10.1 Tangents to Circles 599

1. Sketch a circle. Then sketch and label a radius, a diameter, and a chord.

2. How are chords and secants of circles alike? How are they different?

3. XY¯̆

is tangent to ›C at point P. What is m™CPX? Explain.

4. The diameter of a circle is 13 cm. What is the radius of the circle?

5. In the diagram at the right, AB = BD = 5 and AD = 7. Is BD

¯̆tangent to ›C? Explain.

AB¯̆

is tangent to ›C at A and DB¯̆

is tangent to ›C at D. Find the value of x.

6. 7. 8.

FINDING RADII The diameter of a circle is given. Find the radius.

9. d = 15 cm 10. d = 6.7 in. 11. d = 3 ft 12. d = 8 cm

FINDING DIAMETERS The radius of ›C is given. Find the diameter of ›C.

13. r = 26 in. 14. r = 62 ft 15. r = 8.7 in. 16. r = 4.4 cm

17. CONGRUENT CIRCLES Which two circles below are congruent? Explainyour reasoning.

MATCHING TERMS Match the notation with the term that best describes it.

18. ABÆ

A. Center

19. H B. Chord

20. HF¯̆

C. Diameter

21. CHÆ

D. Radius

22. C E. Point of tangency

23. HBÆ

F. Common external tangent

24. AB¯̆

G. Common internal tangent

25. DE¯̆

H. Secant

PRACTICE AND APPLICATIONS

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STUDENT HELP

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STUDENT HELP

HOMEWORK HELPExample 1: Exs. 18–25,

42–45Example 2: Exs. 26–31Example 3: Exs. 32–35Example 4: Exs. 36–39Example 5: Exs. 40, 41Example 6: Exs. 49–53Example 7: Exs. 46–48

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600 Chapter 10 Circles

IDENTIFYING TANGENTS Tell whether the common tangent(s) are internalor external.

26. 27. 28.

DRAWING TANGENTS Copy the diagram. Tell how many common tangentsthe circles have. Then sketch the tangents.

29. 30. 31.

COORDINATE GEOMETRY Use the diagram at the right.

32. What are the center and radius of ›A?

33. What are the center and radius of ›B?

34. Describe the intersection of the two circles.

35. Describe all the common tangents of the two circles.

DETERMINING TANGENCY Tell whether AB¯̆

is tangent to ›C. Explain yourreasoning.

36. 37.

38. 39.

GOLF In Exercises 40 and 41, use the following information.A green on a golf course is in the shape of a circle. A golfball is 8 feet from the edge of the green and 28 feet from a point of tangency on the green, as shown at the right.Assume that the green is flat.

40. What is the radius of the green?

41. How far is the golf ball from the cup at the center?

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TIGER WOODS Atage 15 Tiger Woods

became the youngest golferever to win the U.S. JuniorAmateur Championship, andat age 21 he became theyoungest Masters championever.

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10.1 Tangents to Circles 601

MEXCALTITLÁN The diagram shows the layout of the streets on Mexcaltitlán Island.

42. Name two secants.

43. Name two chords.

44. Is the diameter of the circle greater than HC?Explain.

45. If ¤LJK were drawn, one of its sides would betangent to the circle. Which side is it?

USING ALGEBRA AB¯̆

and AD¯̆

are tangent to ›C. Find the value of x.

46. 47. 48.

49. PROOF Write a proof.

GIVEN � PS¯̆

is tangent to ›X at P.

PS¯̆

is tangent to ›Y at S.

RT¯̆

is tangent to ›X at T.

RT¯̆

is tangent to ›Y at R.

PROVE � PSÆ

£ RTÆ

PROVING THEOREM 10.1 In Exercises 50–52, you will use an indirect argument to prove Theorem 10.1.

GIVEN � l is tangent to ›Q at P.

PROVE � l fi QPÆ

50. Assume l and QPÆ

are not perpendicular. Then the perpendicular segmentfrom Q to l intersects l at some other point R. Because l is a tangent, Rcannot be in the interior of ›Q. So, how does QR compare to QP? Write an inequality.

51. QRÆ

is the perpendicular segment from Q to l, so QRÆ

is the shortest segmentfrom Q to l. Write another inequality comparing QR to QP.

52. Use your results from Exercises 50 and 51 to complete the indirect proof ofTheorem 10.1.

53. PROVING THEOREM 10.2 Write an indirect proof of Theorem 10.2.(Hint: The proof is like the one in Exercises 50–52.)

GIVEN � l is in the plane of ›Q.

l fi radius QPÆ

at P.

PROVE � l is tangent to ›Q.

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602 Chapter 10 Circles

LOGICAL REASONING In ›C, radii CAÆ

and CBÆ

are perpendicular. BD¯̆

and AD¯̆

are tangent to ›C.

54. Sketch ›C, CAÆ

, CBÆ

, BDÆ

, and ADÆ

.

55. What type of quadrilateral is CADB? Explain.

56. MULTI-STEP PROBLEM In the diagram, line j is tangent to ›C at P.

a. What is the slope of radius CPÆ

?

b. What is the slope of j? Explain.

c. Write an equation for j.

d. Writing Explain how to find an equation for a line tangent to ›C at a point other than P.

57. CIRCLES OF APOLLONIUS The Greek mathematician Apollonius (c. 200 B.C.) proved that for any three circles with no common points orcommon interiors, there are eight ways to draw a circle that is tangent to thegiven three circles. The red, blue, and green circles are given. Two ways todraw a circle that is tangent to the given three circles are shown below.Sketch the other six ways.

58. TRIANGLE INEQUALITIES The lengths of two sides of a triangle are 4 and 10.Use an inequality to describe the length of the third side. (Review 5.5)

PARALLELOGRAMS Show that the vertices represent the vertices of aparallelogram. Use a different method for each proof. (Review 6.3)

59. P(5, 0), Q(2, 9), R(º6, 6), S(º3, º3)

60. P(4, 3), Q(6, º8), R(10, º3), S(8, 8)

SOLVING PROPORTIONS Solve the proportion. (Review 8.1)

61. �1x1� = �

35� 62. �6

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69. 70. 71.

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EXTRA CHALLENGE

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6.1 Polygons 325

1. What is the plural of vertex?

2. What do you call a polygon with 8 sides? a polygon with 15 sides?

3. Suppose you could tie a string tightly around a convex polygon. Would thelength of the string be equal to the perimeter of the polygon? What if thepolygon were concave? Explain.

Decide whether the figure is a polygon. If it is not, explain why.

4. 5. 6.

Tell whether the polygon is best described as equiangular, equilateral,regular, or none of these.

7. 8. 9.

Use the information in the diagram to find m™A.

10. 11.

RECOGNIZING POLYGONS Decide whether the figure is a polygon.

12. 13. 14.

15. 16. 17.

PRACTICE AND APPLICATIONS

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STUDENT HELP

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326 Chapter 6 Quadrilaterals

CONVEX OR CONCAVE Use the number of sides to tell what kind ofpolygon the shape is. Then state whether the polygon is convex or concave.

18. 19. 20.

PARACHUTES Some gym classes use parachutes that look like the polygon at the right.

21. Is the polygon a heptagon, octagon, or nonagon?

22. Polygon LMNPQRST is one name for the polygon. State two other names.

23. Name all of the diagonals that have vertex M as anendpoint. Not all of the diagonals are shown.

RECOGNIZING PROPERTIES State whether the polygon is best described asequilateral, equiangular, regular, or none of these.

24. 25. 26.

TRAFFIC SIGNS Use the number of sides of the traffic sign to tell what kindof polygon it is. Is it equilateral, equiangular, regular, or none of these?

27. 28.

29. 30.

DRAWING Draw a figure that fits the description.

31. A convex heptagon 32. A concave nonagon

33. An equilateral hexagon that is not equiangular

34. An equiangular polygon that is not equilateral

35. LOGICAL REASONING Is every triangle convex? Explain your reasoning.

36. LOGICAL REASONING Quadrilateral ABCD is regular. What is the measure of ™ABC? How do you know?

L M

N

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ROAD SIGNSThe shape of a sign

tells what it is for. Forexample, triangular signslike the one above are usedinternationally as warningsigns.

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STUDENT HELP

HOMEWORK HELPExample 1: Exs. 12–17,

48–51Example 2: Exs. 18–20,

48–51Example 3: Exs. 24–30,

48–51Example 4: Exs. 36–46

6.1 Polygons 327

ANGLE MEASURES Use the information in the diagram to find m™A.

37. 38. 39.

40. TECHNOLOGY Use geometry software to draw a quadrilateral. Measureeach interior angle and calculate the sum. What happens to the sum as you

drag the vertices of the quadrilateral?

USING ALGEBRA Use the information in the diagram to solve for x.

41. 42. 43.

44. 45. 46.

47. A decagon has ten sides and a decade has tenyears. The prefix deca- comes from Greek. It means ten. What does theprefix tri- mean? List four words that use tri- and explain what they mean.

PLANT SHAPES In Exercises 48–51, use the following information.Cross sections of seeds and fruits often resemble polygons. Next to each crosssection is the polygon it resembles. Describe each polygon. Tell what kind ofpolygon it is, whether it is concave or convex, and whether it appears to beequilateral, equiangular, regular, or none of these. � Source: The History and Folklore of

N. American Wildflowers

48. Virginia Snakeroot 49. Caraway

50. Fennel 51. Poison Hemlock

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cross section shaped like a 5 pointed star. The fruitcomes from an evergreentree whose leaflets may fold at night or when the tree is shaken.

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SOFTWARE HELPVisit our Web site

www.mcdougallittell.comto see instructions forseveral softwareapplications.

INTE

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STUDENT HELP

328 Chapter 6 Quadrilaterals

52. MULTI-STEP PROBLEM Envelope manufacturers fold a specially-shapedpiece of paper to make an envelope, as shown below.

a. What type of polygon is formed by the outside edges of the paper before itis folded? Is the polygon convex?

b. Tell what type of polygon is formed at each step. Which of the polygonsare convex?

c. Writing Explain the reason for the V-shaped notches that are at the endsof the folds.

53. FINDING VARIABLES Find the values of x and y in the diagram at the right.Check your answer. Then copy the shape and write the measure of eachangle on your diagram.

PARALLEL LINES In the diagram, j ∞ k. Find the value of x. (Review 3.3 for 6.2)

54. 55. 56.

57. 58. 59.

COORDINATE GEOMETRY You are given the midpoints of the sides of atriangle. Find the coordinates of the vertices of the triangle. (Review 5.4)

60. L(º3, 7), M(º5, 1), N(º8, 8) 61. L(º4, º1), M(3, 6), N(º2, º8)

62. L(2, 4), M(º1, 2), N(0, 7) 63. L(º1, 3), M(6, 7), N(3, º5)

64. USING ALGEBRA Use the Distance Formula to find the lengths of thediagonals of a polygon with vertices A(0, 3), B(3, 3), C(4, 1), D(0, º1), andE(º2, 1). (Review 1.3)

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11.1 Angle Measures in Polygons 665

1. Name an interior angle and an exterior angle of the polygon shown at the right.

2. How many exterior angles are there in an n-gon? Are they all consideredwhen using the Polygon Exterior Angles Theorem? Explain.

Find the value of x.

3. 4. 5.

SUMS OF ANGLE MEASURES Find the sum of the measures of the interiorangles of the convex polygon.

6. 10-gon 7. 12-gon 8. 15-gon 9. 18-gon

10. 20-gon 11. 30-gon 12. 40-gon 13. 100-gon

ANGLE MEASURES In Exercises 14–19, find the value of x.

14. 15. 16.

17. 18. 19.

20. A convex quadrilateral has interior angles that measure 80°, 110°, and 80°. What is the measure of the fourth interior angle?

21. A convex pentagon has interior angles that measure 60°, 80°, 120°, and 140°.What is the measure of the fifth interior angle?

DETERMINING NUMBER OF SIDES In Exercises 22–25, you are given themeasure of each interior angle of a regular n-gon. Find the value of n.

22. 144° 23. 120° 24. 140° 25. 157.5°

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21Example 2: Exs. 17–19,

22–28Example 3: Exs. 29–38Example 4: Exs. 39, 40,

49, 50Example 5: Exs. 51–54

Extra Practiceto help you masterskills is on p. 823.

STUDENT HELP

666 Chapter 11 Area of Polygons and Circles

CONSTRUCTION Use a compass, protractor, and ruler to check the results of Example 2 on page 662.

26. Draw a large angle that measures 140°. Mark congruent lengths on the sidesof the angle.

27. From the end of one of the congruent lengths in Exercise 26, draw the secondside of another angle that measures 140°. Mark another congruent lengthalong this new side.

28. Continue to draw angles that measure 140° until a polygon is formed. Verifythat the polygon is regular and has 9 sides.

DETERMINING ANGLE MEASURES In Exercises 29–32, you are given thenumber of sides of a regular polygon. Find the measure of each exteriorangle.

29. 12 30. 11 31. 21 32. 15

DETERMINING NUMBER OF SIDES In Exercises 33–36, you are given themeasure of each exterior angle of a regular n-gon. Find the value of n.

33. 60° 34. 20° 35. 72° 36. 10°

37. A convex hexagon has exterior angles that measure 48°, 52°, 55°, 62°, and68°. What is the measure of the exterior angle of the sixth vertex?

38. What is the measure of each exterior angle of a regular decagon?

STAINED GLASS WINDOWS In Exercises 39 and 40, the purple andgreen pieces of glass are in the shape of regular polygons. Find themeasure of each interior angle of the red and yellow pieces of glass.

39. 40.

41. FINDING MEASURES OF ANGLESIn the diagram at the right, m™2 = 100°,m™8 = 40°, m™4 = m™5 = 110°. Find the measures of the other labeled angles and explain your reasoning.

42. Writing Explain why the sum of the measures of the interior angles of anytwo n-gons with the same number of sides (two octagons, for example) is thesame. Do the n-gons need to be regular? Do they need to be similar?

43. PROOF Use ABCDE to write a paragraphproof to prove Theorem 11.1 for pentagons.

44. PROOF Use a paragraph proof to provethe Corollary to Theorem 11.1.

A

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EB

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STAINED GLASSis tinted glass that

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11.1 Angle Measures in Polygons 667

45. PROOF Use this plan to write a paragraph proof of Theorem 11.2.Plan for Proof In a convex n-gon, the sum of the measures of an interiorangle and an adjacent exterior angle at any vertex is 180°. Multiply by n toget the sum of all such sums at each vertex. Then subtract the sum of theinterior angles derived by using Theorem 11.1.

46. PROOF Use a paragraph proof to prove the Corollary to Theorem 11.2.

TECHNOLOGY In Exercises 47 and 48, use geometry software toconstruct a polygon. At each vertex, extend one of the sides of the

polygon to form an exterior angle.

47. Measure each exterior angle and verify that the sum of the measures is 360°.

48. Move any vertex to change the shape of your polygon. What happens to themeasures of the exterior angles? What happens to their sum?

49. HOUSES Pentagon ABCDE is 50. TENTS Heptagon PQRSTUVan outline of the front of a house. is an outline of a camping tent. Find the measure of each angle. Find the unknown angle measures.

POSSIBLE POLYGONS Would it be possible for a regular polygon to haveinterior angles with the angle measure described? Explain.

51. 150° 52. 90° 53. 72° 54. 18°

USING ALGEBRA In Exercises 55 and 56, you are given a function and itsgraph. In each function, n is the number of sides of a polygon and ƒ(n) ismeasured in degrees. How does the function relate to polygons? Whathappens to the value of ƒ(n) as n gets larger and larger?

55. ƒ(n) = �180nnº 360� 56. ƒ(n) = �36

n0

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57. LOGICAL REASONING You are shown part of a convex n-gon. The pattern of congruent angles continues around the polygon. Use the Polygon Exterior Angles Theorem to find the value of n.

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QUANTITATIVE COMPARISON In Exercises 58–61, choose the statementthat is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

58.

59.

60.

61.

62. Polygon STUVWXYZ is a regular octagon. Suppose sides STÆ

and UVÆ

areextended to meet at a point R. Find the measure of ™TRU.

FINDING AREA Find the area of the triangle described. (Review 1.7 for 11.2)

63. base: 11 inches; height: 5 inches 64. base: 43 meters; height: 11 meters

65. vertices: A(2, 0), B(7, 0), C(5, 15) 66. vertices: D(º3, 3), E(3, 3), F(º7, 11)

VERIFYING RIGHT TRIANGLES Tell whether the triangle is a right triangle.(Review 9.3)

67. 68. 69.

FINDING MEASUREMENTS GDÆ

and FHÆ

are diameters of circle C. Find the indicated arc measure. (Review 10.2)

70. mDH� 71. mED�72. mEH� 73. mEHG�

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Column A Column B

The sum of the interior angle The sum of the interior angle measures of a decagon measures of a 15-gon

The sum of the exterior angle 8(45°)measures of an octagon

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