CHAPTER 1Review of Geometry

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

1.1 Polygons

CLASSIFICATION

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

TRIANGLESA D 1

2bh The sum of the measures of theangles of a triangle is 180°.

P D a C b C c

Pythagorean theorem: The sum of the squares of the lengths ofthe 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

p2 times the length of one of those legs.

1

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 the

hypotenuse.3. The leg opposite the 60° angle (the longer leg) equals the length

of the other (shorter) leg timesp

3.4. The leg opposite the 30° angle equals the length of the other leg

divided byp

3.

Equilateral triangle: For any equilateral triangle:

˛ D ˇ D � D 60° A D 14b

2p

3 h D 12b

p3

QUADRILATERALS

Rectangle SquareA D �w A D s2

P D 2� C 2w P D 4sDiagonal D p

�2 C w2 Diagonal D sp

2

Parallelogram TrapezoidA D bh D ab sin � A D 1

2h�a C b�P D 2a C 2b P D a C b

Ch�csc � C csc ��

Chapter 1 3

REGULAR POLYGON OF n SIDES

central angle:2�

nA D 1

4na2 cot�

nP D an

1.2 Circles

TERMINOLOGY

Definition: 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 D r�.

To intercept an arc is to cut off the arc; in Figure 1.1, 6 COB interceptsBC.

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 twopoints.

FIGURE 1.1

4 Chapter 1

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 apolygon.

BASIC FORMULAS

Circle

A D �r2

C D 2�r D �d

Sector

A D 12r

2�s D r�

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

Segment

A D 12r

2�� � sin ��

Chapter 1 5

1.3 Solid Geometry

Rectangular parallelepiped (box)

V D abcDiagonal D p

a2 C b2 C c2.

PrismV D BhB is area of the base

PyramidV D 1

3BhB is area of the base

This formula is derived in Example 2, Section 6.2, of the text.

6 Chapter 1

Tetrahedron(a pyramid with a triangular base)V D 1

3hp

s�s � a��s � b��s � c�where s D 1

2 �a C b C c�

Right circular cylinderV D �r2hLateral surface D 2�rhS D 2�rh C 2�r2

Right circular coneV D 1

3�r2hLateral surface D �rlS D �rl C �r2

The formula for the lateral surface is derived in Section 6.4of the text.

Frustum of a right circular coneV D 1

3�h�r2 C rR C R2� orV D 1

3h�B1 C pB1B2 C B2�

Note: h D h1 � h2

This formula is derived in Problem 66 of Problem Set 6.2.

Chapter 1 7

Frustum of a pyramidV D 1

3h�B1 C pB1B2 C B2�

Note: h D h1 � h2

TorusS D 4�2Rr

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

V D 2�2Rr2

This formula is derived in Example 7 of Section 6.5.

Spherical segmentS D 2�rh, with radius rV D 1

6�h�3r12 C 3r2

2 C h2�, with cross sections of radii r1 and r2

For the ‘‘cap,’’S D 2�r�r � h�, with radius r

V D �

3�2r2 � 3r2h C h3�, with cross sections of radii r1 and r2

This volume for the cap is Problem 65 of Problem Set 6.2.

8 Chapter 1

Cylinder with a cross-sectional area AV D Ah; S D p� C 2A

Prismatoid, pontoon, wedgeV D 1

6h�B0 C 4B1 C B2�

Quadric Surfaces

Sphere

x2 C y2 C z2 D r2

V D 43�r3

S D 4�r2

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

Chapter 1 9

Ellipsoidx2

a2C y2

b2C z2

c2D 1

V D 43�abc

A special case of this formula is derived in Problem 46 of thesupplementary problems for Chapter 6, where r D 0 and c D b.

Elliptic Paraboloidx2

a2C y2

b2D z

Hyperboloid of One Sheetx2

a2C y2

b2� z2

c2D 1

10 Chapter 1

Hyperboloid of Two Sheetsx2

a2C y2

b2� z2

c2D �1

Hyperbolic paraboloidy2

a2� x2

b2D z

An oblate spheroid is formed by the rotation of the ellipsex2

a2C y2

b2D 1 about its minor axis, b. Let ! be the eccentricity.

V D 4

3�a2b

S D 2�a2 C �b2

!ln

(1 C !

1 � !

)

A prolate spheroid is formed by the rotation of the ellipsex2

a2C y2

b2D 1 about its minor axis, a. Let ! be the eccentricity.

V D 4

3�ab2

S D 2�b2 C �ab

!sin�1 !

Chapter 1 11

1.4 Congruent Triangles

We say that two figures are congruent if they have the same sizeand shape. For congruent triangles ABC and DEF, denoted byABC ' 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 ' A0B0C0 b. RST ' UST

Solution

a. AB corresponds to A0B

0b. RS corresponds to US

AC corresponds to A0C

0RT corresponds to UT

BC corresponds to B0C

0ST corresponds to ST

6 A corresponds to 6 A0 6 R corresponds to 6 U6 B corresponds to 6 B0 6 RTS corresponds to 6 UTS6 C corresponds to 6 C0 6 RST corresponds to 6 UST

�

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 m 6 A D m 6 A0 to mean that the angles have the same measure,or in other words, the same size and shape.

Line segments, angles, triangles, or other geometric figures arecongruent if they have the same size and shape. In Example 1.1, sincem 6 A D m 6 A0, we say that angles A and A0 are congruent, and we write6 A ' 6 A0.

In this section, we focus on triangles.

12 Chapter 1

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 sixparts (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

SIDE–SIDE–SIDE (SSS)If three sides of one triangle are congruent to three sides ofanother triangle, then the two triangles are congruent.

SIDE–ANGLE–SIDE (SAS)If two sides of one triangle and the angle between thosesides are congruent to the corresponding sides and angle ofanother triangle, then the two triangles are congruent.

ANGLE–SIDE–ANGLE (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

Chapter 1 13

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 wantto 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 figures to have the same shape, but not necessarilythe same size. These figures are called similar figures. We will nowfocus on similar triangles. If ABC is similar to DEF, we write

ABC ¾ DEF

Similar triangles are shown in Figure 1.2.You should note that congruent triangles must be similar, but similar

triangles are not necessarily congruent. Since similar figures 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 oftriangles with capital letters and the sides opposite the angles at those

14 Chapter 1

Figure 1.2 Similar triangles

vertices with corresponding lowercase letters. It is easy to see that, ifthe 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; and6 C and 6 F 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; andc 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

a

bD d

e

a

cD d

f

b

cD e

fb

aD e

d

c

aD f

d

c

bD f

e

SIMILAR TRIANGLES

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

Chapter 1 15

FIGURE 1.3

EXAMPLE 1.3 Similar triangles

Identify pairs of triangles that are similar in Figure 1.3.

Solution ABC ¾ JKL; DEF ¾ GIH; SQR ¾ STU. �

EXAMPLE 1.4 Finding unknown lengths in similar triangles

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

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

a0

aD b0

b

a

cD a0

c04

8D b0

12

8

14.4D 4

c0

b0 D 4�12�

8c0 D 14.4�4�

8D 6 D 7.2

�

16 Chapter 1

EXAMPLE 1.5 Finding a perimeter by using similar triangles

In equilateral ABC, suppose DE D 2 and is parallel to AB, as shownat the right. If AB is three times as long as DE, what is the perimeterof quadrilateral ABED?

Solution ABC ¾ 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

jABj C jBEj C jDEj C jADj D 6 C 4 C 2 C 4 D 16 �

Finding similar triangles is simplified even further if we know that thetriangles are right triangles, because then the triangles are similar ifone of the acute angles has the same measure as an acute angle of theother triangle.

EXAMPLE 1.6 Using similar triangles to find 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

Chapter 1 17

Solution Since 6 G and 6 G0 are right angles, and since S and S0 are thesame size, we see that SGT ¾ S0G0T0. Therefore, correspondingsides are proportional. Hence,

1

3D h

12

h D 1�12�

3

D 4

The 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 Hunits 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 tofind 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 aresimilar. Obviously, m 6 C D m 6 C in both triangles. Since the linesegment AB is parallel to the line segment DE, and we can considerthe line passing through A and D to be a transversal, we conclude thatm 6 A D m 6 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 ¾ DEC. Itfollows that if the triangles are similar, then their corresponding sides

Figure 1.5 Pyramid with a square base

18 Chapter 1

are proportional.

�

LD H � h

HNote that the height of DEC is H � h

� D(

1 � h

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 jTRj DjANj. Name other pairs you would need to know in order toshow that the triangles are congruent bya. SSS b. SAS c. ASA

2. In ABC and DEF shown at right, 6 A ' 6 D and jACj DjDFj. 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 3–6.

3. 4.

Chapter 1 19

5. 6.

In Problems 7–10, determine whether each pair of triangles iscongruent. If so, cite one of the congruent-triangle properties.

7. 8.

9. 10.

In Problems 11–16, tell whether it is possible to conclude that thepairs of triangles are similar.11.

12.

20 Chapter 1

13.

14.

15.

16.

Given the two similar triangles shown, find the unknown lengths inProblems 17–22.

17. a D 4, b D 8, a0 D 2; find b0.18. b D 5, c D 15, b0 D 2; find c0.19. c D 6, a D 4, c0 D 8; find a0.20. a0 D 7, b0 D 8, a D 5; find b.21. b0 D 8, c0 D 12, c D 4; find b.22. c0 D 9, a0 D 2, c D 5; find 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?

Chapter 1 21

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 wouldyou 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. Ifthe 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 shouldbe purchased if it cannot be bought in fractions of a foot?

30. In equilateral ABC, shown below, D is the midpoint of segmentAB. What is the length of CD?

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

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

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

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

22 Chapter 1

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

casts a shadow of 21

2ft, find 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 person’sshadow at the instant when the person is 10 ft from the base ofthe 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

Chapter 1 23

39. A tank filled 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.