angle when you know the lengths of all three sides. This method is useful for constructing scalene triangles, which have three different angles and sides of three different lengths. Figure 5-21A shows given triangle sides AB, BC, and AC.

1. Draw base line AB in the desired location.2. Construct arcs from the ends of line AB

with radii equal to lines BC and AC to locate point C (Figure 5-21B).

3. Connect both ends of line AB with point C to complete the triangle (Figure 5-21C).

Compare What is the diff erence between an isosceles triangle and an equilateral triangle?

HYPOTENUSE

SIDE

A

A

C

B

A C

12 AC

A C

B

A B C

Figure 5-20

Constructing a right triangle given the length of one side and the length of the hypotenuse.

A B

B C

A C

A

A B

R = AC

R = BC

C

B

A B

C

C

Figure 5-21

Constructing a triangle given the lengths of all three sides


Construct a CircleThis construction describes a method for

creating a circle given three points that lie on the circle. Refer to Figure 5-22.

1. Given points A, B, and C, draw lines AB and BC (Figure 5-22A).

2. Draw perpendicular bisectors of AB and BC to intersect at point O (Figure 5-22B).

3. Draw the required circle with point O as the center and radius R = OA = OB =OC (Figure 5-22C).

Construct Lines Tangent to a Circle

The constructions that follow present methods of creating lines tangent to a circle. As you may recall, a line that touches a circle at one point only is said to be tangent to the circle.

Method 1Use this method to construct a line tangent

to a given point on a circle without using a triangle or T-square. Refer to Figure 5-23.

1. Given a circle with center point O and tangent point P (Figure 5-23A), draw line OA from the center of the circle to extend beyond the circle through point P.

2. Draw a line perpendicular to line OA at P (Figure 5-23B). The perpendicular line is the tangent line.

A

B

C

A

B

CO

A

B

CO

A B

C

Figure 5-22

Constructing a circle given three points that lie on the circle

A

P

O

A

P

O

A B

Figure 5-23

Constructing a line tangent to a circle through a given point on the circle (Method 1)

O

P FIRST POSITION

SECOND POSITION

Figure 5-24

Constructing a line tangent to a given point on a circle (Method 2)


to a given point on a circle using a 30°-60° tri-angle and a T-square. See Figure 5-24.

1. Given a circle with center point O and tan-gent point P, place a T-square and triangle so that you can construct the hypotenuse of the triangle through points P and O.

2. Hold the T-square, turn the triangle to the second position at point P, and draw the tangent line.


Method 3This method creates lines tangent to a circle

from a given point outside the circle. See Figure 5-25.

1. Draw a circle with center point O and point P outside the circle (Figure 5-25A).

2. Draw line OP and bisect it to locate point A (Figure 5-25B).

3. Draw a circle with center A and radius R = AP = AO to locate tangent points T1

and T2 (Figure 5-25B). 4. Draw lines PT1 and PT2 (Figure 5-25C).

These lines are tangent to the circle.


to the exterior of two circles. Refer to Figure 5-26.

1. Draw the two given circles with centers O1 and O2 and radii R1 and R2 (Figure 5-26A).

2. Draw a circle with center O1 and a radius R, where R = R1 − R2. Refer again to Figure 5-26A.

3. From center point O2, draw a tangent O2T to the circle of radius (Figure 5-26B).

4. Draw radius O1T as shown in Figure5-26B, and extend it to locate point T1.

5. Draw the needed tangent T1T2 parallel to TO2 (Figure 5-26C).

O

P

O

T2

APT1

O

T2

APT1

A B C

Figure 5-25

Constructing a line tangent to a circle from a given point outside the circle (Method 3)

O2

R2

O1

RR1

O2 O1

R

T2

T1

T

O2 O1

T2

T1

T

A B C

Figure 5-26

Constructing an exterior common tangent to two circles of unequal radii (Method 4)


A B

O2

R2

O1

R

R1O2

O1

T1

T2

T

C

O2O1

T1

T2

T

Figure 5-27

Constructing an interior common tangent to two circles of unequal radii (Method 5)

Method 5Use this method to construct a line tangent to

the interior of two circles. Refer to Figure 5-27.

1. Draw the two given circles with centers O1

and O2 and radii R1 and R2 (Figure 5-27A). 2. Draw a circle with center O1 and a radius

R, where R = R1 + R2. Refer again to Figure 5-27A.

3. From center point O2, draw a tangent O2T to the circle of radius R (Figure 5-27B).

4. Draw radius O1T to locate point T1 (Figure 5-27B).

5. Draw O2T2 parallel to O1T. 6. Draw the needed tangent T1T2 parallel to

TO2 (Figure 5-27C).

Identify What two tools are used in some of the methods described in Section 5.1?

Construct Arcs Tangent to Straight Lines and Other Arcs

The following are methods for drawing arcs tangent to other geometric fi gures, such as straight lines and other arcs.

Construct an Arc Tangent to

Two Straight LinesThe technique is shown for two lines at

an acute angle, an obtuse angle, and a right angle. Refer to Figure 5-28.

1. Given lines AB and CD (Figure 5-28A), draw lines parallel to AB and CD at a dis-tance R from them on the inside of the angle. The intersection O will be the cen-ter of the arc you need.

2. Draw perpendicular lines from O to AB and CD to locate the points of tangency T (Figure 5-28B).

3. With O as the center and radius R, draw the needed arc (Figure 5-28C).



Two Given ArcsRefer to Figure 5-29 for the steps in con-

structing this arc.

1. Draw two arcs having radii R1 and R2

(Figure 5-29A). The radii R1 and R2 may be equal or unequal.

2. Draw an arc with center O1 and radius =R + R1, where R is the radius of the desired tangent arc. See Figure 5-29B. The intersection O is the center of the tangent arc.

3. Draw lines O1O and O2O to locate tan-gent points T1 and T2 (Figure 5-29C).

4. With point O as the center and radius R, draw the tangent arc needed.

ACUTE ANGLE

A

B

R

OR

D

C

R

A

T

B

O

C

T

D

R

A

T

B

O

T

C

D

D

O

B C

R

R

A

OBTUSE ANGLE

D

TO

CBTA

D

R

O

TCBTA

R

R

A

B

O

D

RIGHT ANGLE

C T D

A

T O

B

R

D

A

T

B

C T

O

A B C

Figure 5-28

Constructing an arc tangent to two straight lines at an acute angle, an obtuse angle, and a right angle

R1

O1R2

O2

R+R1

O1 O2

R+R2

O

O1 O2

R

O

T1 T2

RADIUS OFTANGENT ARC

A B C

Figure 5-29

Constructing an arc tangent to two given arcs


Construct an Arc Tangent to a

Line and an ArcUse this method to construct an arc tan-

gent to a line and an arc, given the line, the arc, and the radius R of the desired tangent arc. Refer to Figure 5-30.

1. Draw given line AB and arc CD as shown in Figure 5-30A.

2. Draw a line parallel to line AB, at dis-tance R, toward arc CD. (Figure 5-30B).

3. Use radius R1 + R to locate point O1. Refer again to Figure 5-30B.

4. Draw a line from O1 perpendicular to AB to locate tangent point T.

5. Draw a line from O to O1 to locate tan-gent point T1 on CD (Figure 5-30C).

6. With point O1 as the center and radius R, draw the tangent arc.

Recall What three types of angles do you create constructing an arc tangent to two lines?

A

R1

C O

D

B

A

B

RADIUS R

R1C

O D

A

R + R1

C O

D

BT

O1

A

B

C

O D

R + R1

O1

T

A O

D

BT

O1

A

R1

T1

B

C

O D

T1

O1

TR1

A B C

Figure 5-30

Constructing an arc tangent to line and an arc

3

1 2

A B

Construct a SquareA square is a rectangle with all four sides

equal. You can construct a square in several ways. The method you choose depends on the other geometry in the drawing.

Construct a Square When the

Length of One Side Is KnownUse this method to construct a square

when you know the length of a side. Refer to Figure 5-31.

1. Given the length of the side AB, draw line AB.

2. Construct 45° diagonals from the ends of line AB. Refer again to Figure 5-31.

3. Complete the square by drawing perpen-dicular lines at each end of line AB to intersect the diagonals. Draw the last line from the intersection of the diagonal and the vertical lines. Draw the lines in the order shown by the numbered arrows.

Figure 5-31

Constructing a square given the length of a side


Construct a Square Inscribed in

a CircleA square or other polygon is inscribed in

a circle when its four corners are tangent to the circle. Refer to Figure 5-32. 1. Draw the given circle with center point O. 2. Draw 45° diagonals through the center

point O to locate points A, B, C, and D. Refer again to Figure 5-32.

3. Connect points A and B, B and C, C and D, and D and A to complete the square.

Construct a Square

Circumscribed in a CircleA square or other polygon is circumscribed

about a circle when the square fully encloses the circle and the circle is tangent to the square on all four sides. Refer to Figure 5-33.

1. Draw the given circle with center point O. 2. Draw 45° diagonals through the center

point O. 3. Draw sides tangent to the circle, inter-

secting at the 45° diagonals, to complete the square.

A B

O

CD

Figure 5-32

Constructing a square inscribed within a circle

Construct a PentagonA pentagon is a fi ve-sided polygon. When its

fi ve sides are exactly the same length and all of its angles are equal, it is called a regular polygon. The following methods demon-strate the construction of regular pentagons.

Regular Pentagon When the

Length of One Side Is KnownTo use this method, refer to Figure 5-34.

1. Given line AB, construct a perpendicular line AC equal to one-half of the length of AB.

O

Figure 5-33

Constructing a square circumscribed about a circle

A B

CD

A B

C

D O O

F

E

BA

G

A B C

2. Draw line BC and extend it to make line CD equal to AC. Refer to Figure 5-34Afor steps 1 and 2.

3. With radius AD and points A and B as centers, draw intersecting arcs to locate point O (Figure 5-34B).

4. With the same radius and O as the cen-ter, draw a circle.

5. Step off AB as a chord to locate points E, F, and G. Connect the points to complete the pentagon (Figure 5-34C).

Figure 5-34

Constructing a regular pentagon given the length of one side


A BO

C

A BO

C

D

R

E

F

R

A BO

C

F

J H

G

A B C

Inscribe a Pentagon within a

CircleRefer to Figure 5-35 for this method.

1. Draw the given circle with diameter AB and radius OC (Figure 5-35A). The diameter of a circle is the distance across the circle through its center point. (The symbol for diameter is Ø.)

2. Bisect radius OB to locate point D (Figure 5-35B).

3. With D as center and radius DC, draw an arc to locate point E.

4. With C as center and radius CE, draw an arc to locate point F.

5. Draw chord CF. This chord is one side of the pentagon.

6. Step off chord CF around the circle to locate points G, H, and J. Draw the chords to complete the pentagon (Figure 5-35C).

Explain What is a regular polygon?

Figure 5-35

Inscribing a regular pentagon within a circle

Construct a HexagonA hexagon is a six-sided polygon. The fol-

lowing methods demonstrate construction for regular hexagons, which have six sides of equal length, six internal angles of equal size, and six external angles of equal size.

Construct a Regular Hexagon

When the Distance across the

Flats Is KnownThis method constructs a regular hexagon

when you know the distance across the fl ats, or sides. The distance across the fl ats is the distance from the midpoint of one side through the center point to the midpoint of the opposite side of the polygon. Refer to Figure 5-36.

1. Given the distance across the fl ats of a regular hexagon, draw centerlines and a circle with a diameter equal to the dis-tance across the fl ats.

DISTANCEACROSS FLATS

3

4

1

2

5

6

Figure 5-36

Constructing a regular hexagon given the distance across the fl ats

2. With the T-square and 30°-60° triangle, draw the tangents in the order shown inFigure 5-36.


Construct a Regular Hexagon

When the Distance across the

Corners Is Known

Method 1Use this method to construct a regular

hexagon when you know the distance across the corners. The distance across the corners is the distance from one vertex through the center point to the opposite vertex. Refer to Figure 5-37.

1. Given the distance AB across the corners, draw a circle with AB as the diameter.

2. With A and B as centers and the same radius, draw arcs to intersect the circle at points C, D, E, and F.

3. Connect the points to complete the hexagon.

A B

C D

EF

DISTANCE ACROSSCORNERS

Figure 5-37

Constructing a regular hexagon given the distance across the corners (Method 1)

Method 2 This construction demonstrates another

method of constructing a regular hexagon given the distance across the corners. Refer to Figure 5-38.

1. Given the distance AB across the corners, draw lines from points A and B at 30° to line AB. The lines can be any convenient length.

2. With the T-square and 30°-60° triangle, draw the sides of the hexagon in the order shown.

Explain What is a fl at of a hexagon?

A B

1

2

3

4

5

6

60°

30°60°

30°

Figure 5-38

Constructing a regular hexagon given the distance across the corners (Method 2)

3

1

2

6

4

5 7

8

Figure 5-39

Constructing a regular octagon circumscribed about a circle given the distance across the fl ats

Construct an OctagonAn octagon is an eight-sided polygon. The

following methods demonstrate the construc-tion of regular octagons.

Construct an Octagon

Circumscribed about a CircleRefer to Figure 5-39 as you follow the steps

in constructing an octagon circumscribed about a circle.

1. Given the distance across the fl ats, draw centerlines and a circle with a diameter equal to the distance across the fl ats.

2. With the T-square and 45° triangle, draw lines tangent to the circle in the order shown to complete the octagon.


Construct an Octagon Inscribed

within a CircleRefer to Figure 5-40 as you follow the

steps for constructing an octagon inscribed within a circle.

1. Given the distance across the corners, draw centerlines AB and CD and a circle with a diameter equal to the distance across the corners.

2. With the T-square and 45° triangle, draw diagonals EF and GH.

3. Connect the points to complete the octagon.

Construct an Octagon Inscribed

within a SquareRefer to Figure 5-41 as you follow the

steps to construct an octagon inscribed within a square.

1. Given the distance across the flats, con-struct a square having sides equal to AB.

2. Draw diagonals AD and BC with their intersection at O. With A, B, C, and D as centers and radius R = AO, draw arcs to intersect the sides of the square.

3. Connect the points to complete the octagon.

Describe How many sides does an octagon have?

45°A

G

F

D

H

B

EC

Figure 5-40

Inscribing a regular octagon within a circle given the distance across the corners of the octagon

A B

O

C D

Figure 5-41

Inscribing a regular octagon within a square given the distance across the fl ats

(Figure 5-42A). The major axis AB and minor axis CD are given. They intersect at O.

1. With C as center and radius R = AO, draw an arc to locate points F1 and F2 (Figure 5-42A).

2. Place pins at points F1, C, and F2 (Figure 5-42B).

3. Tie a string around the three pins and remove pin C.

4. Put the point of a pencil in the loop and draw the ellipse. Keep the string tight when moving the pencil (Figure 5-42C).

Construct an EllipseAn ellipse is a regular oval. It is sym-

metrical around two axes that form a right angle. The shorter axis is the minor axis, and the longer one is the major axis. This sec-tion demonstrates methods for drawing an ellipse.

Pin-and-String Method to

Construct an EllipseThis illustration demonstrates the use of the

pin-and-string method of drawing a large ellipse


AF1 R = AO

D

O

C

BF2

AF1

D

O

C

BF2

A

D

O

C

B

A B C

Figure 5-42

Constructing an ellipse by the pin-and-string method

Trammel Method to Construct

an EllipseThis method demonstrates the use of the

trammel to draw an ellipse. A trammel is a piece of paper or plastic on which specifi c distances have been marked off. Figure 5-43A shows the major axis AB and minor axis CD, intersecting at O.

1. Cut a strip of paper or plastic to use as a trammel. Mark off distances AO and OD on the trammel (Figure 5-43A).

2. On the trammel, move point O along minor axis CD and point D along major axis AB and mark points at A (Figure 5-43B).

3. Use a French curve or irregular curve to connect the points to draw the ellipse (Figure 5-43C).

FIRSTPOSITION

SECONDPOSITION

A

C

O

DD

Bo

o

da

aA

C

O

D

B

o

d

a

A B C

Figure 5-43

Constructing an ellipse by the trammel method

Use of Major and Minor Axes to

Construct an EllipseThis method constructs an approximate

ellipse by using its major and minor axes. This method works when the minor axis is at least two-thirds the size of the major axis. Figure 5-44A shows the major axis AB and minor axis CD, intersecting at O.

1. Lay off OF and OG, each equal to AB –CD Refer again to Figure 5-44A.

2. Lay off OJ and OH, each equal to three-fourths of OF.

3. Draw and extend lines GJ, GH, FJ, and FH (Figure 5-44B).


D

G

OJA

F

C

H B

D

G

O

JA

F

C

HB

D

G

O

JA

F

C

HB

T

T T

T

A B C

Figure 5-44

Constructing an approximate ellipse when the minor axis is at least two-thirds the size of the major axis

4. Draw arcs with centers F and G and radii FD and GC to the points of tangency (Figure 5-44C).

5. Draw arcs with centers J and H and radii JA and HB to complete the ellipse. The points of tangency are marked T in (Figure 5-44C).

Identify What tool is used with the trammel method?

Reduce or Enlarge a Drawing

The following techniques reduce or enlarge an existing drawing.

Reduce or Enlarge a Square or

Rectangular DrawingIf a drawing is square or rectangular, use a

diagonal line method to reduce or enlarge it. Refer to Figure 5-45.

1. Draw a diagonal through corners D and B. 2. Measure the width or height you need

along DC or DA (example: DG). 3. Draw a perpendicular line from that point

(G) to the diagonal. 4. Draw a line perpendicular to DE intersect-

ing at point F.

E

A

D

F

B

CG

ENLARGED SIZE

ORIGINAL SIZE

REDUCED SIZE

DIAGONAL

Figure 5-45

Reducing or enlarging a square or rectangular area



That Is Not Square or RectangularUse this method to reduce or enlarge a

drawing that is not square or rectangular. Refer to Figure 5-46.

1. Draw a grid larger or smaller than the one shown at B. The size of the grid depends on the amount of enlargement or reduc-tion needed.

2. Use dots to mark key points on the sec-ond grid corresponding to points on the original drawing at A.

3. Connect the points and darken the lines to complete the new drawing.

1 2 3 4 5 6 71 2 3 4 5 6 70

1

2

3

4

5

6

8

7

0123456

87

A

B C

Figure 5-46

Reducing or enlarging a drawing of a sailboat

Section 5.1 AssessmentAfter You Read

Self-Check 1. List various geometric shapes and con-

structions used by drafters. 2. Describe one method for constructing

a geometric shape. 3. Explain how to solve technical and

mathematical problems through geometric constructions using drafting instruments.

Academic Integration Mathematics

4. Calculate Circumference Calculate the circumference of a circle with a dia-meter of 2.50 inches.

Calculating Circumference

To fi nd the circumference of a circle:

DIAMETER (d) = 2.50

Multiply pi (π) times the diameter of the circle. The approximate decimal equivalent of pi is 3.1416.Circumference = πdCircumference = 3.1416 × 2.50″Circumference = 7.85″

Drafting Practice 5. Draw the gasket shown in Figure 5-47.Before beginning, determine an appropriate scale and sheet size. Do not dimension.

Ø24

Ø64 R48

178

R24

2x

METRIC

Figure 5-47

Go to glencoe.com for this book’s OLC for help with this drafting practice.



Preview In this section, you will learn to construct geomteric shapes using CAD techniques.

Content Vocabulary• object snap• ogee curve

Academic VocabularyLearning these words while you read this section will also help you in your other subjects and tests.• intervals • specify

Graphic Organizer

Use a diagram like the one below to organize the CAD commands discussed in the section.

Academic Standards

English Language Arts

Students employ a wide rage of strategies as they write and use diff erent writing process elements

appropriately to communicate with diff erent audiences for a variety of purposes (NCTE)

Use information resources to gather information and create and communicate knowledge (NCTE)

Mathematics

Geometry Specify location and describe spatial relationships using coordinate geometry and other

representational systems (NCTM)

NCTE National Council of Teachers of English


Go to glencoe.com for this book’s OLC for a downloadable version of this graphic organizer.

Applied Geometry for CAD Systems

5.2

Inscribe a polygon

POLYGON

Section 5.2 Applied Geometry for CAD Systems 159


A

B

A

B

A

B

A

B

C

D

A B

C

D

SNAP TOMIDPOINT

Using Geometry with CAD SystemsWhat do object snaps allow a drafter to do?

The techniques for creating geometry in AutoCAD and other CAD programs differ sig-nifi cantly from board drafting techniques. With CAD programs, the software creates the geometry, but you must understand the geo-metric principles before you can direct the software to create the geometry to achieve the correct result.

This section consists of a series of exam-ple exercises in which you will use CAD techniques to create the same geometry described in the fi rst section of this chapter. You can also use the same techniques described in that section. However, drafters who use CAD systems usually take advan-tage of the streamlined methods when the software offers them. By working through these constructions, you will begin to under-stand how to draw the basic geometry in AutoCAD.

To work through the constructions, open a new drawing in AutoCAD. Use the tem-plate specifi ed by your instructor, or start a new drawing using AutoCAD’s default acad.dwt template. Your instructor will advise you on how many constructions to include in each drawing fi le. Be sure to save your work frequently.

Object SnapsAutoCAD has a set of features known as

object snaps that allow you to “snap” auto-matically to important points on any Auto-CAD object. Object snaps you will use in this section include:

MidpointNearestEndpointCenter

••••

IntersectionQuadrantPerpendicularTangent

••••

Specifying the Intersection object snap, for example, allows you to snap to the intersec-tion of two existing lines or arcs. This can be useful if you have used two arcs to locate the beginning of a new line. Object snaps have many other uses, too, as you will see as you work through the following constructions. To

specify an object snap, type the fi rst three let-ters of its name.

Explain How do you specify an object snap?

Bisect or Divide a Line, an Arc, or an AngleWhat actions do the LINE, TRIM, and DIVIDE commands perform?

Lines and arcs are usually bisected to fi nd a beginning point for a new line or arc.

Bisect a Line or an ArcIn AutoCAD, the point that lies at the

exact middle of a line or arc is known as the midpoint. Because AutoCAD has a Midpoint object snap, bisecting a line or arc—fi nding its midpoint—is simply part of the construction of the new line or arc.

1. Draw a line and an arc (Figure 5-48A). 2. Enter the LINE command, but do not

enter a fi rst point. Instead, type MID (for midpoint) and press Enter.

3. At the “of” prompt, select the line you drew in step 1. Depending on the version of AutoCAD you are using, you may see a yellow triangle appear at the midpoint of the line. In any case, the fi rst point of the new line you are creating begins at the exact midpoint of the original line, shown as point C in Figure 5-48B.

Figure 5-48

Bisecting a line or arc in AutoCAD


A

O

B

A

A

O

B

C

D

B

A

O

B

C

D

C

A B

A B

4. Pick another point anywhere in the drawing area and press Enter to end the LINE command.

5. Repeat steps 2 through 4, but this time select the arc in step 3. This results in a line that starts at point C and bisects the arc (Figure 5-48B).

Bisect an AngleThe CAD method for bisecting an angle is

very similar to the board drafting method. Refer to Figure 5-49.

1. Use the LINE command to draw two con-nected line segments to create angle AOB (Figure 5-49A).

2. Enter the CIRCLE command and specify point O as its center point. Use the cur-

sor to draw a radius similar to the one in Figure Figure 5-49B.

3. Enter the TRIM command and press Enter to select all of the objects on the screen automatically. Then pick any point on the circle outside angle AOB. This procedure trims away all of the cir-cle except for an arc that extends from one arm of angle AOB to the other. See Figure 5-49C.

4. Enter the LINE command. Use the Inter-section object snap to place the fi rst point of the line at point O. Then use the Midpoint object snap to place the second point of the line at the exact midpoint of the arc. Refer again to Figure 5-49C. This line bisects angle AOB.

Figure 5-49

Bisecting an angle in AutoCAD

Figure 5-50

Dividing a line into equal parts in AutoCAD

Divide a Line into Eight Equal

PartsAutoCAD includes a DIVIDE command

that divides lines, arcs, and other geometry into equal parts. The following procedure divides a line into eight equal parts. Refer to Figure 5-50.

1. Draw a line of any length as in Figure 5-50A.

2. Enter the DIVIDE command. 3. When prompted for the number of seg-

ments, type 8 and press Enter.Markers appear at equal intervals along

the line to divide it into eight parts. If you cannot see these markers, you will need to change the point style. To do so, enter DDP-TYPE at the keyboard and select a different

point style from the dialog box that appears. See Figure 5-50B.

Explain Why might a line or arc need to be bisected?


B

O

A

USE THEPERPENDICULAROBJECT SNAP FORSECOND POINT

B

A

B

A

D

C

Construct Lines with a CAD System

Most CAD systems can construct a full vari-ety of lines.

Construct a Perpendicular LineFollow these steps to create a line perpen-

dicular to a given line. Refer to Figure 5-51.

1. Draw given line AB. 2. Reenter the LINE command and pick

point O as the fi rst point of the new line.

Presetting Object Snaps

If you know that you will be using certain object snaps frequently for a particular drawing, you can set AutoCAD to use them automatically, without having to specify them each time you use them. Object snaps that have been preset in this way are known as running object snaps. To set running object snaps, enter the OSNAP com-mand. A dialog box appears. Pick the Object Snap tab of the dialog box to see the available object snaps. Pick the check boxes next to the object snaps you want to run automatically and pick OK to close the dialog box.

3. Before specifying the second point of the line, type PER to enter the Perpendicular object snap. Then pick a point on line AB and press Enter. The resulting line is per-pendicular to line AB.

Construct Lines Parallel to a

Given LineTo create parallel lines in AutoCAD, use the

OFFSET command. Refer to Figure 5-52.

1. Draw the line AB. 2. Enter the OFFSET command and enter

an offset distance of 1. This will place the second line 1 unit away from line AB.

3. When prompted to select the object to offset, pick line AB.

4. When prompted for the side to offset, pick a point anywhere above line AB. The parallel line CD appears.

Notice that the OFFSET command is still active. You can offset as many lines or arcs as you want without reentering the command. This can save time when you are working on a technical drawing. 5. Press Enter to end the command.

Figure 5-51

Creating a line perpendicular to a given line through a point that does not lie on the given line

Figure 5-52

Creating a line parallel to a given line using the OFFSET command

Construct a PolygonAutoCAD provides a POLYGON command

to create regular polygons with 3 to 1,024 sides. Equilateral triangles and squares are examples of regular polygons that have three and four sides, respectively. The constructions in this section use the following geometry:

square, or four-sided polygonpentagon, or fi ve-sided polygonhexagon, or six-sided polygon

Create a SquareUse this method to construct a polygon, in

this case a square, when you know the length

•••


of one of its sides. It can be very useful when you need to construct a polygon that shares a line with other geometrical shapes in the drawing.

1. Enter the POLYGON command, and specify 4 as the number of sides. Press E (Edge) and pick a point on the screen.

2. Either pick another point on the screen for the second endpoint of the edge or use polar coordinates to specify where the endpoint should be. If you use polar coordinates, the length of the line you specify becomes the length of one side of the square. The square appears on the screen.

Inscribe a Pentagon in a CircleUse this method to inscribe a pentagon in

a circle with a known center point and radius. Refer to Figure 5-53.

1. Create the given circle. 2. Enter the POLYGON command and spec-

ify 5 sides. 3. Use the Center object snap to select the

center of the circle as the center point of the pentagon.

4. Enter I (Inscribed) to inscribe the poly-gon in the circle. When prompted for the radius of the circle, use the Nearest object snap to snap to a point on the circle. The pentagon appears inside the circle with the point you picked using the Nearest object snap as one of the vertices.

the center of the polygon to be. Instead of pick-ing a point on the circle to defi ne the radius, enter a numerical value at the keyboard.

Circumscribe a Hexagon about

a CircleThis method circumscribes a hexagon

about a circle with a known center point and radius. Refer to Figure 5-54.

1. Create the circle. 2. Enter the POLYGON command and

specify 6 sides. 3. Use the Center object snap to select the

center of the circle as the center point of the hexagon.

4. Enter C (Circumscribed) to circumscribe the polygon about the circle. When prompted for the radius of the circle, use the Nearest object snap to snap to a point on the circle. The hexagon appears inside the circle, with the point you picked on the circle as one of the vertices.

Figure 5-53

Using the POLYGON command to inscribe a regular pentagon within a circle

Figure 5-54

Using the POLYGON command to circumscribe a hexagon about a circle

You can use this method to “circumscribe” a polygon about a circle even if the circle does not exist. Follow the preceding four steps, but for the center point, pick a point where you want the center of the polygon to be. Instead of picking a point on the circle to defi ne the radius, enter a numerical value at the keyboard.

Construct an EllipseOf the two axes of an ellipse, the shorter

axis is the minor axis, and the longer one is the major axis. In AutoCAD, the ELLIPSE

You can use this method to “inscribe” a polygon in a circle even if the circle does not exist. Follow the preceding four steps, but for the center point, pick a point where you want


THIRDPOINT

SECONDPOINT

FIRSTPOINT

SECOND AXIS(MINOR)

FIRST AXIS(MAJOR)

A

O

B

A B CA

O

B

ORIGINAL

C

P

D

COPY

A

O

B

ORIGINAL

C

P

D

ROTATEDCOPY

command allows you to create ellipses of any size by defining the axes. Refer to Figure 5-55.

1. Enter the ELLIPSE command and pick a point anywhere in the drawing area as the fi rst endpoint of the fi rst axis.

2. Pick another point as the second end-point of the fi rst axis.

3. As the ellipse begins to appear on the screen, select a third point to specify the other axis.

Notice that you do not have to specify two points for the second axis. When you specify the third point, AutoCAD calculates the last point automatically, so that the second axis is at right angle to the fi rst.

Figure 5-55

Using the ELLIPSE command.

Figure 5-56

Copying and changing the orientation of an angle in AutoCAD.

Copy an AngleThis construction demonstrates a method

of copying a given angle to a new location and orientation. Refer to Figure 5-56.

1. Draw the angle AOB (Figure 5-56A).2. Enter the COPY command and use a

window to select both arms of the angle. To do this, pick a point below and to the right of the angle, and then pick another point above and to the left of the angle. The selected lines become dashed to show that they are selected. Press Enter to proceed to the next prompt.

3. For the point of displacement, pick point O.

4. When asked for the second point of dis-placement, pick another point anywhere on the screen. An exact copy of angle AOB appears (Figure 5-56B). Press Enter to end the COPY command.

5. To change the orientation of the sec-ond angle, enter the ROTATE command, select both legs of the second angle, and press Enter.

6. Specify a point anywhere on the angle as the base point. This is the point about which the angle will rotate.

7. Move the cursor to reposition the angle at a new orientation (Figure 5-56C).

Note that you can control the orientation of the angle by entering a numerical value for the angle of rotation instead of using the cursor.

Contrast How can you control the orientation of an angle other than by using the cursor?


BASEBA BA

R R

VERTEXUSE INTERSECTIONOBJECT SNAP

A

B C

BA

A B

C

@2.50<0

USE ENDPOINTOBJECT SNAP

@3.

25<9

0

1

2

3

Construct a TriangleThis type of polygon can be created using

the LINE command as described in the fol-lowing methods. Note: The POLYGON com-mand is usually used to create an equilateral triangle.

Construct an Isosceles TriangleThe following method is for constructing

an isosceles triangle. Refer to Figure 5-57.

1. Draw the given base line AB (Figure 5-57A).

2. Create a circle with its center point at point A and a radius equal to the length of the sides you want. See Figure 5-57B.

3. Create a second circle with the same radius, placing its center point at point BRefer again to Figure 5-57B.

4. Enter the LINE command and enter END to use the Endpoint object snap to place the fi rst point of the line at point A. Use the Intersection object snap to place the second point of the line at the upper intersection of the two circles. Then use the Endpoint object snap for point B. See Figure 5-57C.

5. Erase the two circles. The remaining tri-angle is an isosceles triangle.

Figure 5-57

Constructing an isosceles triangle using AutoCAD

Figure 5-58

Constructing a right triangle in AutoCAD given the length of two sides

Construct a Right TriangleConstruct a right triangle using this method

when you know the length of two sides of the triangle. In this construction, sides AB and BC are given. Side AB is 2.50 units long, and side BC is 3.25 units long. Refer to Figure 5-58.

1. Draw side AB using the LINE command and polar coordinates: @2.50<0. Leave the LINE command active.

2. Specify the coordinates for side BC: @3.25<90. This creates line BC perpendic-ular, to side AB. Leave the LINE command active.

3. Use the Endpoint object snap to place the third point at point A, completing the right triangle.


(2.50,5.50)

(5.00,7.50)

(6.00,7.00)

(6.50,5.00)

PTANGENTPOINT

Construct TangentsYou already know several methods for creating

a circle:

specify a center point and a radiussubstitute the diameter for the radius by pressing the D key before entering the numerical valuespecify two points on the diameter of the circlespecify three points on the diameter of the circle

AutoCAD also allows you to create a circle that is tangent to two other objects in AutoCAD by specifying the tangent objects and a value for the radius of the circle. As you may recall, a line is tangent to a circle if the line touches the circle at one point only. Refer to Figure 5-59.

1. Before you can use this option, you must have at least two lines in the drawing to specify as tangents. Use the LINE com-mand to create the two lines. Use coor-dinate values to place the endpoints of the lines at the coordinates shown in the illustration.

2. Enter the CIRCLE command. Enter T at the keyboard to select the tan tan radius (Ttr) option.

3. At the appropriate prompts, pick any-where on the two lines as the two tan-gents. Specify a radius of 1.00. The circle appears as in Figure 5-59.

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•

•

Figure 5-59

Creating a circle tangent to two other objects given the radius of the circle.

Figure 5-60

Using the Tangent object snap to create a line tangent to a circle.

Object TrackingThe process for extending a

line that is described in step 4 of Construct a Tangent Line is known as object tracking. If this does not seem to work for you, enter the OSNAP command, go to the Object Snap tab, and make sure the Object Snap Track-ing On box is checked. If this option is not available in your version of Auto-CAD, you can achieve the same eff ect by using the EXTEND command.

Construct a Tangent LineBecause AutoCAD has a Tangent object

snap, creating tangent lines is fairly easy. Refer to Figure 5-60.

1. Draw a circle anywhere in the drawing area using the cursor to specify any radius.

2. Enter the LINE command. Pick any point outside the circle as the fi rst point of the line.

3. Enter the Tangent object snap and move the cursor near the circle. Select a point on the circle. The line automatically snaps to the tangent point on the circle.

4. To extend the line beyond the tangent point, keep moving the cursor in the

same general direction. AutoCAD displays an “Extension” message that shows the length and angle of the extended line.


A

B

O1

D

C

1

A

B

O

C

D

A B

TANGENTPOINT 1

TANGENTPOINT 2

BREAKPOINTS

A

B

C

D

Summarize How do you create a circle that is tangent to two other objects in AutoCAD?

Construct a Tangent ArcAutoCAD has an ARC command that gives

CAD users great fl exibility in creating arcs. However, sometimes the best solution is to use the tan tan radius option of the CIRCLE com-mand, trimming away the unneeded parts of the circle. This section illustrates a few of the ways to create arcs in AutoCAD.


Two LinesThe procedure for constructing an arc tan-

gent to two lines in AutoCAD is similar to the board drafting procedure. In CAD, the pro-cedure is the same whether the angle is an acute, obtuse, or right angle. Therefore, only an acute angle is shown in Figure 5-61.

1. Draw given lines AB and CD (Figure 5-61A).

2. Use the OFFSET command to offset both lines 1 unit to the inside.

3. Enter the ARC command. At the prompt, enter C (Center), and use the Intersec-tion object snap to snap to the intersec-tion of the two lines you offset in step 2.

4. Use the Perpendicular object snap to place the ends of the arc perpendicular to lines AB and CD (Figure 5-61B).

Figure 5-61

Using AutoCAD to create an arc tangent to two lines

Figure 5-62

Using the CIRCLE command to construct an arc tangent to two given arcs


Two Given ArcsThis method uses the CIRCLE command

to construct an arc tangent to two given arcs. Refer to Figure 5-62.

1. Enter the ARC command and follow the prompts to enter the start point, second point, and endpoint of arcs AB and CD. The radii of the arcs may be equal or unequal.

2. Enter the CIRCLE command. At the prompt, enter T (tan tan radius). Select points on the given arcs near the tangent locations. Note that you have only to pick a point somewhere near the tangent point. AutoCAD calculates the exact tan-gents for you.

3. Specify a radius of 1.50 to make the tan-gent circle appear.

4. Enter the BREAK command, and pick two points on the circle to break the arc out of the circle. Use the ERASE com-mand to erase the unwanted portion of the circle. The remaining arc is tangent to the two given arcs.

Determine How does the CAD procedure for constructing an arc tangent to two lines diff er for acute, obtuse, and right angles?


C

A B

B

E

C D

A

B

E

C D

A

D

B

E

C D

A

B

E

C D

A

BREAKPOINT

Construct an Ogee CurveAn ogee curve is a reverse curve that looks

something like an S. The CAD procedure for drawing an ogee curve is similar to the board drafting procedure. Refer to Figure 5-63.

1. With Ortho on, draw lines AB and CD (Figure 5-63A). Then turn Ortho off and use the Endpoint object snap with the LINE command to draw line BC.

2. Enter the BREAK command. This com-mand is used to “break” a single line, arc, circle, or other geometry into two distinct objects. At the prompt, enter F (First), and use the Nearest object snap to pick a point E on line BC through which the curve is to pass. Refer again to Figure 5-63. Line BC becomes two lines: BE and EC.

3. Construct perpendiculars at the mid-points of lines BE and EC. The length of the perpendicular lines does not matter. Erase any circles or arcs used for con-struction before continuing to step 4 (Figure 5-63B).

4. Create two circles. For the fi rst, use the intersection of the vertical line from point B and the lower perpendicular as the center point. For the radius, enter the Endpoint object snap and snap to point E. For the second circle, use the intersection of the vertical line from point C and the upper perpendicular as the center point. For the radius, use the Endpoint object snap to snap to point E. It does not matter if the circles extend off the screen. (Figure 5-63C).

5. Notice that the two circles are tangent to each other at point E. One circle is also tangent to line AB, and the other is tan-gent to line CD. To fi nish the ogee curve, enter the TRIM command, press Enter to select all of the objects, and trim away the unwanted parts of the circles. Erase lines BE, EC, and the vertical and perpen-dicular lines. See Figure Figure 5-63Dfor the fi nished curve.

Figure 5-63

Creating an ogee curve in AutoCAD.


A B

Figure 5-64

Using the SCALE command to enlarge or reduce the physical size of a drawing in AutoCAD.


To change the size of objects in an Auto-CAD drawing, you can use the SCALE com-mand. Note that this process is different from using the ZOOM command to make objects on the screen appear larger or smaller. It is also different from choosing a standard scale in paper space to scale a drawing for printing. When you use the SCALE command, you change the actual dimensions of the objects you see on the screen. You can scale all of the objects in the drawing at once or scale only those objects that you select.

This construction demonstrates the effect of scaling objects in AutoCAD. Refer to Figure 5-64.

1. Set the snap and grid to .50. Use the snap, grid, and coordinate display to create two concentric circles (both with the same center point). Make the radius of one circle 2.00 units, and make the radius of the second circle 1.00 unit (Figure 5-64A).

2. Enter the SCALE command. Pick both circles to scale, and press Enter.

Section 5.2 AssessmentAfter You Read

Self-Check 1. Describe how technical and mathemat-

ical problems related to geometric con-structions can be solved using CAD.

2. Explain how to reduce or enlarge the physical size (dimensions) of a drawingusing CAD.

Academic IntegrationEnglish Language Arts

3. Read the following content vocabulary and technical terms from this chapter. Organize the terms using one of them as the heading under which the others are listed as examples.

isosceles equilateral triangle scalene

Drafting Practice Repeat the board drafting practice

in Section 5.1, this time using CAD techniques.

Go to glencoe.com for this book’s OLC for help with this drafting practice.

3. The base point is the point around which the scaling will occur. Use the Center object snap to select the center of the cir-cles for the base point.

4. Enter a scale factor of .75 to scale the cir-cles to 75% of their original size (Figure 5-64B). You can check their size by using the grid, remembering that the dots on the grid are spaced at intervals of .50.

Notice that you must enter a decimal frac-tion. The number 1 stands for 100%, or full size. If you enter 75, the circles will enlarge to 75 times their original size.



Section 5.1 Geometry is the study of the size and shape of objects and their relationship to each other.Drafters, surveyors, engineers, architects, scientists, mathematicians, and design-ers use geometric constructions to show proper relationships between individual lines and points.Geometric shapes discussed in this chapter include lines, triangles, squares, circles, arcs, angles, pentagons, hexagons, polygons.The most important principles of drafting include accuracy. Work that is not accurate may give designers wrong information.

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Section 5.2 Using CAD object snaps for geometric constructions greatly increase the effi -ciency of the drawing process and reduces the time involved in preparing accurate, high-quality drawings.In CAD, many commands are available for drawing basic geometric shapes. Examples include CIRCLE, POLYGON, ARC, and ELLIPSE.

•

•

Chapter Summary

Review Content Vocabulary and Academic Vocabulary 1. Use each of these content and academic vocabulary words in a sentence or drawing.

Content Vocabularygeometry (p. 135)geometric construction (p. 137)vertex (p. 138)bisect (p. 138)perpendicular (p. 139)parallel (p. 139)

••••••

polygon (p. 144) inscribe (p. 152) circumscribe (p. 152) regular polygon (p. 152)ellipse (p. 155) object snap (p. 160) ogee curve (p. 168)

•••••••

Academic Vocabularyaccurate (p. 135)methods (p. 139) intervals (p. 161)specify (p. 163)

••••

Review Key Concepts 2. List geometric shapes that drafters use. 3. Demonstrate how to construct various geometric shapes accurately. 4. Describe how technical and mathematical problems related to geometric constructions can

be solved using board-based drafting. 5. Describe how technical and mathematical problems related to geometric constructions can

be solved in a computer environment. 6. List the steps involved in using geometry to enlarge or to change a drawing’s proportions.

Review and Assessment5


6"

2"

ALTITUDE (h)

BASE (b)

Engineering 7. What do Engineers Do?

Webster’s Dictionary defi nes engineering as “the application of science and mathemat-ics by which the properties of matter and the sources of energy in nature are made useful to people.” According to the National Academy of Engineering (NAE), there are more than two million practicing engineers in the United States. What are all these engineers doing? In what ways are engineers making things use-ful for people? Using the Internet or library, research a type of engineering, such as archi-tectural or biomedical engineering. Then write a one-page paper, summarizing what makes the fi eld of engineering important and name one major innovator working in the fi eld.

8. Productivity and AccountabilityYou and a classmate have been assigned a

project that represents a signifi cant part of your grade. You are both to participate equally in completing it. The two of you agree to the parts for which each will be responsible. You fi nished your work, but your partner did not. How do you handle this situation? Prepare a bulleted list to show your options, to use as a basis for a class discussion.

Mathematics 9. Calculate Area

Determine the area of a triangle with a base of 6 inches and a height of 2 inches.

Calculating Area

To fi nd the area of a triangle, multiply the base (b) times the height (h) and divide by two. Area = bh/2.

Multiple Choice QuestionDirections Choose the letter of the best answer. Write the letter for the answer on a separate piece of paper. 10. Which of the following is an example

of a polygon? A. Circle B. Angle C. Triangle D. Parallel Line

TEST-TAKING TIP

In a multiple-choice test, the answers should be specifi c and precise. Read the question fi rst, then read all the answer choices before you choose. Eliminate answers that you know are incorrect.

Prep For

Win Competitive Events

11. Technical MathOrganizations such as SkillsUSA offer a

variety of architectural, career, and draft-ing competitions. Completing activities such as the one below will help you pre-pare for these events.

Activity Complete the STEM Math-ematics exercise on this page. Then team with a partner and check each other’s work, going over any concepts that might be unclear.

Go to glencoe.com for this book’s OLC for more information about competitive events.

Review and Assessment 171


Problems5

Drafting ProblemsThe problems in this chapter can be per-

formed using board drafting or CAD tech-niques. The problems are presented in order of diffi culty, from least to most diffi cult.

Problems 1 through 18 are designed for work-ing four problems on an A-size sheet, laid out as shown in Figure 5-65. Draw each problem three times the size shoown. If you are using board drafting, use dividers to pick up the dimensions from the problems, and step off each measurement three times. If you are using a CAD system, use a scale to measure the dimen-sions, and create the geometry in the CAD sys-tem at three times the measured size.

1.

Draw and bisect line AB Figure 5-66A.

2.

Draw line AB Figure 5-66B. Construct a perpendicular at point P.

3.

Draw line AB Figure 5-66C. Divide line AB into fi ve equal parts.

4.

Draw line AB Figure 5-66D. Construct line CD through point P so that CD is par-allel to AB and equal in length to line AB.

5.

Draw angle ABC Figure 5-66E. Bisect angle ABC.

6.

Draw angle ABC Figure 5-66F. Copy the angle in a new location, beginning with line A1B1.

7.

Draw base line AB Figure 5-66G. Con-struct an isosceles triangle using base line AB and sides equal to line CD.

8.

Draw base line AB Figure 5-66H. Con-struct a triangle on base AB with sides equal to BC and AC.

9. Draw a circle with a 3″ diameter Figure 5-66I. Inscribe a square in the circle.

10. Draw a circle with a 3″ diameter Figure 5-66I. Inscribe a regular pentagon in the circle.

11. Draw a circle with a 3″ diameter Figure 5-66I. Circumscribe a regular hexagon about the circle.

12. Draw a circle with a 3″ diameter Figure 5-66I. Circumscribe a regular octagon about the circle.

13. Draw a circle with a 3″ diameter Figure 5-66J. Construct a tangent line through point P.

14. Locate points A, B, and C on the drawing sheet Figure 5-66K. Construct a circle through these three points.

15. Draw the two lines shown in Figure 5-66L. Construct an arc having a radius R tangent to the two lines.

16. Draw the two arcs shown in Figure 5-66M. Construct an arc having a radius R tangent to the fi rst two arcs.

Figure 5-65


A

B

30°

PA B

P

A

B

A

B

C

B

B

A

A

1

1

BBASE

D

A C BBASE

C

A A

C

B

P

AB

C

R

A B C

D E F

G H I

J K L

R

MAJOR AXIS

M N O

Figure 5-66

17. Draw a 3.00″ square Figure 5-66N. Construct a regular octagon within the square.

18.

Construct an ellipse that has a 4.00″ major axis and a 2.50″ minor axis Figure 5-66O.

Problems 173

J

K

HG

F

C

B

A

E D

I

R1.25

2.50 3.50 3.00 3.00

12.00

F

H

DC

AB

E

G

5.50

7.62

R3.50Ø1.063 HOLES

2.75

R1.00

Problems5

Problems 19 through 24: These problems provide additional practice in geometric con-structions. They are designed to be drawn one per drawing sheet. Before beginning each drawing, determine an approximate scale and sheet size. Do not add dimensions to your drawing.

19. Draw the handwheel shown in Figure 5-67. Use the following dimensions: A = Ø7.00″; B = Ø6.12″; C = Ø5.50″; D = R1.25″; E = Ø2.00″; F = Ø1.00″; G ( keyway) = .20″ wide × .10″ deep; H = Ø.38″; I = R.38″; J = R.20″; K = 1.00″.

Figure 5-67

20. Draw the combination wrench shown in Figure 5-68. Use the following dimen-sions: square: 1.00″; octagon: 1.38″ across fl ats; isosceles triangle: 2.75″ base, 2.00″sides; pentagon: inscribed within Ø1.38�

circle; hexagon: 1.25″ across fl ats. If you are using board drafting techniques, do not erase construction lines.

Figure 5-68

21.

Draw the adjustable fork shown inFigure 5-69. Use the following dimen-sions: A = 220 mm; B = 80 mm; C =40 mm; D = 27 mm; E = 64 mm; F =20 mm; G = 8 mm; H = 10 mm.

Figure 5-69

22.

Draw the rod support shown in Figure 5-70.

Figure 5-70

METRIC


R1.00

R1.00

R.38

R.38

60?

R7.00R1.62

R.75

15?

8.004.00Ø.75

90 0 1 70

20 60 30

50 40

45

50°

45°

45°

A

GF

E

DJ

K

H

X

I

B

C

2 2

24.

Draw the tilt scale shown in Figure 5-72. Use the following dimensions: AB = 44 mm; AX = 66 mm; AC = 140 mm; AD = 184 mm; AE = 216 mm; AF = 222 mm; AG = 236 mm; H = R24 mm; I = R16 mm; J = R5 mm; K = Ø12 mm.

Figure 5-72

23.

Draw the adjustable table support shown in Figure 5-71.

Figure 5-71

METRIC

Design ProblemsDesign problems have been prepared to challenge individual students or

teams of students. In these problems, you will apply skills learned mostly in this chapter but also in other chapters throughout the text. The problems are designed to be completed using board drafting, CAD, or a combination of the two. Be creative and have fun!

Challenge Your Creativity

1. Design an educational toy to help tod-dlers develop manual dexterity, spatial relationships, and color association. The toy should be similar to Figure 5-73, but expanded to include at least six geomet-ric shapes of different colors. Material: 1″thick pine.

2. Design an octagon-shaped jewelry box with a hinged lid. The overall size should not exceed 160 mm across the corners of the octagon by 90 mm high. Material: optional. Do not dimension.

Teamwork

3. Design and draw a cover for your 8.50″ ×11.00″ or 11.003 × 17.00″ set of technical drawings. Use various geometric shapes in the design. Geometric shapes, such as cir-cles, squares, hexagons, octagons, ellipses, etc., can be used to enhance the design. Use colors where desired. Use block letters to add information on the cover, such as your name, the school name, the course title, the instructor’s name, and the year.

Figure 5-73

Problems 175

Create a Logo for Your Own Business

Your Project AssignmentExplore the opportunities to become an

entrepreneur by working as a freelance draft-sperson from your home. Create a logo for your business.

Use what you have learned in Chapters 1–5 to create a plan for starting your own freelance business. Your challenge is to:

Identify opportunities for employment as an independent draftsperson by researching local classifi ed ads and regional and national online job search sites.

Choose a focus for your home-based business based on your interests and abilities. Will your clients be manufacturing companies, engineering fi rms, or architects? Are you stronger at board drafting, or computer-aided drafting?

Explain the educational requirements for the kind of work you have chosen.

Create fi nished drawings for a business logo to use on stationery, business cards brochures, etc.

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TIP! A logo will be used to promote your business and give it an identity. It should be eye-catching, simple, and speak to the needs of your potential customers.

Prepare three fi nished drawings of your logo at three different sizes.

Applied Skills

List and categorize the opportunities you uncover in your research. Include contact information for each company, information about qualifi cations and requirements, type of company, and the nature of the work.

Outline the educational requirements, and identify schools or programs where you could obtain the necessary training.

Write a paragraph or two about your interests and abilities. Discuss why you chose the focus for your business that you did.

List the steps, materials, and tools you used to create the drawings for your logo. Explain the procedure you used to reduce or enlarge your initial drawing to create three versions.

The Math Behind the ProjectThe primary math skills you will use to com-

plete this project are geometry modeling, algebra, and measurement. To get you started, remember these key concepts, and follow this example:

Geometry—Ratio, Proportion, and Scale

To understand how to reduce or enlarge the size of a drawing, think about the terms ratio, proportion, and scale. A ratio is a comparison of two numbers. For example, a rectangle has a

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Math Standards

Geometry Use visualization, spatial reasoning, and

geometric modeling to solve problems (NCTM)

Problem Solving Solve problems that arise in

mathematics and other contexts (NCTM)


Hands-On Math ProjectUNIT 1


length of 2 inches and width of 3 inches. The ratio of length to width is 2/3.

When two ratios are equal, they form a pro-portion. One way to determine whether two ratios form a proportion is to check their cross products. For example, to fi nd out if a 2/3 rect-angle is proportional to a 6/8 rectangle, multiply the numerator of each ratio by the denominator of the other. If the resulting products are equal, the fi gures are proportional.

2 __ 3 ? 6 __

8 2 × 8 ? 6 × 3 16 ≠ 18

Since the products are not equal, the rectan-gles are not proportional. In other words, they are not drawn to scale.

Solving Proportion ProblemsTo use proportions to solve problems, set

up two ratios using a letter symbol for the

unknown quantity. For example, consider this problem:

What is the length of a rectangle 6 incheswide that is proportional to another rectangle 2 inches wide and 3 inches long?

Use l to represent the length of the enlarged rectangle drawn to scale.

2 __ 3 = 6 __

l 2l = 6 × 3 2l = 18 l = 9

The length should be 9 inches.To determine the scale factor of the enlarged

drawing, write a ratio comparing similar sides, and reduce. For example, 6/2 or 9/3. In both cases, the ratios reduce to 3. When a fi gure is enlarged, the scale factor is greater than one. When two similar fi gures are identical in size, the scale factor is equal to one. When a fi gure is reduced, the scale factor is less than one.

Designers of Famous LogosIn the United States and around the world, famil-

iar corporate logos dot the landscape. You can spot

your favorite fast food restaurant or gas station from

far away because their powerful logos are easy to rec-

ognize and prominently displayed. What makes these

symbols so eff ective? Who designed them?

One of the most infl uential logo designers of the

twentieth century is Milton Glaser. He designed the

famous “I Love New York” logo. He also designed the

“bullet” you see on DC Comics. A good logo catches

the eye. It may also say something about the product

or service off ered, or make the observer curious.

Research Activity Find out more about Milton Gla-

ser and the things he has designed. What characteris-

tics do his logos and other objects have in common?

Also research the principles behind good logo design.

Write a one-page summary of your fi ndings.

Bonus! Incorporate the principles of good logo

design into your creation.

Unit 1 Hands-On Math Project 177Car Culture/Corbis

UNIT 1 Hands-On Math Project

Project Steps:

Design Your Future!

STEP 1 Research

Explain the type of drafting work you are best suited for and most interested in pursuing.

Look for job opportunities in local classifi ed ads and on the Internet. Make phone calls to these companies to fi nd out more about available opportunities.

Phone other similar fi rms in your area and ask if they ever hire outside fi rms to handle any of their drafting needs.

Find out more about logo design and think about what you want to communicate with your logo.

TIP! Write a script, and practice your phone inquiry skills before you call prospective clients.

STEP 2 Plan

Defi ne and write out your overall goal for this project.

Gather the appropriate supplies and tools for board drafting.Set up to prepare your drawing fi le with AutoCAD.

Refer to the Math Concepts on the previous page, or go to glencoe.com for this book’s OLC for more informa-tion on the math concepts used in this project.

STEP 3 Apply

Make several preliminary sketches of ideas you have for your logo.

Complete one version of your logo, then

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enlarge or reduce it so you have three ver-sions: one should be sized for use on a busi-ness card, one for use on stationery, and one for use on a Web site.

TEAMWORK Collaborate: Ask a classmate to review the design of your logo before you continue. Ask for feedback on the technical aspects of your drawing as well as the overall concept.

STEP 4 Present

Prepare a presentation combining your research with your completed drawings using the checklist below.

Presentation ChecklistDid you remember to…

state your objectives for your business?

describe the services you will off er?

show and discuss your logo?

explain the process you used to create the logo and what you hope to achieve with it?

describe the services you will off er?

show and discuss your logo?

explain the process you used to create the logo and what you hope to achieve with it?

show your preliminary sketches and explain how you created your logo?

demonstrate the basic sketch or CAD drawing?

show the three versions of your logo and explain how they could be used?

review the drafting principles involved in completing your logo?

explain any problems you encountered and how you overcame them?

turn in your research and planning notes to your teacher?

✓

✓

✓

✓

✓

✓

✓

✓

✓

✓

✓

✓

✓



The purpose of a portfolio is to showcase your education and examples of your work and accomplishments. A typical portfolio might include the following:

Career summary and goalsRésuméList of accomplishmentsEducation and certifi cationsSamples of your work Job or Job-shadowing experience

Getting Started To prepare the written components for your portfolio, you will need access to a computer with Microsoft Word, Pages, or other word processing application. Use this software to create the written components of your portfolio.

Career summary and goals: Prepare a brief summary of your specifi c career goals. Describe the industry or job that interests you.

Résumé: If you have not already done so, use the information from Chapter 1 to prepare your résumé. Include in your résumé a list of accomplishments, education, and certifi cations you hold.

Samples of your work: Now that you have completed your business planning and design project for this unit, include your drawings as sam-ples of your work in your portfolio.

Save Your Work In the following Units, you will add more elements to your portfolio. Keep items you want to save for your portfolio in a special folder as you progress through this class.

••••••

1.

2.

3.

STEP 5 Build Your Portfolio

The purpose of a portfolio is to showcase your education and examples of your work and accomplishments.

Organize your drawings in a manner that will show your ideas well.

Attach a written introduction and a descrip-tion of your design.

STEP 6 Evaluate Your Technical

Skills

Assess yourself before and after your presentation.

Is your research thorough?Did you plan your steps carefully?Did you organize your visuals so that they showcase your ideas?Is your presentation creative and effective?During your presentation, do you make eye contact and speak clearly enough?

Rubrics Go to glencoe.com to this book’s OLC for a printable evaluation rubric and Academic Assessment.

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4.5.

Unit 1 Hands-On Math Project 179Image Source Black/Alamy


Documents

5 Geometry for DraftingDrafting Career Architect Zaha Hadid’s designs for the Cincinnati Contemporary Art Center were “like a rollercoaster, a little scary, but exhilarating,”