^ CHAPTER ^

Get Ready!Skills

Handbook,

p. 889

Skills

Handbook,p. 890

Skills

Handbook,p. 890

Skills

Handbook,p. 892

Skills

Handbook,p. 894

^ Squaring NumbersSimplify.

1. 3^ 2. 4' 3. 11'

^ Simplifying ExpressionsSimplify each expression. Use 3.14 for tt.

4.2-7.5 + 2-11 5.

Evaluating ExpressionsEvaluate the following expressions for a = 4 and b

7.a + b

8.a-7

3-b

6. VPTT?

-2.

9. V(7-fl)2 + [2-fo)2

^ Finding Absolute ValueSimplify each absolute value expression.

10. |-8| 11. |2-6|

^ Solving EquationsAlgebra Solve each equation.

13. 2x+7 = 13 14. 5x-12 = 2x + 6

12. 1-5-(-8)1

15. 2(a: + 3) - 1 = 7a;

Looking Ahead Vocabulary16. A child can construct models of buildings by stacking and arranging colored blocks.

What might the term construction mean in geometry?

17. The Mid-Autumn Festival, celebrated in China, falls exactly in the middle ofautumn, according to the Chinese lunar calendar. What would you expect amidpoint to be in geometry?

18. Artists often use long streaks to show rays of light coming from the sun. A ray is alsoa geometric figure. What do you think the properties of a ray are?

19. You and your friend work with each other. In other words, you and your friend areco-workers. What might the term collinear mean in geometry?

C I chapter 1 Tools of Geometry^

CHAPTER

f _ > Tools of Geometry

ChapfeDownload videos

connecting mathto your world.

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Vary numbers,graphs, and figuresto explore mathconcepts.

The online

Solve it will getyou in gear foreach lesson.

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In English andSpanish

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to stepped-outproblems alignedto Common Core

Get and view

your assignmentsonline.

Extra practiceand review

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1-1

1-2

1-3

1-4

1-5

1-6

1-7

1-8

r PreviewNets and Drawings for Visualizing

Geometry

Points, Lines, and Planes

Measuring Segments

Measuring Angles

Exploring Angle Pairs

Basic Constructions

Midpoint and Distance in the

Coordinate Plane

Perimeter, Circumference, and Area

BIG ideas

VocabularyEnglish/Spanish Vocabulary Audio Online:

English Spanish

angle bisector, p. 37 bisectriz de un anqulo

construction, p. 43 construccion

isometric drawing, p. 5 dibujo isometrico

linear pair, p. 36 par lineal

net, p. 4 plantilla

orthographic drawing, p, 6 dibujo ortoqrMico

perpendicular bisector, p. 44 mediatriz

postulate, p. 13 postulado

1 Visualization

Essential Question How can yourepresent a three-dimensional figure witha two-dimensional drawing?

2 ReasoningEssential Question What are the

building blocks of geometry?

3 Measurement

Essential Question How can you

describe the attributes of a segment or

angle?

DOMAINS

• Congruence

congruent segments, p. 22 segmentos conqruentes

segment bisector, p. 2i bisectriz de un seqmento

supplementary angles, p. 34

vertical angles, p. 34

angulos suplementarlos

anqulos verticales

Virtual Nerd™

tutorials withbuilt-in support

Common Core Performance Task

Solving a RiddleWhile browsing in an antique store, Cameron found a page that came from an oldbook of riddles.

What sits in a corner but travels

around the world?

(8m - 18)(7m + 3

(21s + 6)

5a + 12 9a - 12

Solve the riddle, today at the latest.Arrange the variables from least to greatest.

Task DescriptionFind the value of each variable and answer the riddle.

Connecting the Task to the Math PracticesAs you complete the task, you'll apply several Standards for MathematicalPractice.

• You'll make sense of the problem as you analyze the information in a complexdiagram. (MP 1)

• You'll attend to precision as you write and solve an equation. (MP 6)

• You'll explain relationships between angles in the diagram. (MP 3)

MATHEMATICAL

PRACTICES

0PqwerGMmet^com I Chapter 1 Tools of Geometry

Nets and Drawings forVisualizing Geometry

Mathematics Fiorlda StandardsPrepares for MAFS.912.G-C0.1.1 Know precisedefinitions of angle, circle, perpendicular line, parallelline, and line segment, based on the undefined notionsof point, line, distance along a line, and distance arounda circular arc.

ivip 3, MP 4,"MP"7

Objective To make nets and drawings of three-dimensional figures

Getting Ready! <>■ X ^

Try to visualizewhat the figuremight look likefrom differentperspectives.

When you shine a flashlight onan object, you can see a shadowon the opposite wall. What shapewould you expect the shadows inthe diagram to hove? Explain yourreasoning.

MATHEMATICALPRACTICES

LessonVocabulary

• net

isometricdrawingorthographicdrawing

In the Solve It, you had to "see" the pro]'ection of one side of an object onto a flatsurface. Visualizing figures is a key skill that you will develop in geometry.

Essential Understanding You can represent a three-dimensional object with atwo-dimensional figure using special drawing techniques.

A net is a two-dimensional diagram that you can fold to form a three-dimensionalfigure. A net shows all of the surfaces of a figure in one view.

How can you seethe 3-D figure?Visualize folding the netat the seams so that theedges join together. Trackthe letter positions byseeing one surface movein relation to another.

Identifying a Solid From a Net

Hie net at the right folds into the cube shown beside it.Which letters will be on the top and front of the cube?

A, C, E, and F all share an edge with Dwhen you fold the net, but only two ofthose sides are visible in the cube shown.

A B C D

A wraps around and joins with D to become the back of the cube.B becomes the left side. F folds back to become the bottom.

E folds down to become the top of the cube. C becomes the front.

Chapter 1 Tools of Geometry

Got It? 1. The net in Problem 1 folds into the cube shown at the right.Which letters will be on the top and right side of the cube?

T;r-i ■

How can you seethe net?

Visualize opening the topand bottom flaps of thebox. Separate one of theside seams. Then unfold

and flatten the box

completely.

J

Packaging designers use nets to design boxes and other containers like the box

in Problem 2.

Problem 2 Drawing a Net From a Solid QS)

Package Design What is a net for the graham cracker boxto the right? Label the net with its dimensions. 20 cm

6 cm

14 cm

cfacHcrs

CraviCfatvJ

14 cm

20 cm

^^1^ Got It? 2. a. What is a net for the figure at the right? Label the net withits dimensions,

b. Reasoning Is there another possible net for the figure inpart (a)? If so, draw it.

10 cm

7 cm

10 cm

4 cm

I

JAn isometric drawing shows a corner view of a three-dimensional

figure. It allows you to see the top, front, and side of the figure. You candraw an isometric drawing on isometric dot paper. The simple drawingof a file cabinet at the right is an isometric drawing.

A net shows a three-dimensional figure as a folded-out flat surface.

An isometric drawing shows a three-dimensional figure using slantedlines to represent depth.

C PowerGeometry.com I Lesson 1-1 Nets and Drawings for Visualizing Geometry

Is there more than

one way to make anisometric drawing?Yes. You can start with

any edge of the structure.Use that edge as areference to draw the

other edges.

Problem 3 Isometric Drawing

What is an isometric drawing of the cube structure at the right?

Step 1

Draw the front edges.

Step 2

Draw the right edges.

Step 3

Draw the back edges.

Got It? 3. What is an isometric drawing of this cube structure?

How can you

determine the

three views?

Rotate the structure in

your head so that youcan "see" each of the

three sides straight on.

An orthographic drawing is another way to represent a three-dimensional figure.An orthographic drawing shows three separate views: a top view, a front view, anda right-side view.

Although an orthographic drawing may take more time to analyze, it providesimique information about the shape of a structure.

Orthographic Drawing

What is the orthographic drawing for the isometric drawing at the right?

Solid lines show -

visible edges. i—

□rTop

Dashed linesshow hiddenedges. -rent Right

Oi:

An isometricdrawing shows thesame three views.

4. What is the orthographic drawing for this^ isometric drawing?

Chapter 1 Tools of Geometry

& Lesson Check

Do you know HOW?

1. What is a net for the figure below? Label the net withits dimensions.

2 in. h/1 in.

5 in.

2. What is an isometric drawing of

the cube structure?

3. What is the orthographic

drawing of the isometric

drawing at the right? Assume

there are no hidden cubes.

MATHEMATICAL

PRACTICESDo you UNDERSTAND?

4. Vocabulary Tell whether each drawing is isomefn'c,orthographic, a net, or none.

a.

7\<n>

b.

Top

V Front

&Right

5. Compare and Contrast What are the differences and

similarities between an isometric drawing and anorthographic drawing? Explain.

Practice and Problem-Solving Exercises

© Practice Match each three-dimensional figure with its net.

6. 7. ZEI>

A.

MATHEMATICAL

PRACTICES

C.

^ See Problem 1.

Draw a net for each figure. Label the net with its dimensions.

9.

2 in.

2 in.4 in.

4^ See Problem 2.

11.

30 mmp i

36 mm

C PowerGeometry.com I Lesson 1-1 Nets and Drawings for Visualizing Geometry

Make an isometric drawing of each cube structure on isometric dot paper.

12. 13. 14.

^ See Problem 3.

For each isometric drawing, make an orthographic drawing. Assume there are ^ See Problem 4.no hidden cubes.

16. 17.

Apply @ 20. Multiple Representations There are eight different nets for the solidshown at the right. Draw as many of them as you can. [Hint: Two netsare the same if you can rotate or flip one to match the other.)

21. a. Open-Ended Make an isometric drawing ofa structure that you

can build using 8 cubes.

b. Make an orthographic drawing of this structure.

22. Think About a Plan Draw a net of the can at the right.

• What shape are the top and bottom of the can?• If you uncurl the body of the can, what shape do you get?

23. History In 1525, German printmaker Albrecht Diirer first used

the word net to describe a printed pattern that folds up into a

three-dimensional shape. Why do you think he chose to use

the word netl

^025) Manufacturing Match the package with its net.

24.

A. B.

N

8 Chapter 1 Tools of Geometry

27. Error Analysis Miquela and Gina drew orthographic drawings for the cubestructure at the right. Who is correct?

Miquela Gina

Top Top

■ront Right "rent□

Right

Make an orthographic drawing for each isometric drawing.

28.. • • 29.' . •

w-* .

30.. *

W •

31. Fort Use the diagram of the fort at the right.a. Make an isometric drawing of the fort.b. Make an orthographic drawing of the fort.

^25) 32. Aerial Photography Another perspective in aerialphotography is the "bird's-eye view," which shows an objectfrom direcdy overhead. What type of drawing that you havestudied in this lesson is a bird's-eye view?

' 33. Writing Photographs of buildings are typically not takenfrom a bird's-eye view. Describe a situation in which youwould want a photo showing a bird's-eye view.

Visualization Think about how each net can be folded to form a cube. What isthe color of the face that will be opposite the red face?

34.

R35. 37.

jT38. Multiple Representations There are 11 different nets for a cube. Four of them are

shown above.

a. Draw the other seven nets.

b. Writing Suppose you want to make 100 cubes for an art project. Which of the11 nets would you use? Explain why.

c j PowerGeometry.com Lesson 1-1 Nets and Drawings for Visualizing Geometry

Challenge 39. The net at the right folds into a cube. Sketch the cube so that its front face isshaded as shown below.

40. Architecture What does the net of the staircase shown look like? Draw the net

{Hint. Visualize stretching the stairs out flat.)

41. A hexomino is a two-dimensional figure formed with she squares. Each square

shares at least one side with another square. The 11 nets of a cuhe that you found

in Exercise 38 are hexominoes. Draw as many of the remaining 24 hexominoes

as you can.

42. Visualization Use the orthographic drawing at the right.

a. Make an isometric drawing of the structure.

b. Make an isometric drawing of the structure from part (a) after it has beenturned on its base 90° counterclockwise.

c. Make an orthographic drawing of the structure from part (b).d. Turn the structure from part (a) 180°. Repeat parts (b) and (c).

Top

Front Right

Short

.Response

Standardized Test Prep

43. How many possible nets does the solid at the right have?

®1 ®2 C03

44. Solve lOfl - 5h = 25 for b.

CT:)h=10fl + 25 C^h=10fl-25 CS:)b = 2a + 5

45. Graph the equation x + 2y= -3. Label the x- andy-intercepts.

CX) b = 2a-5

Mixed Review

For Exercises 46 and 47, use the diagram at the right.

46. Measure DE and EFto the nearest millimeter.

47. Measure each angle to the nearest degree.

48. Draw a triangle that has sides of length 6 cm and 5 cm with a90° angle between those two sides.

Get Ready! To prepare for Lesson 1-2, do Exercises 49-51

Coordinate Geometry Graph the points on the coordinate plane.

49. (0, 0), (2, 2), (0, 3} 50. (1, 2), (-4, 3), (-5, 0)

^ See p. 884.

^ See p. 893.

51. (-4, -5), (0,-1), (3, -2)

10 Chapter 1 Tools of Geometry