Review of Geometry

8/12/2019 Review of Geometry

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CHAPTER 1Review of Geometry

In this book, we use the following variables when stating formulas: A Darea , P D perimeter , C Dcircumference , S Dsurface area, and V Dvolume. Also, r denotes radius, h altitude, l slant height, b base, B area of base, and central angle expressed in radians.

1.1 PolygonsCLASSIFICATION

Type Number of sidestriangle 3quadrilateral 4 pentagon 5hexagon 6heptagon 7octagon 8nonagon 9decagon 10undecagon 11dodecagon 12

TRIANGLES A D 12 bh The sum of the measures of theangles of a triangle is 180 .P Da Cb Cc

Pythagorean theorem : The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of thehypotenuse.

45

45

90

triangle theorem : For any right triangle with acuteangles measuring 45 , the legs are the same length, and the hypotenusehas a length equal to p 2 times the length of one of those legs.

1


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2 Chapter 1

30

60

90

triangle theorem : For any right triangle with acuteangles measuring 30 and 60 ,

1. The hypotenuse is twice as long as the leg opposite the 30 angle(the shorter leg).

2. The leg opposite the 30 angle (the shorter leg) is 12 as long as thehypotenuse.

3. The leg opposite the 60 angle (the longer leg) equals the lengthof the other (shorter) leg times p 3.

4. The leg opposite the 30 angle equals the length of the other legdivided by p 3.

Equilateral triangle : For any equilateral triangle:

D D D60

A D 14 b2p 3 h D 12 bp 3QUADRILATERALS

Rectangle Square A D w ADs2P D2 C2w P D4sDiagonal Dp 2 Cw2 Diagonal Dsp 2

Parallelogram Trapezoid A Dbh Dab sin AD 12 h a CbP D2a C2b P Da Cb

Ch csc Ccsc


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Chapter 1 3

REGULAR POLYGON OF n SIDES

central angle:2n

A D 14 na

2cot n

P Dan

1.2 Circles

TERMINOLOGY

Denition : In a plane, a circle is the set of all points a given distance,called the radius , from a given point, called the center .

Circumference : distance around a circle.Chord : a line joining two points of a circle. Diameter : a chord through the center: AB in Figure 1.1. Arc: part of a circle: BC, AC , or ACB in Figure 1.1. The length s of an

arc of a circle of radius r with central angle (measured in radians)is s Dr .To intercept an arc is to cut off the arc; in Figure 1.1, 6COB intercepts BC .

A tangent of a circle is a line that intersects the circle at one and onlyone point.

A secant of a circle is a line that intersects the circle at exactly two points.

FIGURE 1.1


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An inscribed polygon is a polygon, all of whose sides are chords of acircle. A regular inscribed polygon is a polygon, all of whose sidesare the same length.

An inscribed circle is a circle to which all the sides of a polygon aretangents.

A circumscribed polygon is a polygon, all of whose sides are tangentsto a circle.

A circumscribed circle is a circle passing through each vertex of a polygon.

BASIC FORMULAS

Circle A D r 2C D2 r D d

Sector A D 12 r 2 s Dr

We rst use this in the text in Section 2.2 proving a veryimportant limit property.

Segment

A D 12 r 2 sin


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1.3 Solid GeometryRectangular parallelepiped (box)V DabcDiagonal Dp a 2 Cb2 Cc2.

PrismV D Bh B is area of the base

PyramidV D

13 Bh

B is area of the baseThis formula is derived in Example 2, Section 6.2, of the text .


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Tetrahedron(a pyramid with a triangular base)V D 13 hp s s a s b s cwhere s D 12 aCb Cc

Right circular cylinderV D r 2hLateral surface D2 rhS D2 rh C2 r 2

Right circular coneV D 13 r 2hLateral surface D rlS D rl C r 2The formula for the lateral surface is derived in Section 6.4

of the text.

Frustum of a right circular coneV D 13 h r 2 CrR C R2 or V D

13 h B1 Cp B1 B2 C B2 Note: h Dh1 h2This formula is derived in Problem 66 of Problem Set 6.2.


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Frustum of a pyramidV D 13 h B1 Cp B1 B2 C B2 Note: h Dh1 h2

TorusS

D4 2 Rr

This formula is derived in Problem 55 of Problem Set 12.6.

V D2 2 Rr 2This formula is derived in Example 7 of Section 6.5.

Spherical segmentS D2 rh , with radius r V D

16 h 3r 1

2 C3r 22 Ch2 , with cross sections of radii r 1 and r 2For the cap,S D2 r r h , with radius r V D3 2r

2 3r 2h Ch3 , with cross sections of radii r 1 and r 2This volume for the cap is Problem 65 of Problem Set 6.2 .


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Cylinder with a cross-sectional area AV D Ah; S Dp C2 A

Prismatoid, pontoon, wedgeV D 16 h B0 C4 B1 C B2

Quadric Surfaces

Sphere

x 2 Cy 2 C z2 Dr 2V D

43 r

3

S D4 r 2

We derive the formula for the volume of a sphere in Example 3 of Section 12.3 and again in Example 5 of Section 12.7.


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Ellipsoid x 2

a 2 Cy 2

b2 C z2

c2 D1V D

43 abc

A special case of this formula is derived in Problem 46 of the supplementary problems for Chapter 6, where r D0 and c Db.

Elliptic Paraboloid x 2

a 2 Cy 2

b2 D z

Hyperboloid of One Sheet x 2

a 2 Cy 2

b2 z2

c2 D1


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10 Chapter 1

Hyperboloid of Two Sheets x 2

a 2 Cy 2

b2 z2

c2 D 1

Hyperbolic paraboloidy 2

a 2 x 2

b2 D z

An oblate spheroid is formed by the rotation of the ellipse x 2

a 2 Cy 2

b2 D1 about its minor axis, b. Let be the eccentricity.

V D43

a 2b

S

D2 a 2

C

b2ln

1 C1

A prolate spheroid is formed by the rotation of the ellipse x 2

a 2 Cy 2

b2 D1 about its minor axis, a. Let be the eccentricity.

V D43

ab 2

S D2 b2

Cab

sin 1


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1.4 Congruent Triangles

We say that two gures are congruent if they have the same sizeand shape. For congruent triangles ABC and DEF , denoted by

ABC ' DEF , we may conclude that all six corresponding parts(three angles and three sides) are congruent.

EXAMPLE 1.1 Corresponding parts of a triangle

Name the corresponding parts of the given triangles.a. ABC ' A 0 B0C 0 b. RST ' UST

Solution

a. AB corresponds to A0 B0 b. RS corresponds to US AC corresponds to A0C 0 RT corresponds to UT BC corresponds to B0C 0 ST corresponds to ST6 A corresponds to 6 A0 6 R corresponds to 6U6 B corresponds to 6 B0 6 RTS corresponds to 6UTS6C corresponds to 6C 0 6 RST corresponds to 6UST

Two angles are equal if they have the same measure. For thetriangles in Example 1.1, we see that 6 A corresponds to 6 A0. Thismeans that these angles are the same size, or have the same measure .We write m6 A Dm6 A0to mean that the angles have the same measure,or in other words, the same size and shape.

Line segments, angles, triangles, or other geometric gures arecongruent if they have the same size and shape. In Example 1.1, sincem6 A Dm6 A0, we say that angles A and A0 are congruent, and we write6 A ' 6 A0.In this section, we focus on triangles.


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CONGRUENT TRIANGLES

Two triangles are congruent if their corresponding sideshave the same length and their corresponding angles havethe same measure.

To prove that two triangles are congruent, you must show that theyhave the same size and shape. It is not necessary to show that all six parts (three sides and three angles) are congruent; if certain of thesesix parts are congruent, it necessarily follows that the other parts arecongruent. Three important properties are used to show congruenceof triangles:

CONGRUENT-TRIANGLE PROPERTIES

SIDESIDESIDE (SSS)If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

SIDEANGLESIDE (SAS)If two sides of one triangle and the angle between thosesides are congruent to the corresponding sides and angle of another triangle, then the two triangles are congruent.

ANGLESIDEANGLE (ASA)If two angles and the side that connects them on onetriangle are congruent to the corresponding angles and sideof another triangle, then the two triangles are congruent.

EXAMPLE 1.2 Finding congruent triangles

Determine whether each pair of triangles is congruent. If so, cite oneof the congruent-triangle properties.

Solution

a . b .

Congruent; SAS Not congruent


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c. d .

Congruent; ASA Congruent; SSS

A side that is in common to two triangles obviously is equal in lengthto itself and does not need to be marked.

In geometry, the main use of congruent triangles is when we want

to know whether an angle from one triangle is congruent to an anglefrom a different triangle or when we want to know whether a sidefrom one triangle is the same length as the side from another triangle.In order to do this, we often prove that one triangle is congruent to theother (using one of the three congruent-triangle properties) and thenuse the following property:

CONGRUENT-TRIANGLE PROPERTY

Corresponding parts of congruent triangles are congruent.

1.5 Similar Triangles

See Sections 3.7 and 4.6 of the text for examples in which we usethese ideas in calculus.

It is possible for two gures to have the same shape, but not necessarilythe same size. These gures are called similar gures. We will nowfocus on similar triangles . If ABC is similar to DEF , we write

ABC DEFSimilar triangles are shown in Figure 1.2.

Youshouldnote that congruent triangles must be similar, but similar triangles are not necessarily congruent. Since similar gures have thesame shape, we talk about corresponding angles and correspondingsides . The corresponding angles of similar triangles are the anglesthat have the same measure. It is customary to label the vertices of triangles with capital letters and the sides opposite the angles at those


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Figure 1.2 Similar triangles

vertices with corresponding lowercase letters. It is easy to see that, if the triangles are similar, the corresponding sides are the sides oppositeequal angles. In Figure 1.2, we see that

6 A and 6 D are corresponding angles;6 B and 6 E are corresponding angles; and 6C and 6F are corresponding angles.

Side a BC is opposite 6 A, and d EF is opposite 6 D, so we say thata corresponds to d ;b corresponds to e; and c corresponds to f .

Even though corresponding angles are the same size, correspondingsides do not need to be the same length. If they are the same length,then the triangles are congruent. However, when they are not the samelength, we can say they are proportional . As Figure 1.2 illustrates,when we say the sides are proportional, we mean that

ab D

de

ac D

df

bc D

ef

ba D

ed

ca D

fd

cb D

fe

SIMILAR TRIANGLES

Two triangles are similar if two angles of one triangle havethe same measure as two angles of the other triangle. If the triangles are similar, then their corresponding sides are proportional.


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FIGURE 1.3

EXAMPLE 1.3 Similar triangles

Identify pairs of triangles that are similar in Figure 1.3.

Solution ABC JKL ; DEF GIH ; SQR STU.

EXAMPLE 1.4 Finding unknown lengths in similar triangles

Given the following similar triangles, nd the unknown lengthsmarked b0 and c0:

Solution Since corresponding sides are proportional (other propor-tions are possible), we have

a0a D

b0b

ac D

a0c0

4

8 Db012

8

14 .4 D4

c0b0D

4 128

c0D14 .4 4

8

D6 D7.2


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EXAMPLE 1.5 Finding a perimeter by using similar triangles

In equilateral ABC , suppose DE D2 and is parallel to AB, as shownat the right. If AB is three times as long as DE , what is the perimeter of quadrilateral ABED?

Solution ABC DEC , so DEC is equilateral. This means thatCE and DC both are of length 2; thus, EB and AD both are of length4. The perimeter of the quadrilateral is

j ABj C j BEj C j DE j C j ADj D6 C4 C2 C4 D16 Finding similar triangles is simplied even further if we know that thetriangles are right triangles, because then the triangles are similar if one of the acute angles has the same measure as an acute angle of theother triangle.

EXAMPLE 1.6 Using similar triangles to nd an unknownlength

Suppose that a tree and a yardstick are casting shadows as shownin Figure 1.4. If the shadow of the yardstick is 3 yards long and theshadow of the tree is 12 yards long, use similar triangles to estimatethe height of the tree if you know that angles S and S0 are the samesize.

FIGURE 1.4


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Solution Since 6G and 6G0are right angles, and since S and S0are thesame size, we see that SGT S 0G0T0. Therefore, correspondingsides are proportional. Hence,13 D

h12

h D1 12

3

D4The tree is 4 yards tall.

EXAMPLE 1.7 Similar triangles in a pyramid

This example is adapted from Example 2 of Section 6.2

A regular pyramid has a square base of side L, and its apex is located H units above the center of its base. The pyramid is shown in Figure 1.5.

Suppose a cut is made h units from the bottom, thus forming twotriangles, as shown at the right in Figure 1.5. Use similar triangles tond the length of the cut.

Solution Let be the length of the cut. We recognize two triangles:ABC and DEC , and we want to show that these triangles are

similar. Obviously, m6C Dm6C in both triangles. Since the linesegment AB is parallel to the line segment DE , and we can consider the line passing through A and D to be a transversal, we conclude thatm6 A Dm6 D because 6 A and 6 D are corresponding angles. Since twoangles of one triangle have the same measure as the correspondingangles in the other triangle, we conclude that ABC DEC . Itfollows that if the triangles are similar, then their corresponding sides

Figure 1.5 Pyramid with a square base


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are proportional.

L D H h

H Note that the height of DEC is H h

D 1h

H L Multiply both sides by L.

1.6 PROBLEM SET 1

1. In TRI and ANG shown at right, 6 R ' 6 N and jT Rj D

j A N

j. Name other pairs you would need to know in order to

show that the triangles are congruent bya. SSS b. SAS c. ASA

2. In ABC and DEF shown at right, 6 A ' 6 D and j A Cj Dj D F j. Name other pairs you would need to know in order toshow that the triangles are congruent bya. SSS b. SAS c. ASA

Name the corresponding parts of the triangles in Problems 36.

3. 4.


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5. 6.

In Problems 710, determine whether each pair of triangles iscongruent. If so, cite one of the congruent-triangle properties.

7. 8.

9. 10.

In Problems 1116, tell whether it is possible to conclude that the pairs of triangles are similar.11.

12.


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13.

14.

15.

16.

Given the two similar triangles shown, nd the unknown lengths in Problems 1722.

17. a

D4, b

D8, a 0

D2; nd b0.

18. b D5, c D15 , b 0 D2; nd c0.19. c D6, a D4, c 0 D8; nd a0.20. a0 D7, b 0 D8, a D5; nd b.21. b0 D8, c 0 D12 , c D4; nd b.22. c0 D9, a 0 D2, c D5; nd a.23. How far from the base of a building must a 26-ft ladder be

placed so that it reaches 10 ft up the wall?24. How high up a wall does a 26-ft ladder reach if the bottom of

the ladder is placed 6 ft from the building?


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25. A carpenter wants to be sure that the corner of a building issquare and measures 6 ft and 8 ft along the sides. How longshould the diagonal be?

26. What is the exact length of the hypotenuse if the legs of a righttriangle are 2 in. each?

27. What is the exact length of one leg of an isosceles right triangleif the hypotenuse is 3 ft?

28. An empty rectangular lot is 40 ft by 65 ft. How many feet would you save by walking diagonally across the lot instead of walkingthe length and width? Round your answer to the nearest foot.

29. A television antenna is to be erected and held by guy wires. If the guy wires are 15 ft from the base of the antenna and theantenna is 10 ft high, what is the exact length of each guy wire?What is the length of each guy wire, rounded to the nearest foot?If three guy wires are attached, how many feet of wire should be purchased if it cannot be bought in fractions of a foot?

30. In equilateral ABC , shown below, D is themidpointof segment A B. What is the length of C D ?

31. In the gure shown below, AB and DE are parallel. What is thelength of AB?

32. Ben walked diagonally across a rectangular eld that measures100 ft by 240 ft. How far did Ben walk?

33. Use similar triangles and a proportion to nd the length of thelake shown in the following gure:

34. Use similar triangles and a proportion to nd the height of thehouse shown in the following gure:


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35. If a tree casts a shadow of 12 ft at the same time a 6-ft person

casts a shadow of 212

ft, nd the height of the tree to the nearest

foot.36. If an inverted circular cone (vertex at the bottom) of height 10

cm and a radius of 4 cm contains liquid with height measuring3.8 cm, what is the volume of the liquid?

37. A person 6 ft tall is walking away from a streetlight 20 ft highat the rate of 7 ft/s. How long (to the nearest ft) is the personsshadow at the instant when the person is 10 ft from the base of the lamppost?

Example 2, Section 3.7

38. A bag is tied to the top of a 5-m ladder resting against a verticalwall. Suppose the ladder begins sliding down the wall in such away that the foot of the ladder is moving away from the wall.How high is the bag at the instant when the base of the ladder is4 m from the base of the wall?

Example 3, Section 3.7


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39. A tank lled with water is in the shape of an inverted cone 20ft high with a circular base (on top) whose radius is 5 ft. Howmuch water does the tank hold when the water level is 8 ft deep?

See Example 5 of Section 3.7 of the text.

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Review of Geometry