jacobcd52/hs_math_cleaned · Datasets at Hugging Face

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the height of the original cone. a. Find the surface area of the original cone. b. Find the lateral area of the top of the cone. c. Find the area of the top base of the frustum. d. Use your results from parts a, b, and c to find the �� surface area of the frustum of the cone. �� 42. A frustum of a pyramid is a part of the pyramid with two parallel bases. The lateral faces of the frustum are trapezoids. Use the area formula for a trapezoid to derive a formula for the lateral area of a frustum of a regular square pyramid with base edge lengths b 1 and b 2 and slant height ℓ. 43. Use the net to derive the formula for the lateral area of a right cone with radius r and slant height ℓ. a. The length of the curved edge of the lateral surface must equal the circumference of the base. Find the circumference c of the base in terms of r. b. The lateral surface is part of a larger circle. Find the circumference C of the larger circle. c. The lateral surface area is c __ times the area of C the larger circle. Use your results from parts a and b to find c __. C �� d. Find the area of the larger circle. Use your result and the result from part c to find the lateral area L. SPIRAL REVIEW State whether the following can be described by a linear function. (Previous course) 44. the surface area of a right circular cone with height h and radius r 45. the perimeter of a rectangle with a height h that is twice as large as its width w 46. the area of a circle with radius r A point is chosen randomly in ACEF. Find the probability of each event. Round to the nearest hundredth. (Lesson 9-6) 47. The point is in △BDG. 48. The point is in ⊙H. 49. The point is in the shaded region. Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth. (Lesson 10-4) � � � � �� 50. 51. �� 52. �� 696 696 Chapter 10 Spatial Reasoning 10-6 Volume of Prisms and Cylinders TEKS G
.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the volume of a prism. Learn and apply the formula for the volume of a cylinder. Vocabulary volume Also G.1.B, G.5.A, G.5.B, G.11.D Who uses this? Marine biologists must ensure that aquariums are large enough to accommodate the number of fish inside them. (See Example 2.) The volume of a threedimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Cavalieri’s principle says that if two three-dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume. Volume of a Prism The volume of a prism with base area B and height h is V = Bh. The volume of a right rectangular prism with length ℓ, width w, and height h is V = ℓwh. The volume of a cube with edge length s is Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A V = ℓwh = (10) (12) (8) = 960 cm 3 Volume of a right rectangular prism Substitute 10 for ℓ, 12 for w, and 8 for h. B a cube with edge length 10 cm V = s 3 = 10 3 = 1000 cm 3 Volume of a cube Substitute 10 for s. 10-6 Volume of Prisms and Cylinders 697 697 �� To review the area of a regular polygon, see page 601. To review tangent ratios, see page 525. Find the volume of each prism. Round to the nearest tenth, if necessary. C a right regular pentagonal prism with base edge length 5 m and height 7 m Step 1 Find the apothem a of the base. First draw a right triangle on one base as shown. The measure of the angle with its vertex at = 36°. 10 the center is 360° _ tan 36° = 2.5 _ a a = 2.5 _ tan 36° The leg of
the triangle is half the side length, or 2.5 m. Solve for a. Step 2 Use the value of a to find the base area. aP =.5 _ tan 36° ) (25) = 31.25 _ tan 36° P = 5 (5) = 25 m Step 3 Use the base area to find the volume. V = Bh = 31.25 _ · 7 ≈ 301.1 m 3 tan 36° 1. Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. E X A M P L E 2 Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon ≈ 0.134 ft 3 ) Step 1 Find the volume of the aquarium in cubic feet. V = ℓwh = (120) (60) (8) = 57,600 ft 3 Step 2 Use the conversion factor to estimate the volume in gallons. 57, 600 ft 3 · 1 gallon _ 0.134 ft 3 ≈ 429,851 gallons 1 gallon _ = 1 0.134 ft 3 1 gallon _ 0.134 ft 3 Step 3 Use the conversion factor of the water. 8.33 pounds __ 1 gallon to estimate the weight 429,851 gallons · ≈ 3,580,659 pounds 8.33 pounds __ 1 gallon 8.33 pounds __ = 1 1 gallon Marine Biology The Islands of Steel habitat at the Texas State Aquarium in Corpus Christi has a volume of 132,000 gallons. Over 150 animals live in the habitat, including a sand tiger shark and nurse sharks. The aquarium holds about 429,851 gallons. The water in the aquarium weighs about 3,580,659 pounds. 2. What if…? Estimate the volume in gallons and the weight of the water in the aquarium above if the height were doubled. 698 698 Chapter 10 Spatial Reasoning 120 ft60 ft8 ftge07se_c10l06003aAB �� Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. Volume of a Cylinder The volume of a cylinder with base area B, radius r, and height h is V = Bh,
or V = π r 2h. E X A M P L E 3 Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. A V = π r 2h = π (8)2(12) Volume of a cylinder Substitute 8 for r and 12 for h. = 768π cm 3 ≈ 2412.7 cm 3 B a cylinder with a base area of 36π in 2 and a height equal to twice the radius Step 1 Use the base area to find the radius. π r 2 = 36π r = 6 Substitute 36π for the base area. Solve for r. Step 2 Use the radius to find the height. The height is equal to twice the radius. h = 2r = 2 (6) = 12 cm Step 3 Use the radius and height to find the volume. Volume of a cylinder V = π r 2h = π (6)2(12)= 432π in 3 ≈ 1357.2 in 3 Substitute 6 for r and 12 for h. 3. Find the volume of a cylinder with a diameter of 16 in. and a height of 17 in. Give your answer both in terms of π and rounded to the nearest tenth. 10-6 Volume of Prisms and Cylinders 699 699 �� E X A M P L E 4 Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by 1 __. Describe the effect on the volume. 2 original dimensions6) 2 (12) = 432π m 3 radius and height multiplied by 1 __ 3) 2 (6) = 54π m 3 Notice that 54π = 1 __ 8 (432π). If the radius and height are multiplied by 1 __ 2, the volume is multiplied by ( 1 __ 2 ), or 1 __ 8. 3 4. The length, width, and height of the prism are doubled. Describe the effect on the volume. E X A M P L E 5 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The base area of the prism is B = 1 __ 2 (6) (8) = 24 m 2. The volume of the prism is V = Bh = 24 (9) = 216 m 3. �� The cylinder’s diameter equals the hypotenuse of
the prism’s base, 10 m. So the radius is 5 m. The volume of the cylinder is V = π r 2 h = π (5) 2 (5) = 125π m 3. �� The total volume of the figure is the sum of the volumes. V = 216 + 125π ≈ 608.7 m 3 �� 5. Find the volume of the composite figure. Round to the nearest tenth. THINK AND DISCUSS 1. Compare the formula for the volume of a prism with the formula for the volume of a cylinder. 2. Explain how Cavalieri’s principle relates to the formula for the volume of an oblique prism. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the formula for the volume. 700 700 Chapter 10 Spatial Reasoning �� 10-6 Exercises Exercises KEYWORD: MG7 10-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a right cylinder, the altitude is? the axis. (longer than, shorter ̶̶̶̶ than, or the same length as Find the volume of each prism. p. 697 2. 3. 4. a cube with edge length 8 ft. Food The world’s largest ice cream cake, built in p. 698 New York City on May 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream cake in gallons. If the density of the ice cream was 4.73 pounds per gallon, estimate the weight of the cake. (Hint: 1 gallon ≈ 0.134 cubic feet. 699 Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. 6. 7. 8. a cylinder with base area 25π cm 2 and height 3 cm more than the radius. 700 Describe the effect of each change on the volume of the given figure. 9. The dimensions are multiplied by 1 _. 4 10. The dimensions are tripled Find the volume of each composite figure. Round to the nearest tenth. p. 700 11. 12. 10-6 Volume of Prisms and Cylinders 701 701 �� PRACTICE AND
PROBLEM SOLVING Find the volume of each prism. 13. 14. Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS 15. a square prism with a base area of 49 ft 2 and a height 2 ft less than the base Skills Practice p. S23 Application Practice p. S37 edge length 16. Landscaping Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle. If he wants to fill it to a depth of 4 in., how many cubic yards of dirt does he need? If dirt costs $25 per yd 3, how much will the project cost? (Hint: 1 yd 3 = 27 ft 3 ) Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. 17. 18. 19. a cylinder with base area 24π cm 2 and height 16 cm Describe the effect of each change on the volume of the given figure. 20. The dimensions are multiplied by 5. 21. The dimensions are multiplied by 3 _. 5 Find the volume of each composite figure. 22. 23. 24. One cup is equal to 14.4375 in 3. If a 1 c cylindrical measuring cup has a radius of 2 in., what is its height? If the radius is 1.5 in., what is its height? 25. Food A cake is a cylinder with a diameter of 10 in. and a height of 3 in. For a party, a coin has been mixed into the batter and baked inside the cake. The person who gets the piece with the coin wins a prize. a. Find the volume of the cake. Round to the nearest tenth. b. Probability Keka gets a piece of cake that is a right rectangular prism with a 3 in. by 1 in. base. What is the probability that the coin is in her piece? Round to the nearest tenth. 702 702 Chapter 10 Spatial Reasoning �� 26. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A cylindrical juice container with a 3 in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw
can be inserted. a. Find the height h of the container to the nearest tenth. b. Find the volume of the container to the nearest tenth. c. How many ounces of juice does the container hold? (Hint: 1 in 3 ≈ 0.55 oz) Math History 27. Find the height of a rectangular prism with length 5 ft, width 9 ft, and volume 495 ft 3. 28. Find the area of the base of a rectangular prism with volume 360 in 3 and height 9 in. 29. Find the volume of a cylinder with surface area 210π m 2 and height 8 m. 30. Find the volume of a rectangular prism with vertices (0, 0, 0), (0, 3, 0), (7, 0, 0), (7, 3, 0), (0, 0, 6), (0, 3, 6), (7, 0, 6), and (7, 3, 6). 31. You can use displacement to find the volume of an irregular object, such as a stone. Suppose the tank shown is filled with water to a depth of 8 in. A stone is placed in the tank so that it is completely covered, causing the water level to rise by 2 in. Find the volume of the stone. �� Archimedes (287–212 B.C.E.) used displacement to find the volume of a gold crown. He discovered that the goldsmith had cheated the king by substituting an equal weight of silver for part of the gold. 32. Food A 1 in. cube of cheese is one serving. How many servings are in a 4 in. by 4 in. by 1 __ 4 in. slice? 33. History In 1919, a cylindrical tank containing molasses burst and flooded the city of Boston, Massachusetts. The tank had a 90 ft diameter and a height of 52 ft. How many gallons of molasses were in the tank? (Hint: 1 gal ≈ 0.134 ft 3 ) 34. Meteorology If 3 in. of rain fall on the property shown, what is the volume in cubic feet? In gallons? The density of water is 8.33 pounds per gallon. What is the weight of the rain in pounds? (Hint: 1 gal ≈ 0.134 ft 3 ) �� 35. Critical Thinking The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a
similar prism. Make a conjecture about the ratio of the surface area of the new prism to its volume. Test your conjecture using a cube with an edge length of 1 and a scale factor of 2. 36. Write About It How can you change the edge length of a cube so that its volume is doubled? 37. Abigail has a cylindrical candle mold with the dimensions shown. If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm, about how many candles can she make after melting the block of wax? 14 31 35 76 10-6 Volume of Prisms and Cylinders 703 703 �� 38. A 96-inch piece of wire was cut into equal segments that were then connected to form the edges of a cube. What is the volume of the cube? 576 in 3 512 in 3 729 in 3 1728 in 3 39. One juice container is a rectangular prism with a height of 9 in. and a 3 in. by 3 in. square base. Another juice container is a cylinder with a radius of 1.75 in. and a height of 9 in. Which best describes the relationship between the two containers? The prism has the greater volume. The cylinder has the greater volume. The volumes are equivalent. The volumes cannot be determined. 40. What is the volume of the three-dimensional object with the dimensions shown in the three views below? �� 160 cm 3 240 cm 3 840 cm 3 1000 cm 3 CHALLENGE AND EXTEND Algebra Find the volume of each three-dimensional figure in terms of x. 41. 42. �� 43. � �� 44. The volume in cubic units of a cylinder is equal to its surface area in square units. Prove that the radius and height must both be greater than 2. SPIRAL REVIEW 45. Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled? (Previous course) 46. Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his
term paper? (Previous course) ABCD is a parallelogram. Find each measure. (Lesson 6-2) 47. m∠ABC 48. BC 49. AB Find the surface area of each figure. Round to the nearest tenth. (Lesson 10-5) 50. a square pyramid with slant height 10 in. and base edge length 8 in. �� 51. a regular pentagonal pyramid with slant height 8 cm and base edge length 6 cm 52. a right cone with slant height 2 ft and a base with circumference of π ft 704 704 Chapter 10 Spatial Reasoning 10-7 Volume of Pyramids and Cones TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of..., pyramids,..., cones,.... Objectives Learn and apply the formula for the volume of a pyramid. Learn and apply the formula for the volume of a cone. Also G.5.A, G.5.B, G.11.D Who uses this? The builders of the Rainforest Pyramid in Galveston, Texas, needed to calculate the volume of the pyramid to plan the climate control system. (See Example 2.) The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube. Volume of a Pyramid The volume of a pyramid with base area B and height h is V = 1 _ 3 Bh. E X A M P L E 1 Finding Volumes of Pyramids Find the volume of each pyramid. A a rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft Bh = 1 _ V = 1 _ (7 · 9) (12) = 252 ft 3 3 3 B the square pyramid The base is a square with a side length of 4 in., and the height is 6 in. Bh = 6) = 32 in 3 3 3 10-7 Volume of Pyramids and Cones 705 705 �� Find the volume of the pyramid. C the trapezoidal pyramid with base ABCD, where and ̶̶ AE ⊥ plane ABC ̶̶
AB ǁ ̶̶ CD Step 1 Find the area of the base9 + 18) 6 2 = 81 m 2 Area of a trapezoid Substitute 9 for b 1, 18 for b 2, and 6 for h. Simplify. Step 2 Use the base area and the height to find the volume. ̶̶ AE is the altitude, so the height Because ̶̶ AE ⊥ plane ABC, Bh is equal to AE81) (10) 3 = 270 m 3 Volume of a pyramid Substitute 81 for B and 10 for h. 1. Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of the base. E X A M P L E 2 Architecture Application The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd 2, 1 story ≈ 10 ft) The base is a square with an area of about 4840 yd 2. The base edge length is √  is about 10 (10) = 100 ft, or about 33 yd. 4840 ≈ 70 yd. The height First find the volume in cubic yards. Bh V = 1 _ 3 = 1 _ ( 70 2 ) (33) = 53,900 yd 3 3 Volume of a regular pyramid Substitute 70 2 for B and 33 for h. Then convert your answer to find the volume in cubic feet. The volume of one cubic yard is (3 ft) (3 ft) (3 ft) = 27 ft 3. Use the conversion factor 27 ft 3 ____ 1 yd 3 to find the volume in cubic feet. 53,900 yd 3 · 27 ft 3 _ 1 yd 3 ≈ 1,455,300 ft 3 2. What if…? What would be the volume of the Rainforest Pyramid if the height were doubled? 706 706 Chapter 10 Spatial Reasoning �� Volume of Cones The volume of a cone with base area B, radius r, and height h is V = 1 _ 3 or V = 1 _ π r 2 h. 3 Bh, E X A M P L E 3 Finding Volumes of Cones Find the volume of each cone. Give your answers both in terms of π and rounded
to the nearest tenth. A a cone with radius 5 cm and height 12 cm 5) 2 (12) 3 = 100π cm 3 ≈ 314.2 cm 3 Volume of a cone Substitute 5 for r and 12 for h. Simplify. B a cone with a base circumference of 21π cm and a height 3 cm less than twice the radius Step 1 Use the circumference to find the radius. 2πr = 21π Substitute 21π for C. r = 10.5 cm Divide both sides by 2π. Step 2 Use the radius to find the height. 2 (10.5) - 3 = 18 cm The height is 3 cm less than twice the radius. Step 3 Use the radius and height to find the volume10.5) 2 (18) 3 Volume of a cone Substitute 10.5 for r and 18 for h. = 661.5π cm 3 ≈ 2078.2 cm 3 Simplify. C Step 1 Use the Pythagorean Theorem to find the height. 7 2 + h 2 = 25 2 h 2 = 576 h = 24 Pythagorean Theorem Subtract 7 2 from both sides. Take the square root of both sides. Step 2 Use the radius and height to find the volume7) 2 (24) 3 Volume of a cone Substitute 7 for r and 24 for h. = 392π ft 3 ≈ 1231.5 ft 3 Simplify. 3. Find the volume of the cone. 10-7 Volume of Pyramids and Cones 707 707 �� E X A M P L E 4 Exploring Effects of Changing Dimensions The length, width, and height of the rectangular pyramid are multiplied by 1 __. 4 Describe the effect on the volume. original dimensions: Bh V = 1 _ 3 = 1 _ (24 · 20) (20) 3 = 3200 ft 3 length, width, and height multiplied by 1 _ : 4 Bh V = 1 _ 3 = 1 _ (6 · 5) (5) 3 = 50 ft 3 Notice that 50 = 1 __ 64 (3200). If the length, width, and height are multiplied by 1 __ 4, the volume is multiplied by ( 1 __ 4 ), or 1 __ 64. 3 4. The radius and height of the cone are doubled. Describe the effect on the volume. E X A M P L E 5 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the
composite figure. Round to the nearest tenth. The volume of the cylinder is V = π r 2 h = π (2) 2 (2) = 8π in 3. The volume of the cone is 2) 2 (3) = 4π in 3. 3 3 The volume of the composite figure is the sum of the volumes. V = 8π + 4π = 12π in 3 ≈ 37.7 in 3 5. Find the volume of the composite figure. THINK AND DISCUSS 1. Explain how the volume of a pyramid is related to the volume of a prism with the same base and height. 2. GET ORGANIZED Copy and complete the graphic organizer. 708 708 Chapter 10 Spatial Reasoning �� 10-7 Exercises Exercises KEYWORD: MG7 10-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The altitude of a pyramid is or oblique)? to the base. (perpendicular, parallel, ̶̶̶̶ Find the volume of each pyramid. Round to the nearest tenth, if necessary. p. 705 2. 3. 4. a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft. 706 5. Geology A crystal is cut into the shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of 3 mm. What is the volume to the nearest cubic millimeter of the crystal. 707 Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. 6. 7. 8. a cone with radius 12 m and height 20 Describe the effect of each change on the volume of the given figure. p. 708 9. The dimensions are tripled. 10. The dimensions are multiplied by Find the volume of each composite figure. Round to the nearest tenth, if necessary. p. 708 11. 12. 10-7 Volume of Pyramids and Cones 709 709 ge07sec10l07003aAB3 mm5.7 mm�� Independent Practice For See
Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the volume of each pyramid. Round to the nearest tenth, if necessary. 13. 14. 15. 15. a regular square pyramid with base edge length 12 ft and slant height 10 ft 16. Carpentry A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards. What is the volume of the attic in cubic feet? In cubic yards? �� Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. 17. 18. 19. a cone with base area 36π ft 2 and a height equal to twice the radius �� Describe the effect of each change on the volume of the given figure. 20. The dimensions are multiplied by 1 _. 3 21. The dimensions are multiplied by 6. Find the volume of each composite figure. Round to the nearest tenth, if necessary. 22. 23. Find the volume of each right cone with the given dimensions. Give your answers in terms of π. 24. radius 3 in. height 7 in. 25. diameter 5 m height 2 m 26. radius 28 ft 27. diameter 24 cm slant height 53 ft slant height 13 cm 710 710 Chapter 10 Spatial Reasoning �� Find the volume of each regular pyramid with the given dimensions. Round to the nearest tenth, if necessary. Number of sides of base Base edge length Height Volume 28. 29. 30. 31. 3 4 5 6 10 ft 15 m 9 in. 8 cm 6 ft 18 m 12 in. 3 cm 32. Find the height of a rectangular pyramid with length 3 m, width 8 m, and volume 112 m 3. 33. Find the base circumference of a cone with height 5 cm and volume 125π cm 3. 34. Find the volume of a cone with slant height 10 ft and height 8 ft. 35. Find the volume of a square pyramid with slant height 17 in. and surface area 800 in 2. 36. Find the surface area of a cone with height 20 yd and volume 1500π yd 3.
37. Find the volume of a triangular pyramid with vertices (0, 0, 0), (5, 0, 0), (0, 3, 0), and (0, 0, 7). 38. /////ERROR ANALYSIS///// Which volume is incorrect? Explain the error. 39. Critical Thinking Write a ratio comparing the volume of the prism to the volume of the composite figure. Explain your answer. 40. Write About It Explain how you would find the volume of a cone, given the radius and the surface area. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice stand sells smoothies in cone-shaped cups that are 8 in. tall. The regular size has a 4 in. diameter. The jumbo size has an 8 in. diameter. a. Find the volume of the regular size to the nearest tenth. b. Find the volume of the jumbo size to the nearest tenth. c. The regular size costs $1.25. What would be a reasonable price for the jumbo size? Explain your reasoning. �� 10-7 Volume of Pyramids and Cones 711 711 �� 42. Find the volume of the cone. 432π cm 3 720π cm 3 1296π cm 3 2160π cm 3 43. A square pyramid has a slant height of 25 m and a lateral area of 350 m 2. Which is closest to the volume? 392 m 3 1176 m 3 404 m 3 1225 m 3 44. A cone has a volume of 18π in 3. Which are possible dimensions of the cone? Diameter 3 in., height 6 in. Diameter 1 in., height 18 in. Diameter 6 in., height 6 in. Diameter 6 in., height 3 in. 45. Gridded Response Find the height in centimeters of a square pyramid with a volume of 243 cm 3 and a base edge length equal to the height. CHALLENGE AND EXTEND Each cone is inscribed in a regular pyramid with a base edge length of 2 ft and a height of 2 ft. Find the volume of each cone. 46. 47. 48. 49. A regular octahedron has 8 faces that are equilateral triangles. Find the volume of a regular octahedron with a side length of 10 cm. 50. A cylinder has a radius
of 5 in. and a height of 3 in. Without calculating the volumes, find the height of a cone with the same base and the same volume as the cylinder. Explain your reasoning. SPIRAL REVIEW Find the unknown numbers. (Previous course) 51. The difference of two numbers is 24. The larger number is 4 less than 3 times the smaller number. 52. Three times the first number plus the second number is 88. The first number times 10 is equal to 4 times the second. 53. The sum of two numbers is 197. The first number is 20 more than 1 __ 2 of the second number. Explain why the triangles are similar, then find each length. (Lesson 7-3) 54. AB 55. PQ Find AB and the coordinates of the midpoint of if necessary. (Lesson 10-3) 56. A (1, 1, 2), B (8, 9, 10) 57. A (-4, -1, 0), B (5, 1, -4) ̶̶ AB. Round to the nearest tenth, 58. A (2, -2, 4), B (-2, 2, -4) 59. A (-3, -1, 2), B (-1, 5, 5) 712 712 Chapter 10 Spatial Reasoning �� Functional Relationships in Formulas Algebra You have studied formulas for several solid figures. Here you will see how a change in one dimension affects the measurements of the other dimensions. See Skills Bank page S63 Example A square prism has a volume of 21 cubic units. Write an equation that describes the base edge length s in terms of the height h. Graph the relationship in a coordinate plane with h on the horizontal axis and s on the vertical axis. What happens to the base edge length as the height increases? First use the volume formula to write an equation. V = Bh 100 = s 2h Volume of a prism Substitute 100 for V and s 2 for B. Then solve for s to get an equation for s in terms of h. s 2 = 100_ h s = √100_ h s = 10_ √  h Divide both sides by h. Take the square root of both sides. √  100 = 10 Graph the equation. First make a table of h- and s-values. Then plot the points and draw a
smooth curve through the points. Notice that the function is not defined for h = 0. h 1 4 9 16 25 s 10 5 ̶ 3 3. 2.5 2 As the height of the prism increases, the base edge length decreases. Try This TAKS Grades 9–11 Obj. 8 1. A right cone has a radius of 10 units. Write an equation that describes the slant height ℓ in terms of the surface area S. Graph the relationship in a coordinate plane with S on the horizontal axis and ℓ on the vertical axis. What happens to the slant height as the surface area increases? 2. A cylinder has a height of 5 units. Write an equation that describes the radius r in terms of the volume V. Graph the relationship in a coordinate plane with V on the horizontal axis and r on the vertical axis. What happens to the radius as the volume increases? On Track for TAKS 713 713 �� 10-8 Spheres TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the volume of a sphere. Learn and apply the formula for the surface area of a sphere. Vocabulary sphere center of a sphere radius of a sphere hemisphere great circle Who uses this? Biologists study the eyes of deep-sea predators such as the giant squid to learn about their behavior. (See Example 2.) A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres. Also G.5.A, G.5.B, G.11.D The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalieri’s Principle. You will prove that the cross sections have equal areas in Exercise 39. V (hemisphere) = V (cylinder) - V (cone = 2_ 3 = 2_ 3 = 2_ 3 π r 2(r) π r 2h π r 3 The height of the hemisphere is equal to the radius. The volume of a sphere with
radius r is twice the volume of the hemisphere, or V = 4__ 3 π r 3. Volume of a Sphere The volume of a sphere with radius r is Finding Volumes of Spheres Find each measurement. Give your answer in terms of π. A the volume of the sphere 9) 3 3 Substitute 9 for r. = 972π cm 3 Simplify. 714 714 Chapter 10 Spatial Reasoning �� Find each measurement. Give your answer in terms of π. B the diameter of a sphere with volume 972π in 3 972π = 4 _ π r 3 3 729 = r 3 r = 9 d = 18 in. Substitute 972π for V. Divide both sides by 4 _ π. 3 Take the cube root of both sides. d = 2r C the volume of the hemisphere 4) 3 = 128π _ = 2 _ 3 3 m 3 Volume of a hemisphere Substitute 4 for r. 1. Find the radius of a sphere with volume 2304π ft Biology Application Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximately a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximately a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball? human eyeball: giant squid eyeball1.25) 3 ≈ 8.18 cm 12.5) 3 ≈ 8181.23 cm 3 3 A giant squid eyeball is about 1000 times as great in volume as a human eyeball. 2. A hummingbird eyeball has a diameter of approximately 0.6 cm. How many times as great is the volume of a human eyeball as the volume of a hummingbird eyeball? In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximately the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r. Br + 1 _ V (sphere) ≈ 1 _ Br + … + Br) 4 _ 3 4π r 2 ≈ nB 3 Br The sphere’s volume is close to the sum of the volumes of the pyramids. Divide both sides by 1 _ 3 πr. If the py
ramids fill the sphere, the total area of the bases is approximately equal to the surface area of the sphere S, so 4π r 2 ≈ S. As the number of pyramids increases, the approximation gets closer to the actual surface area. 10 - 8 Spheres 715 715 �� Surface Area of a Sphere The surface area of a sphere with radius r is S = 4π Finding Surface Area of Spheres Find each measurement. Give your answers in terms of π. A the surface area of a sphere with diameter 10 ft S = 4π r 2 S = 4π (5)2 = 200π ft 2 Substitute 5 for r. B the volume of a sphere with surface area 144π m 2 S = 4π r 2 144π = 4π 6) 3 = 288π m 3 3 Substitute 144π for S. Solve for r. Substitute 6 for r. The volume of the sphere is 288π m 3. C the surface area of a sphere with a great circle that has an area of 4π in 2 π r 2 = 4π Substitute 4π for A in the formula for the area of a circle. Solve for r. r = 2 S = 4π r 2 = 4π (2) 2 = 16π in 2 Substitute 2 for r in the surface area formula. 3. Find the surface area of the sphere. E X A M P L E 4 Exploring Effects of Changing Dimensions The radius of the sphere is tripled. Describe the effect on the volume. original dimensions3) 3 3 = 36π m 3 radius tripled9) 3 3 = 972π m 3 Notice that 972π = 27 (36π). If the radius is tripled, the volume is multiplied by 27. 4. The radius of the sphere above is divided by 3. Describe the effect on the surface area. 716 716 Chapter 10 Spatial Reasoning �� E X A M P L E 5 Finding Surface Areas and Volumes of Composite Figures Find the surface area and volume of the composite figure. Give your answers in terms of π. Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone. S (hemisphere) = 1 _ (4π r 2 ) = 2π (7) 2 = 98π cm 2 2 L (cone) = πr�
�� = π (7) (25) = 175π cm 2 The surface area of the composite figure is 98π + 175π = 273π cm 2. Step 2 Find the volume of the composite figure. First find the height of the cone. h = √  25 2 - 7 2 Pythagorean Theorem = √  576 = 24 cm Simplify. The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the conehemispherecone) = 1 _ π (7) 2 (24) = 392π cm 3 3 3 π (7) 3 = 686π _ 3 3 cm 3 The volume of the composite figure is 686π _ + 392π = 1862π _ cm 3. 3 3 5. Find the surface area and volume of the composite figure. THINK AND DISCUSS 1. Explain how to find the surface area of a sphere when you know the area of a great circle. 2. Compare the volume of the sphere with the volume of the composite figure. 3. GET ORGANIZED Copy and complete the graphic organizer. 10 - 8 Spheres 717 717 �� 10-8 Exercises Exercises KEYWORD: MG7 10-8 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe the endpoints of a radius of a sphere Find each measurement. Give your answers in terms of π. p. 714 2. the volume of the hemisphere 3. the volume of the sphere 4. the radius of a sphere with volume 288π cm. Food Approximately how many times as great is the volume of the grapefruit p. 715 as the volume of the lime? �� Find each measurement. Give your answers in terms of π. p. 716 6. the surface area of the sphere 7. the surface area of the sphere 8. the volume of a sphere with surface area 6724π ft Describe the effect of each change on the given measurement of the figure. p. 716 9. surface area 10. volume The dimensions are doubled. The dimensions are multiplied by Find the surface area and volume of each composite figure. p. 717 11
. 12. 718 718 Chapter 10 Spatial Reasoning �� Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find each measurement. Give your answers in terms of π. 13. the volume of the sphere 14. the volume of the hemisphere 15. the diameter of a sphere with volume 7776π in 3 16. Jewelry The size of a cultured pearl is typically indicated by its diameter in mm. How many times as great is the volume of the 9 mm pearl as the volume of the 6 mm pearl? �� Find each measurement. Give your answers in terms of π. �� 17. the surface area of the sphere 18. the surface area of the sphere 19. the volume of a sphere with surface area 625π m 2 Describe the effect of each change on the given measurement of the figure. 20. surface area The dimensions are multiplied by 1 _. 5 21. volume The dimensions are multiplied by 6. Find the surface area and volume of each composite figure. 22. 23. 24. Find the radius of a hemisphere with a volume of 144π cm 3. 25. Find the circumference of a sphere with a surface area of 60π in 2. 26. Find the volume of a sphere with a circumference of 36π ft. 27. Find the surface area and volume of a sphere centered at (0, 0, 0) that passes through the point (2, 3, 6). 28. Estimation A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter. Estimate the surface area and volume of the bead. 10 - 8 Spheres 719 719 �� Marine Biology In 1934, the bathysphere reached a record depth of 3028 feet. The pressure on the hull was about half a ton per square inch. Sports Find the unknown dimensions of the ball for each sport. Sport Ball Diameter Circumference Surface Area Volume 29. Golf 30. Cricket 31. Tennis 32. Petanque 9 in. 1.68 in. 2.5 in. 74 mm 33. Marine Biology The bathysphere was
an early version of a submarine, invented in the 1930s. The inside diameter of the bathysphere was 54 inches, and the steel used to make the sphere was 1.5 inches thick. It had three 8-inch diameter windows. Estimate the volume of steel used to make the bathysphere. 34. Geography Earth’s radius is approximately 4000 mi. About two-thirds of Earth’s surface is covered by water. Estimate the land area on Earth. Astronomy Use the table for Exercises 35–38. 35. How many times as great is the volume of Jupiter as the volume of Earth? 36. The sum of the volumes of Venus and Mars is about equal to the volume of which planet? 37. Which is greater, the sum of the surface areas of Uranus and Neptune or the surface area of Saturn? 38. How many times as great is the surface area of Mercury as the surface area of Pluto? Planet Diameter (mi) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto 3,032 7,521 7,926 4,222 88,846 74,898 31,763 30,775 1,485 39. Critical Thinking In the figure, the hemisphere and the cylinder both have radius and height r. Prove that the shaded cross sections have equal areas. 40. Write About It Suppose a sphere and a cube have equal surface areas. Using r for the radius of the sphere and s for the side of a cube, write an equation to show the relationship between r and s. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A company sells orange juice in spherical containers that look like oranges. Each container has a surface area of approximately 50.3 in 2. a. What is the volume of the container? Round to the nearest tenth. b. The company decides to increase the radius of the container by 10%. What is the volume of the new container? 720 720 Chapter 10 Spatial Reasoning �� 42. A sphere with radius 8 cm is inscribed in a cube. Find the ratio of the volume of the cube to the volume of the sphere. 2 : 1 _ π 3 2 : 3π 43. What is the surface area of a sphere with volume 10 2 _ π in 3? 3 8π in 2 10 2 _ π in 2 3 16π in 2 32π in 2 44. Which expression represents the volume of the
composite figure formed by a hemisphere with radius r and a cube with side length 2r? 3 π + 82π + 12 CHALLENGE AND EXTEND 45. Food The top of a gumball machine is an 18 in. sphere. The machine holds a maximum of 3300 gumballs, which leaves about 43% of the space in the machine empty. Estimate the diameter of each gumball. 46. The surface area of a sphere can be used to determine its volume. a. Solve the surface area formula of a sphere to get an expression for r in terms of S. b. Substitute your result from part a into the volume formula to find the volume V of a sphere in terms of its surface area S. c. Graph the relationship between volume and surface area with S on the horizontal axis and V on the vertical axis. What shape is the graph? Use the diagram of a sphere inscribed in a cylinder for Exercises 47 and 48. 47. What is the relationship between the volume of the sphere and the volume of the cylinder? 48. What is the relationship between the surface area of the sphere and the lateral area of the cylinder? SPIRAL REVIEW Write an equation that describes the functional relationship for each set of ordered pairs. (Previous course)   (0, 1), (1, 2), (-1, 2), (2, 5), (-2, 5) 49. ⎬ ⎨     (-1, 9), (0, 10), (1, 11), (2, 12), (3, 13) 50. ⎬ ⎨   Find the shaded area. Round to the nearest tenth, if necessary. (Lesson 9-3) 51. 52. Describe the effect on the volume that results from the given change. (Lesson 10-6) 53. The side lengths of a cube are multiplied by 3 __ 4. 54. The height and the base area of a prism are multiplied by 5. 10 - 8 Spheres 721 721 �� 10-8 Use with Lesson 10-8 Activity 1 Compare Surface Areas and Volumes In some situations you may need to find the minimum surface area for a given volume. In others you may need to find the maximum volume for a given surface area. Spreadsheet software can help you analyze these
problems. TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed …. 1 Create a spreadsheet to compare surface areas and volumes of rectangular prisms. Create columns for length L, width W, height H, surface area SA, volume V, and ratio of surface area to volume SA/V. In the column for SA, use the formula shown. 2 Create a formula for the V column and a formula for the SA/V column. 3 Fill in the measurements L = 8, W = 2, and H = 4 for the first rectangular prism. 4 Choose several values for L, W, and H to create rectangular prisms that each have the same volume as the first one. Which has the least surface area? Sketch the prism and describe its shape in words. (Is it tall or short, skinny or wide, flat or cubical?) Make a conjecture about what type of shape has the minimum surface area for a given volume. 722 722 Chapter 10 Spatial Reasoning �� Try This 1. Repeat Activity 1 for cylinders. Create columns for radius R, height H, surface area SA, volume V, and ratio of surface area to volume SA/V. What shape cylinder has the minimum surface area for a given volume? (Hint: To use π in a formula, input “PI( )” into your spreadsheet.) 2. Investigate packages such as cereal boxes and soda cans. Do the manufacturers appear to be using shapes with the minimum surface areas for their volume? What other factors might influence a company’s choice of packaging? Activity 2 1 Create a new spreadsheet with the same column headings used in Activity 1. Fill in the measurements L = 8, W = 2, and H = 4 for the first rectangular prism. To create a new prism with the same surface area, choose new values for L and W, and use the formula shown to calculate H. 2 Choose several more values for L and W, and calculate H so that SA = 112. Examine the V and SA/V columns. Which prism has the greatest volume? Sketch the prism and describe it in words. Make a conjecture about what type of shape has the maximum volume for a given surface area. Try This 3. Repeat Activity 2 for cylinders. Create columns for radius R, height H, surface area SA, volume V, and the ratio of surface area to volume SA/
V. What shape cylinder has the maximum volume for a given surface area? 4. Solve the formula SA = 2LW + 2LH + 2WH for H. Use your result to explain the formula that was used to find H in Activity 2. 5. If a rectangular prism, a pyramid, a cylinder, a cone, and a sphere all had the same volume, which do you think would have the least surface area? Which would have the greatest surface area? Explain. 6. Use a spreadsheet to analyze what happens to the ratio of surface area to volume of a rectangular prism when the dimensions are doubled. Explain how you set up the spreadsheet and describe your results. 10-8 Technology Lab 723 723 �� SECTION 10B Surface Area and Volume Juice for Fun You are in charge of designing containers for a new brand of juice. Your company wants you to compare several different container shapes. The container must be able to hold a 6-inch straw so that exactly 1 inch remains outside the container when the straw is inserted as far as possible. 1. One possible container is a cylinder with a base diameter of 4 in., as shown. How much material is needed to make this container? Round to the nearest tenth. 2. Estimate the volume of juice in ounces that the cylinder will hold. Round to the nearest tenth. (Hint: 1 in 3 ≈ 0.55 oz) 3. Another option is a square prism with a 3 in. by 3 in. base, as shown. How much material is needed to make this container? 4. Estimate the volume of juice in ounces that the prism will hold. (Hint: 1 in 3 ≈ 0.55 oz) 5. Which container would you recommend to your company? Justify your answer. �� 724 724 Chapter 10 Spatial Reasoning SECTION 10B Quiz for Lessons 10-4 Through 10-8 10-4 Surface Area of Prisms and Cylinders Find the surface area of each figure. Round to the nearest tenth, if necessary. 1. 2. 3. 4. The dimensions of a 12 mm by 8 mm by 24 mm right rectangular prism are multiplied by 3 __ 4. Describe the effect on the surface area. 10-5 Surface Area of Pyramids and Cones Find the surface area of each figure. Round to the nearest tenth, if necessary. 5. a regular pentagonal pyramid with base edge length 18 yd and sl
ant height 20 yd 6. a right cone with diameter 30 in. and height 8 in. 7. the composite figure formed by two cones 10-6 Volume of Prisms and Cylinders Find the volume of each figure. Round to the nearest tenth, if necessary. 8. a regular hexagonal prism with base area 23 in 2 and height 9 in. 9. a cylinder with radius 8 yd and height 14 yd 10. A brick patio measures 10 ft by 12 ft by 4 in. Find the volume of the bricks. If the density of brick is 130 pounds per cubic foot, what is the weight of the patio in pounds? 11. The dimensions of a cylinder with diameter 2 ft and height 1 ft are doubled. Describe the effect on the volume. 10-7 Volume of Pyramids and Cones Find the volume of each figure. Round to the nearest tenth, if necessary. 12. 13. 14. 10-8 Spheres Find the surface area and volume of each figure. 15. a sphere with diameter 20 in. 16. a hemisphere with radius 12 in. 17. A baseball has a diameter of approximately 3 in., and a softball has a diameter of approximately 5 in. About how many times as great is the volume of a softball as the volume of a baseball? Ready to Go On? 725 725 �� EXTENSION Spherical Geometry EXTENSION TEKS G.1.C Geometric structure: compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. Objective Understand spherical geometry as an example of non-Euclidean geometry. Vocabulary non-Euclidean geometry spherical geometry Euclidean geometry is based on figures in a plane. Non-Euclidean geometry is based on figures in a curved surface. In a non-Euclidean geometry system, the Parallel Postulate is not true. One type of non-Euclidean geometry is spherical geometry, which is the study of figures on the surface of a sphere. A line in Euclidean geometry is the shortest path between two points. On a sphere, the shortest path between two points is along a great circle, so “lines” in spherical geometry are defined as great circles. In spherical geometry, there are no parallel lines. Any two lines intersect at two points. Pil
ots usually fly along great circles because a great circle is the shortest route between two points on Earth. Spherical Geometry Parallel Postulate Through a point not on a line, there is no line parallel to a given line. E X A M P L E 1 Classifying Figures in Spherical Geometry Name a line, a segment, and a triangle on the sphere. The two points used to name a line cannot be exactly opposite each other on the sphere. In Example 1,   AB could refer to more than one line.  AC is a line. ̶̶ AC is a segment. △ACD is a triangle. 1. Name another line, segment, and triangle on the sphere above. In Example 1, the lines   AC and   AD are both perpendicular to   CD. This means that △ACD has two right angles. So the sum of its angle measures must be greater than 180°. Imagine cutting an orange in half and then cutting each half in quarters using two perpendicular cuts. Each of the resulting triangles has three right angles. Spherical Triangle Sum Theorem The sum of the angle measures of a spherical triangle is greater than 180°. 726 726 Chapter 10 Spatial Reasoning �� E X A M P L E 2 Classifying Spherical Triangles Classify each spherical triangle by its angle measures and by its side lengths. A △ABC △ABC is an obtuse scalene triangle. B △NPQ on Earth has vertex N at the North Pole and vertices P and Q on the equator. PQ is equal to 1 __ the circumference of Earth. 3 NP and NQ are both equal to 1 __ 4 the circumference of Earth. The equator is perpendicular to both of the other two sides of the triangle. Thus △NPQ is an isosceles right triangle. 2. Classify △VWX by its angle measures and by its side lengths. The area of a spherical triangle is part of the surface area of the sphere. For the piece of orange on page 726, the area is 1 __ 8 of the surface area of the orange, or 1 __ 8 (4π r 2 ) = π r 2 ___ 2. If you know the radius of a sphere and the measure of each angle, you can
find the area of the triangle. Area of a Spherical Triangle The area of spherical △ABC on a sphere with radius r is A = π r 2 _ 180° (m∠A + m∠B + m∠C - 180°). E X A M P L E 3 Finding the Area of Spherical Triangles Find the area of each spherical triangle. Round to the nearest tenth, if necessary. A △ABC (m∠A + m∠B + m∠C - 180°) (100 + 106 + 114 - 180) ≈ 152.4 cm 2 B △DEF on Earth’s surface with m∠D = 75°, m∠E = 80°, and m∠F = 30°. (Hint: average radius of Earth = 3959 miles) (m∠D + m∠E + m∠F - 180°) (75 + 80 + 30 - 180) ≈ 1,367,786.7 mi 2 A = π r 2 _ 180° π (14) 2 _ 180° A = A = π r 2 _ 180° π (3959) 2 _ 180° = 3. Find the area of △KLM on a sphere with diameter 20 ft, where m∠K = 90°, m∠L = 90°, and m∠M = 30°. Round to the nearest tenth. Chapter 10 Extension 727 727 �� EXTENSION Exercises Exercises Use the figure for Exercises 1–3. 1. Name all lines on the sphere. 2. Name three segments on the sphere. 3. Name a triangle on the sphere. Determine whether each figure is a line in spherical geometry. 4. m 5. n 6. p Classify each spherical triangle by its angle measures and by its side lengths. 7. 9. Find the area of each spherical triangle. 11. 13. 8. 10. 12. 14. 15. △ABC on the Moon’s surface with m∠A = 35°, m∠B = 48°, and m∠C = 100° (Hint: average radius of the Moon ≈ 1079 miles) 16. △RST on a scale model of Earth with radius 6 m, m∠R = 80°, m∠S = 130°,
and m∠T = 150° 728 728 Chapter 10 Spatial Reasoning �� 17. △ABC is an acute triangle. a. Write an inequality for the sum of the angle measures of △ABC, based on the fact that △ABC is acute. b. Use your result from part a to write an inequality for the area of △ABC. c. Use your result from part b to compare the area of an acute spherical triangle to the total surface area of the sphere. 18. Draw a quadrilateral on a sphere. Include one diagonal in your drawing. Use the sum of the angle measures of the quadrilateral to write an inequality. Geography Compare each length to the length of a great circle on Earth. 19. the distance between the North 20. the distance between the North Pole and the South Pole Pole and any point on the equator 21. Geography If the area of a triangle on Earth’s surface is 100,000 mi 2, what is the sum of its angle measures? (Hint: average radius of Earth ≈ 3959 miles) 22. Sports Describe the curves on the basketball that are lines in spherical geometry. 23. Navigation Pilots navigating long distances often travel along the lines of spherical geometry. Using a globe and string, determine the shortest route for a plane traveling from Washington, D.C., to London, England. What do you notice? 24. Write About It Can a spherical triangle be right and obtuse at the same time? Explain. 25. Write About It A 2-gon is a polygon with two edges. Draw two lines on a sphere. How many 2-gons are formed? What can you say about the positions of the vertices of the 2-gons on the sphere? 26. Challenge Another type of non-Euclidean geometry, called hyperbolic geometry, is defined on a surface that is curved like the bell of a trumpet. What do you think is true about the sum of the angle measures of the triangle shown at right? Compare the sum of the angle measures of a triangle in Euclidean, spherical, and hyperbolic geometry. Chapter 10 Extension 729 729 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary altitude................
.... 680 isometric drawing.......... 662 right cone.................. 690 altitude of a cone........... 690 lateral edge................. 680 right cylinder............... 681 altitude of a pyramid........ 689 lateral face................. 680 right prism................. 680 axis of a cone............... 690 lateral surface.............. 681 slant height of a axis of a cylinder............ 681 net........................ 655 center of a sphere........... 714 oblique cone............... 690 cone....................... 654 oblique cylinder............ 681 cross section............... 656 oblique prism.............. 680 cube....................... 654 orthographic drawing....... 661 cylinder.................... 654 perspective drawing........ 662 edge....................... 654 polyhedron................. 670 face........................ 654 prism......
................ 654 great circle................. 714 pyramid................... 654 hemisphere................ 714 radius of a sphere........... 714 horizon.................... 662 regular pyramid............ 689 regular pyramid.......... 689 slant height of a right cone............... 690 space...................... 671 sphere..................... 714 surface area................ 680 vanishing point............. 662 vertex...................... 654 vertex of a cone............. 690 vertex of a pyramid......... 689 volume.................... 697 Complete the sentences below with vocabulary words from the list above. 1. A(n)? has at least one nonrectangular lateral face. ̶̶̶̶ 2. A name given to the intersection of a three-dimensional figure and a plane is?. ̶̶̶̶ 10-1 Solid Geometry (pp. 654–660) TEKS G.2.B, G.6.A, G.6.B, G.9.D E X A M P L E S ■ Classify the figure. Name the vertices, edges, and bases. pentagonal prism vertices: A, B, C, D, E, F, EXERCISES Classify each figure. Name the vertices, edges, and bases. 3. 4. G, H, J,
K ̶̶ AB, ̶̶ EK, edges: ̶̶ ̶̶ AF, KF, ̶̶ BC, ̶̶ DJ, ̶̶ CD, ̶̶ CH, ̶̶ DE, ̶̶ BG ̶̶ AE, ̶̶ FG, ̶̶̶ GH, ̶̶ HJ, ̶̶ JK, bases: ABCDE, FGHJK ■ Describe the three-dimensional figure that can be made from the given net. The net forms a rectangular prism. 730 730 Chapter 10 Spatial Reasoning Describe the three-dimensional figure that can be made from the given net. 5. 6. �� 10-2 Representations of Three-Dimensional Figures (pp. 661–668) TEKS G.6.C, G.9.D E X A M P L E S ■ Draw all six orthographic views of the given object. Assume there are no hidden cubes. EXERCISES Use the figure made of unit cubes for Exercises 7–10. Assume there are no hidden cubes. 7. Draw all six orthographic views. Top: Bottom: 8. Draw an isometric view. Front: Back: Left: Right: ■ Draw an isometric view of the given object. Assume there are no hidden cubes. 9. Draw the object in one-point perspective. 10. Draw the object in two-point perspective. Determine whether each drawing represents the given object. Assume there are no hidden cubes. 11. 12. 10-3 Formulas in Three Dimensions (pp. 670–677) TEKS G.5.A, G.7.C, G.8.C, G.9.D E X A M P L E S ■ Find the number of vertices, edges, and faces of the given polyhedron. Use your results to verify Euler’s formula. V = 12, E = 18, F = 8 12 - 18 + 8 = 2 ■ Find the distance between the points (6, 3, 4) and (2, 7, 9). Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. distance: d = √ �
� (2 - 6) 2 + (7 - 3) 2 + (9 - 4) 2 = √  57 ≈ 7.5 midpoint4, 5, 6.5) EXERCISES Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 13. 14. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 15. (2, 6, 4) and (7, 1, 1) 16. (0, 3, 0) and (5, 7, 8) 17. (7, 2, 6) and (9, 1, 5) 18. (6, 2, 8) and (2, 7, 4) Study Guide: Review 731 731 10-4 Surface Area of Prisms and Cylinders (pp. 680–687) E X A M P L E S EXERCISES TEKS G.5.A, G.5.B, G.6.B, G.8.D, G.11.D Find the lateral area and surface area of each right prism or cylinder. ■ Find the lateral area and surface area of each right prism or cylinder. Round to the nearest tenth, if necessary. 19. L = Ph = 28 (10) = 280 in 2 S = Ph + 2B = 280 + 2 (49) = 378 in 2 20. a cube with side length 5 ft ■ a cylinder with radius 8 m and height 12 m L = 2πrh = 2π (8)(12) = 192π ≈ 603.2 m 2 S = L + 2B = 192π + 2π (8)2 = 320π ≈ 1005.3 m 2 21. an equilateral triangular prism with height 7 m and base edge lengths 6 m 22. a regular pentagonal prism with height 8 cm and base edge length 4 cm 10-5 Surface Area of Pyramids and Cones (pp. 689–696) E X A M P L E S EXERCISES TEKS G.5.A, G.5.B, G.6.B, G.8.D, G.11.D Find the lateral area and surface area of each right pyramid or cone. ■ Find the lateral area and surface area of each right pyramid or
cone. 23. a square pyramid with side length 15 ft and slant height 21 ft 24. a cone with radius 7 m and height 24 m The radius is 8 m, so the slant height is 25. a cone with diameter 20 in. and slant height 15 in. √ 8 2 + 15 2 = 17 m. L = πrℓ = π (8)(17) = 136π m 2 S = πrℓ + π r 2 = 136π + (8)2π = 200π m 2 ■ a regular hexagonal pyramid with base edge length 8 in. and slant height 20 in. L = 1 _ Pℓ = 1 _ (48) (20) = 480 in 2 2 2 S = L + B = 480 + 1 _ (4 √  3 ) (48) ≈ 646.3 in 2 2 Find the surface area of each composite figure. 26. 27. 10-6 Volume of Prisms and Cylinders (pp. 697–704) E X A M P L E S ■ Find the volume of the prism. 2 V = Bh = ( 1 _ aP) h = 1 _ (4 √  3 ) (48) (12) 2 = 1152 √  3 ≈ 1995.3 cm 3 732 732 Chapter 10 Spatial Reasoning EXERCISES Find the volume of each prism. 28. 29. TEKS G.1.B, G.5.A, G.5.B, G.8.D, G.11.D �� ■ Find the volume of the cylinder6) 2 (14) = 504π ≈ 1583.4 ft 3 Find the volume of each cylinder. 30. 31. 10-7 Volume of Pyramids and Cones (pp. 705–712) TEKS G.5.A, G.5.B, G.8.D, G.11.D E X A M P L E S ■ Find the volume of the pyramid. Bh = 1 _ V = 1 _ (8 · 3) (14) 3 3 = 112 in 3 ■ Find the volume of the cone9) 2 (16) 3 3 = 432π
ft 3 ≈ 1357.2 ft 3 EXERCISES Find the volume of each pyramid or cone. 32. a hexagonal pyramid with base area 42 m 2 and height 8 m 33. an equilateral triangular pyramid with base edge 3 cm and height 8 cm 34. a cone with diameter 12 cm and height 10 cm 35. a cone with base area 16π ft 2 and height 9 ft Find the volume of each composite figure. 36. 37. 10-8 Spheres (pp. 714–721) TEKS G.5.A, G.5.B, G.8.D, G.11.D E X A M P L E EXERCISES ■ Find the volume and surface area of the sphere. Give your answers in terms of π9) 3 = 972π m 2 3 3 S = 4π r 2 = 4π (9) 2 = 324π m 2 Find each measurement. Give your answers in terms of π. 38. the volume of a sphere with surface area 100π m 2 39. the surface area of a sphere with volume 288π in 3 40. the diameter of a sphere with surface area 256π ft 2 Find the surface area and volume of each composite figure. 41. 42. Study Guide: Review 733 733 �� Use the diagram for Items 1–3. 1. Classify the figure. Name the vertices, edges, and bases. 2. Describe a cross section made by a plane parallel to the base. 3. Find the number of vertices, edges, and faces of the polyhedron. Use your results to verify Euler’s formula. Use the figure made of unit cubes for Items 4–6. Assume there are no hidden cubes. 4. Draw all six orthographic views. 5. Draw an isometric view. 6. Draw the object in one-point perspective. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 7. (0, 0, 0) and (5, 5, 5) 8. (6, 0, 9) and (7, 1, 4) 9. (-1, 4, 3) and (2, -5, 7) Find the surface area of each figure. Round to
the nearest tenth, if necessary. 10. 13. 11. 14. 12. 15. Find the volume of each figure. Round to the nearest tenth, if necessary. 16. 19. 17. 20. 18. 21. 22. Earth’s diameter is approximately 7930 miles. The Moon’s diameter is approximately 2160 miles. About how many times as great is the volume of Earth as the volume of the Moon? 734 734 Chapter 10 Spatial Reasoning �� FOCUS ON SAT MATHEMATICS SUBJECT TEST SAT Mathematics Subject Test results include scaled scores and percentiles. Your scaled score is a number from 200 to 800, calculated using a formula that varies from year to year. The percentile indicates the percentage of people who took the same test and scored lower than you did. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. The questions are written so that you should not need to do any lengthy calculations. If you find yourself getting involved in a long calculation, think again about all of the information in the problem to see if you might have missed something helpful. 1. A line intersects a cube at two points, A and B. If each edge of the cube is 4 cm, what is the greatest possible distance between A and B? 4. If triangle ABC is rotated about the x-axis, what is the volume of the resulting cone? (A) 2 √  3 cm (B) 4 cm (C) 4 √  2 cm (D) 4 √  3 cm (E) 16 √  3 cm 2. The lateral area of a right cylinder is 3 times the area of its base. What is the height h of the cylinder in terms of its radius r? (A) 1 _ r 2 (B) 2 _ r 3 (C) 3 _ r 2 (D) 3r (E) 3 r 2 3. What is the lateral area of a right cone with radius 6 ft and height 8 ft? (A) 30π ft 2 (B) 48π ft 2 (C) 60π ft 2 (D) 180π ft 2 (E) 360π ft 2 (A) 100π cubic units (B) 144π cubic units (
C) 240π cubic units (D) 300π cubic units (E) 720π cubic units in 3 5. An oxygen tank is the shape of a cylinder with a hemisphere at each end. If the radius of the tank is 5 inches and the overall length is 32 inches, what is the volume of the tank? (A) 500 _ 3π (B) 2275 _ 12 (C) 1900 _ 3 (D) 2150 _ 3 (E) 2900 _ 3 π in 3 π in 3 π in 3 π in 3 College Entrance Exam Practice 735 735 �� Any Question Type: Measure to Solve Problems On some tests, you may have to measure a figure in order to answer a question. Pay close attention to the units of measure asked for in the question. Some questions ask you to measure to the nearest centimeter, and some ask you to measure to the nearest inch. Multiple Choice: The net of a square pyramid is shown below. Use a ruler to measure the dimensions of the pyramid to the nearest centimeter. Which of the following best represents the total surface area of the square pyramid? 9 square centimeters 21 square centimeters 33 square centimeters 36 square centimeters Use a centimeter ruler to measure one side of the square base. The measurement to the nearest centimeter is 3 cm. The base is a square, so all four side lengths are 3 cm. Measure the altitude of a triangular face, which is the slant height of the pyramid. The altitude is 2 cm. Label the drawing with the measurements. To find the total surface area of the pyramid, find the base area and the lateral area. The base of the pyramid is a square. The base area of the pyramid is A = s 2 = (3) 2 = 9 cm 2. The area of one triangular face is A = 1 _ 2 bh = 1 _ (3) (2) = 3 cm 2. 2 The pyramid has 4 faces, so the lateral area is 4 (3) = 12 cm 2. The total surface area is 9 + 12 = 21 cm 2. The correct answer choice is B. 736 736 Chapter 10 Spatial Reasoning �� Read each test item and answer the questions that follow. �� Measure carefully and make sure you are using the correct units to
measure the figure. Item A The net of a cube is shown below. Use a ruler to measure the dimensions of the cube to the nearest 1 __ Which best represents the volume of the cube to the nearest cubic inch? inch. 4 1 cubic inch 2 cubic inches 5 cubic inches 9 cubic inches 1. Measure one edge of the net for the cube. What is the length to the nearest 1__ 4 inch? 2. How would you use the measurement to find the volume of the cube? Item B The net of a cylinder is shown below. Use a ruler to measure the dimensions of the cylinder to the nearest tenth of a centimeter. Which best represents the total surface area of the cylinder to the nearest square centimeter? 6 square centimeters 16 square centimeters 19 square centimeters 42 square centimeters 3. Which part of the net do you need to measure in order to find the height of the cylinder? Find the height of the cylinder to the nearest tenth of a centimeter. 4. What other measurement(s) do you need in order to find the surface area of the cylinder? Find the measurement(s) to the nearest tenth of a centimeter. 5. How would you use the measurements to find the surface area of the cylinder? TAKS Tackler 737 737 KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–10 Multiple Choice 1. If a point (x, y) is chosen at random in the coordinate plane such that -1 ≤ x ≤ 1 and -5 ≤ y ≤ 3, what is the probability that x ≥ 0 and y ≥ 0? 0.1875 0.25 0.375 0.8125 2. △ABC ∼ △DEF, and △DEF ∼ △GHI. If the similarity ratio of △ABC to △DEF is 1 __ and the 2 similarity ratio of △DEF to △GHI is 3 __, what is the 4 similarity ratio of △ABC to △GHI. Which expression represents the number of faces of a prism with bases that are n-gons? n + 1 n + 2 2n 3n 4. Parallelogram ABCD has a diagonal ̶̶ AC with endpoints A (-1, 3) and C (-3, -3). If B has coordinates (x, y), which of the following represents the coordinates for D? D (-3x, -y) D (-x, -y) D (-
x - 4, -y) D (x - 2, y) 5. Right △ABC with legs AB = 9 millimeters and BC = 12 millimeters is the base of a right prism that has a surface area of 450 square millimeters. What is the height of the prism? 4.75 millimeters 9.5 millimeters 6 millimeters 11 millimeters 6. The radius of a sphere is doubled. What happens to the ratio of the volume of the sphere to the surface area of the sphere? It remains the same. It is doubled. It is increased by a factor of 4. It is increased by a factor of 8. 738 738 Chapter 10 Spatial Reasoning 7. ̶̶ AB has endpoints A (x, y, z) and B (-2, 6, 13) and midpoint M (2, -6, 3). What are the coordinates of A? A (-6, 18, 23) A (0, 0, 8) A (2, -6, 19) A (6, -18, -7) 8. If ̶̶ DE bisects ∠CEF, which of the following additional statements would allow you to conclude that △DEF ≅ △ABC? ∠DEF ≅ ∠BAC ∠DEF ≅ ∠CDE ̶̶ ̶̶ CD EF ≅ ̶̶ ̶̶ EC EF ≅ 9. To the nearest tenth of a cubic centimeter, what is the volume of a right regular octagonal prism with base edge length 4 centimeters and height 7 centimeters? 180.3 cubic centimeters 224.0 cubic centimeters 270.4 cubic centimeters 540.8 cubic centimeters 10. Which of the following must be true about a conditional statement? If the inverse is false, then the converse is false. If the conditional is true, then the contrapositive is false. If the conditional is true, then the converse is false. If the hypothesis of the conditional is true, then the conditional is true. �� It may be helpful to include units in your calculations of measures of geometric figures. If your answer includes the proper units, you are less likely to have made an error. 11. A right cylinder has a height of 10 inches. The area of the base is 63.6 square inches. To the nearest tenth of a square inch, what is the lateral area for this cylinder? 53.6 square inches 282.7 square inches 409
.9 square inches 634.6 square inches 12. The volume of the smaller sphere is 288 cubic centimeters. Find the volume of the larger sphere. 864 cubic centimeters 2,592 cubic centimeters 7,776 cubic centimeters 23,328 cubic centimeters Gridded Response 13.  u = 〈3, -7〉, and  v = 〈-6, 5〉. What is the magnitude of the resultant vector to the nearest tenth of  u and  v? STANDARDIZED TEST PREP Short Response 17. The area of trapezoid GHIJ is 103.5 square centimeters. Find each of the following. Round answers to the nearest tenth. Show your work or explain in words how you found your answers. a. the height of trapezoid GHIJ b. m∠J 18. The figure shows the top view of a stack of cubes. The number on each cube represents the number of stacked cubes. Draw the bottom, back, and right views of the object. 19. △ABC has vertices A (1, -2), B (-2, -3), and C (-2, 2). a. Graph △A'B'C', the image of △ABC, after a dilation with a scale factor of 3 __. 2 b. Show that   AB ǁ   A'B',   BC ǁ   B'C', and   CA ǁ   C'A'. Use slope to justify your answer. 14. If a polyhedron has 12 vertices and 8 faces, how many edges does the polyhedron have? Extended Response 15. If Y is the circumcenter of △PQR, what is the value of x? 20. A right cone has a lateral area of 30π square inches and a slant height of 6 inches. 16. How many cubes with edge length 3 centimeters will fit in a box that is a rectangular prism with length 12 centimeters, width 15 centimeters, and height 24 centimeters? a. Find the height of the cone. Show your work or explain in words how you determined your answer. Round
your answer to the nearest tenth. b. Find the volume of this cone. Round your answer to the nearest tenth. c. Given a right cone with a lateral area of L and a slant height of ℓ, find an equation for the volume in terms of L and ℓ. Show your work. Cumulative Assessment, Chapters 1–10 739 739 �� T E X A S TAKS Grades 9–11 Obj. 10 Reliant Stadium When Houston’s Reliant Stadium opened in 2002, it was the first NFL stadium to have a retractable roof. In addition to football games, Reliant Stadium hosts rodeos, concerts, and other events. When configured for football, up to 72,000 fans can sit around 97,000 square feet of playing field. Choose one or more strategies to solve each problem. 1. Inside the playing field, the football field is 160 ft by 360 ft. Approximately how many acres of land surround the football field? (Hint: 1 acre = 43,560 ft 2 ) �� For 2 and 3, use the table. 2. There are two scoreboards in Reliant Stadium. Suppose the ratio of each scoreboard’s length to its width were 46 : 7. What would the dimensions of each scoreboard be? 3. Each scoreboard is equipped with a large video screen. A screen’s width is 72.5 ft greater than its height. What are the dimensions of the video screens on the scoreboards? Approximate Scoreboard Measurements Perimeter (ft) Scoreboard Video Screen 636 241 Area ( ft 2 ) 11,592 2,316 4. Reliant Stadium can be divided into 12 seating sections of equal size. When the roof is opened at a certain time of day, 4 of these sections are in the sunlight. Suppose you choose a seat in the arena at random when the roof is retracted. What is the probability that you are sitting in sunlight? What is the probability that you are sitting in the shade? 740 740 Chapter 10 Spatial Reasoning �� Texas Coins Although Texas does not have its own mint, Texas has a rich history in coins. Starting in the mid-1800s, many Texas businesses issued tokens that were redeemable for merchandise in place of U.S. money. In 1935, the U.S. Mint issued a coin to celebrate explorer Al
var Nunez Cabeza de Vaca’s travels across southern America and Texas. In 2004, the U.S. Mint issued the Texas state quarter. Choose one or more strategies and use the table to solve each problem. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List 1. Quarters are stamped out of a rectangular metal strip that is 13 in. wide by 1500 ft long. Given that the diameter of a quarter is just under an inch (0.955 in.), what is the minimum number of strips needed for 700,000 Texas state quarters? 2. A Spanish Trail memorial half dollar contains a small amount of copper, but most of the metal in the coin is silver. The volume of copper in a Spanish Trail memorial half dollar is about 6.58 mm 3. What percent of the half dollar is copper? 3. Many Texas trade tokens were made from aluminum. About how many tokens could be made from a block of aluminum with a volume of 1 m 3? Texas State Quarter Spanish Trail Memorial Half Dollar Texas Trade Token Coin Specifications Diameter (mm) Thickness (mm) 24.26 1.75 30.60 2.15 44.00 1.42 4. 4. The regular octagonal token shown was issued by a 4. barber shop in Fort Worth. If the distance from the midpoint of one side to the midpoint of the opposite side is 25 mm, what is the area of the face of the coin? Problem Solving on Location 741 741 Circles 11A Lines and Arcs in Circles 11-1 Lines That Intersect Circles 11-2 Arcs and Chords 11-3 Sector Area and Arc Length 11B Angles and Segments in Circles 11-4 Inscribed Angles Lab Explore Angle Relationships in Circles 11-5 Angle Relationships in Circles Lab Explore Segment Relationships in Circles 11-6 Segment Relationships in Circles 11-7 Circles in the Coordinate Plane Ext Polar Coordinates KEYWORD: MG7 ChProj On the floor of the Capitol rotunda are six seals, commemorating the flags that have flown over Texas. 742 742 Chapter 11 Vocabulary Match each term on the left with a definition on the right. A. the distance around a circle 1. radius 2. pi
3. circle 4. circumference B. the locus of points in a plane that are a fixed distance from a given point C. a segment with one endpoint on a circle and one endpoint at the center of the circle D. the point at the center of a circle E. the ratio of a circle’s circumference to its diameter Tables and Charts The table shows the number of students in each grade level at Middletown High School. Find each of the following. 5. the percentage of students who are freshman 6. the percentage of students who are juniors 7. the percentage of students who are sophomores or juniors Year Freshman Sophomore Junior Senior Number of Students 192 208 216 184 Circle Graphs The circle graph shows the age distribution of residents of Mesa, Arizona, according to the 2000 census. The population of the city is 400,000. 8. How many residents are between the ages of 18 and 24? 9. How many residents are under the age of 18? 10. What percentage of the residents are over the age of 45? 11. How many residents are over the age of 45? Solve Equations with Variables on Both Sides Solve each equation. 12. 11y - 8 = 8y + 1 14. z + 30 = 10z - 15 16. -2x - 16 = x + 6 13. 12x + 32 = 10 + x 15. 4y + 18 = 10y + 15 17. -2x - 11 = -3x - 1 Solve Quadratic Equations Solve each equation. 18. 17 = x 2 - 32 20. 4 x 2 + 12 = 7 x 2 19. 2 + y 2 = 18 21. 188 - 6 x 2 = 38 Circles 743 743 �� Key Vocabulary/Vocabulario arc arco arc length longitud de arco central angle ángulo central chord secant cuerda secante sector of a circle sector de un círculo segment of a circle segmento de un círculo semicircle semicírculo tangent of a circle tangente de un círculo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, answer the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word semicircle begins with
the prefix semi-. List some other words that begin with semi-. What do all of these words have in common? 2. The word central means “located at the center.” How can you use this definition to understand the term central angle of a circle? 3. The word tangent comes from the Latin word tangere, which means “to touch.” What does this tell you about a line that is a tangent to a circle? Geometry TEKS Les. 11-1 Les. 11-2 Les. 11-3 Les. 11-4 G.1.A Geometric structure* develop an awareness of the ★ ★ ★ ★ 11-5 Tech. Lab 11-6 Tech. Lab Les. 11-5 ★ Les. 11-6 Les. 11-7 Ext ★ ★ ★ structure of a mathematical system,... G.1.B Geometric structure* recognize the historical development of geometric systems... ★ G.2.A Geometric structure* use constructions to explore ★ ★ ★ ★ ★ attributes of geometric figures and to make conjectures... G.2.B Geometric structure* make conjectures about... circles,... and determine the validity of the conjectures,... ★ ★ ★ ★ ★ ★ ★ ★ G.3.B Geometric structure* construct and justify statements ★ about geometric figures... G.5.A Geometric patterns* use... patterns to develop algebraic ★ ★ ★ ★ ★ expressions representing geometric properties G.5.B Geometric patterns* use numeric and geometric ★ ★ ★ patterns to make generalizations about geometric properties, including... angle relationships in... circles G.8.B Congruence and the geometry of size* find areas of ★ sectors and arc lengths of circles using proportional reasoning G.8.C Congruence and the geometry of size* … use the ★ Pythagorean Theorem G.9.C Congruence and the geometry of size* formulate and test conjectures about the properties and attributes of circles and the lines that intersect them... ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 744 744 Chapter 11 Reading Strategy: Read to Solve Problems A word problem may be overwhelming at first. Once you identify the important parts of the problem and translate the words into math language, you will find that the problem is similar to others you have solved. Reading Tips: ✔ Read each phrase slowly. Write down ✔ Translate the words or
phrases what the words mean as you read them. into math language. ✔ Draw a diagram. Label the diagram so it ✔ Highlight what is being asked. makes sense to you. ✔ Read the problem again before finding your solution. From Lesson 10-3: Use the Reading Tips to help you understand this problem. 14. After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The elevation of the camp is 0.6 km higher than the starting point. What is the distance from the camp to the starting point? After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The starting point can be represented by the ordered triple (0, 0, 0). The elevation of the camp is 0.6 km higher than the starting point. The camp can be represented by the ordered triple (3, 7, 0.6). What is the distance from the camp to the starting point? Distance can be found using the Distance Formula. Use the Distance Formula to find the distance between the camp and the starting point = √  = √  (3 - 0) 2 + (7 - 0) 2 + (0.6 - 0) 2 ≈ 7.6 km Try This For the following problem, apply the following reading tips. Do not solve. • Identify key words. • Translate each phrase into math language. • Draw a diagram to represent the problem. 1. The height of a cylinder is 4 ft, and the diameter is 9 ft. What effect does doubling each measure have on the volume? Circles 745 745 �� 11-1 Lines That Intersect Circles TEKS G.9.C Congruence and the geometry of size: … test conjectures about the properties and attributes of circles and the lines that intersect …. Also G.1.A, Objectives Identify tangents, secants, and chords. Use properties of tangents to solve problems. Vocabulary interior of a circle exterior of a circle chord sec
ant tangent of a circle point of tangency congruent circles concentric circles tangent circles common tangent Why learn this? You can use circle theorems to solve problems about Earth. (See Example 3.) This photograph was taken 216 miles above Earth. From this altitude, it is easy to see the curvature of the horizon. Facts about circles can help us understand details about Earth. Recall that a circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. A circle with center C is called circle C, or ⊙C. The interior of a circle is the set of all points inside the circle. The exterior of a circle is the set of all points outside the circle. �� Lines and Segments That Intersect Circles TERM DIAGRAM A chord is a segment whose endpoints lie on a circle. Also G.2.A, G.2.B A secant is a line that intersects a circle at two points. A tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency. E X A M P L E 1 Identifying Lines and Segments That Intersect Circles Identify each line or segment that intersects ⊙A. chords: ̶̶ EF and ̶̶ BC tangent: ℓ ̶̶ AC and radii: ̶̶ AB secant:   EF diameter: ̶̶ BC 746 746 Chapter 11 Circles �� 1. Identify each line or segment that intersects ⊙P. Remember that the terms radius and diameter may refer to line segments, or to the lengths of segments. Pairs of Circles TERM DIAGRAM Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles. ̶̶ ⊙A ≅ ⊙B if BD. ̶̶ ̶̶ BD if ⊙A ≅ ⊙B. AC ≅ ̶̶ AC ≅ Internally tangent circles Externally tangent circles E X A M P L E
2 Identifying Tangents of Circles Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. radius of ⊙A : 4 radius of ⊙B : 2 Center is (-1, 0). Pt. on ⊙ is (3, 0). Dist. between the 2 pts. is 4. Center is (1, 0). Pt. on ⊙ is (3, 0). Dist. between the 2 pts. is 2. point of tangency: (3, 0) Pt. where the ⊙s and tangent line intersect equation of tangent line: x = 3 Vert. line through (3, 0) 2. Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 11- 1 Lines That Intersect Circles 747 747 �� A common tangent is a line that is tangent to two circles. Lines ℓ and m are common external tangents to ⊙A and ⊙B. Lines p and q are common internal tangents to ⊙A and ⊙B. Construction Tangent to a Circle at a Point    Draw ⊙P. Locate a point on the circle and label it Q. Draw  PQ. Construct the perpendicular ℓ to  PQ at Q. This line is tangent to ⊙P at Q. Notice that in the construction, the tangent line is perpendicular to the radius at the point of tangency. This fact is the basis for the following theorems. Theorems THEOREM HYPOTHESIS CONCLUSION 11-1-1 Theorem 11-1-2 is the converse of Theorem 11-1-1. 11-1-2 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to ⊙ → line ⊥ to radius) If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line ⊥ to radius → line tangent to ⊙) ℓ is tangent
to ⊙A m is ⊥ to ̶̶ CD at D ℓ ⊥ ̶̶ AB m is tangent to ⊙C. You will prove Theorems 11-1-1 and 11-1-2 in Exercises 28 and 29. 748 748 Chapter 11 Circles �� E X A M P L E 3 Problem Solving Application The summit of Mount Everest is approximately 29,000 ft above sea level. What is the distance from the summit to the horizon to the nearest mile? Understand the Problem The answer will be the length of an imaginary segment from the summit of Mount Everest to Earth’s horizon. Make a Plan Draw a sketch. Let C be the center of Earth, E be the summit of Mount Everest, and H be a point on the horizon. You need to find the length of tangent to ⊙C at H. By Theorem 11-1-1, △CHE is a right triangle. ̶̶ EH, which is ̶̶ EH ⊥ ̶̶ CH. So 5280 ft = 1 mi Earth’s radius ≈ 4000 mi Solve ED = 29,000 ft Given = 29,000 _ 5280 ≈ 5.49 mi EC = CD + ED Change ft to mi. Seg. Add. Post. = 4000 + 5.49 = 4005.49 mi Substitute 4000 for CD and 5.49 for ED. EC 2 = EH 2 + CH 2 4005.49 2 = EH 2 + 4000 2 43,950.14 ≈ EH 2 210 mi ≈ EH Look Back Pyth. Thm. Substitute the given values. Subtract 4000 2 from both sides. Take the square root of both sides. The problem asks for the distance to the nearest mile. Check if your answer is reasonable by using the Pythagorean Theorem. Is 210 2 + 4000 2 ≈ 4005 2? Yes, 16,044,100 ≈ 16,040,025. 3. Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the summit of Kilimanjaro to the horizon to the nearest mile? Theorem 11-1-3 THEOREM HYPOTHESIS CONCLUSION If two segments are tangent to a circle from the same external point, then the segments are congruent
. (2 segs. tangent to ⊙ from same ext. pt. → segs. ≅) ̶̶ AB ≅ ̶̶ AC ̶̶ AB and ̶̶ AC are tangent to ⊙P. You will prove Theorem 11-1-3 in Exercise 30. 11- 1 Lines That Intersect Circles 749 749 12��34�� You can use Theorem 11-1-3 to find the length of segments drawn tangent to a circle from an exterior point. E X A M P L E 4 Using Properties of Tangents ̶̶ DE and ̶̶ DF are tangent to ⊙C. Find DF. DE = DF 2 segs. tangent to ⊙ from same ext. pt. → segs. ≅. 5y - 28 = 3y Substitute 5y - 28 for DE and 3y for DF. 2y - 28 = 0 Subtract 3y from both sides. 2y = 28 y = 14 DF = 3 (14) = 42 Add 28 to both sides. Divide both sides by 2. Substitute 14 for y. Simplify. ̶̶ RS and 4a. ̶̶ RT are tangent to ⊙Q. Find RS. 4b. THINK AND DISCUSS 1. Consider ⊙A and ⊙B. How many different lines are common tangents to both circles? Copy the circles and sketch the common external and common internal tangent lines. 2. Is it possible for a line to be tangent to two concentric circles? Explain your answer. 3. Given ⊙P, is the center P a part of the circle? Explain your answer. 4. In the figure, ̶̶ RQ is tangent to ⊙P at Q. Explain how you can find m∠PRQ. 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a definition and draw a sketch. 750 750 Chapter 11 Circles �� 11-1 Exercises Exercises KEYWORD: MG7 11-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A? is a line in the plane of a
circle that intersects the circle at two points. ̶̶̶̶ (secant or tangent) 2. Coplanar circles that have the same center are called?. ̶̶̶̶ (concentric or congruent) 3. ⊙Q and ⊙R both have a radius of 3 cm. Therefore the circles are?. ̶̶̶̶ (concentric or congruent Identify each line or segment that intersects each circle. p. 746 4. 5. 747 Multi-Step Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 6. 7. 749 8. Space Exploration The International Space Station orbits Earth at an altitude of 240 mi. What is the distance from the space station to Earth’s horizon to the nearest mile The segments in each figure are tangent to the circle. Find each length. p. 750 9. JK 10. ST 11- 1 Lines That Intersect Circles 751 751 �� PRACTICE AND PROBLEM SOLVING Identify each line or segment that intersects each circle. 11. 12. Multi-Step Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 13. 14. Independent Practice For See Exercises Example 11–12 13–14 15 16–17 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 Astronomy 15. Astronomy Olympus Mons’s peak rises 25 km above the surface of the planet Mars. The diameter of Mars is approximately 6794 km. What is the distance from the peak of Olympus Mons to the horizon to the nearest kilometer? Olympus Mons, located on Mars, is the tallest known volcano in the solar system. The segments in each figure are tangent to the circle. Find each length. 16. AB 17. RT Tell whether each statement is sometimes, always, or never true. 18. Two circles with the same center are congruent. 19. A tangent to a circle intersects the circle at two points. 20. Tangent circles have the same center. 21. A tangent to a circle will form a right angle with a radius that is drawn to the point of tangency. 22. A chord of
a circle is a diameter. Graphic Design Use the following diagram for Exercises 23–25. The blue topaz was adopted as the Texas state gemstone in 1969. Identify the following. � 23. diameter 24. radii 25. chord � � � � 752 752 Chapter 11 Circles �� In each diagram, ̶̶ PR and 26. m∠Q ̶̶ PS are tangent to ⊙Q. Find each angle measure. 27. m∠P 28. Complete this indirect proof of Theorem 11-1-1. Given: ℓ is tangent to ⊙A at point B. Prove: ℓ ⊥ ̶̶ AB ̶̶ AC such that ̶̶ AB. Then it is possible to ̶̶ AC ⊥ ℓ. If this is true, then △ACB is?. Since ℓ is a ̶̶̶̶ Proof: Assume that ℓ is not ⊥ draw a right triangle. AC < AB because a. tangent line, it can only intersect ⊙A at b.?, and C must be in the ̶̶̶̶ exterior of ⊙A. That means that AC > AB since contradicts the fact that AC < AB. Thus the assumption is false, and d.?. This ̶̶̶̶ ̶̶ AB is a c.?. ̶̶̶̶ 29. Prove Theorem 11-1-2. ̶̶ Given: m ⊥ CD Prove: m is tangent to ⊙C. (Hint: Choose a point on m. Then use the Pythagorean Theorem to prove that if the point is not D, then it is not on the circle.) 30. Prove Theorem 11-1-3. ̶̶ AB and ̶̶ ̶̶ AB ≅ AC Given: Prove: ̶̶ AC are tangent to ⊙P. Plan: Draw auxiliary segments the triangles formed are congruent. Then use CPCTC. ̶̶ PA, ̶̶ PB, and ̶̶ PC. Show that Algebra Assume the segments that appear to be tangent are tangent. Find each length. 31. ST 32. DE 33. JL 34. ⊙M has center M (2
, 2) and radius 3. ⊙N has center N (-3, 2) and is tangent to ⊙M. Find the coordinates of the possible points of tangency of the two circles. 35. This problem will prepare you for the Multi-Step TAKS Prep on page 770. The diagram shows the gears of a bicycle. AD = 5 in., and BC = 3 in. CD, the length of the chain between the gears, is 17 in. a. What type of quadrilateral is BCDE? Why? b. Find BE and AE. c. What is AB to the nearest tenth of an inch? � � � � � 11- 1 Lines That Intersect Circles 753 753 �� 36. Critical Thinking Given a circle with diameter ̶̶ BC, is it possible to draw tangents to B and C from an external point X? If so, make a sketch. If not, explain why it is not possible. 37. Write About It ̶̶ PR and Explain why ∠P and ∠Q are supplementary. ̶̶ PS are tangent to ⊙Q. 38. ̶̶ AB and is closest to AD? ̶̶ AC are tangent to ⊙D. Which of these 9.5 cm 10 cm 10.4 cm 13 cm 39. ⊙P has center P (3, -2) and radius 2. Which of these lines is tangent to ⊙P? x = 4 y = -4 y = -2 x = 0 40. ⊙A has radius 5. ⊙B has radius 6. What is the ratio of the area of ⊙A to that of ⊙B? 125 _ 216 25 _ 36 5 _ 6 36 _ 25 CHALLENGE AND EXTEND ̶̶̶ 41. Given: ⊙G with GH ⊥ ̶̶ KH ̶̶ JH ≅ Prove: ̶̶ JK 42. Multi-Step ⊙A has radius 5, ⊙B has radius 2, ̶̶ CD is a common tangent. What is AB? and (Hint: Draw a perpendicular segment from B to E, a point on ̶̶ AC.) 43. Manufacturing A company builds metal stands for bicycle wheels. A new design calls for a V-shaped
stand that will hold wheels with a 13 in. radius. The sides of the stand form a 70° angle. To the nearest tenth of an inch, what should be the length XY of a side so that it is tangent to the wheel? SPIRAL REVIEW 44. Andrea and Carlos both mow lawns. Andrea charges $14.00 plus $6.25 per hour. Carlos charges $12.50 plus $6.50 per hour. If they both mow h hours and Andrea earns more money than Carlos, what is the range of values of h? (Previous course) ̶̶ LR. A point is chosen randomly on Use the diagram to find the probability of each event. (Lesson 9-6) ̶̶̶ MP. 45. The point is not on ̶̶̶ MN or 47. The point is on ̶̶ PR. 754 754 Chapter 11 Circles 46. The point is on 48. The point is on ̶̶ LP. ̶̶ QR. �� Circle Graphs Data Analysis A circle graph compares data that are parts of a whole unit. When you make a circle graph, you find the measure of each central angle. A central angle is an angle whose vertex is the center of the circle. See Skills Bank page S80 Example Make a circle graph to represent the following data. Step 1 Add all the amounts. 110 + 40 + 300 + 150 = 600 Step 2 Write each part as a fraction of the whole. fiction: 110 _ ; nonfiction: 40 _ ; children’s: 300 _ ; audio books: 150 _ 600 600 600 600 Books in the Bookmobile Fiction Nonfiction Children’s 110 40 300 150 Step 3 Multiply each fraction by 360° to calculate the central Audio books angle measure. 110 _ 600 (360°) = 66°; 40 _ 600 (360°) = 24°; 300 _ (360°) = 180°; 150 _ (360°) = 90° 600 600 Step 4 Make a circle graph. Then color each section of the circle to match the data. The section with a central angle of 66° is green, 24° is orange, 180° is purple, and 90° is yellow. Try This TAKS Grades 9–11 Obj. 8, 9 Choose the circle graph that best represents the data. Show each step. 1. Books in Linda’s Library Novels
Reference Textbooks 18 10 8 2. Vacation Expenses ($) 3. Puppy Expenses ($) Travel Meals Lodging Other 450 120 900 330 Food Health Training Other 190 375 120 50 On Track for TAKS 755 755 �� 11-2 Arcs and Chords TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system …. Also G.2.A, G.2.B, G.8.C, G.9.C Objectives Apply properties of arcs. Apply properties of chords. Who uses this? Market analysts use circle graphs to compare sales of different products. Vocabulary central angle arc minor arc major arc semicircle adjacent arcs congruent arcs Minor arcs may be named by two points. Major arcs and semicircles must be named by three points. A central angle is an angle whose vertex is the center of a circle. An arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them. Arcs and Their Measure ARC MEASURE DIAGRAM A minor arc is an arc whose points are on or in the interior of a central angle. The measure of a minor arc is equal to the measure of its central angle. m ⁀ AC = m∠ABC = x° A major arc is an arc whose points are on or in the exterior of a central angle. The measure of a major arc is equal to 360° minus the measure of its central angle. m ⁀ ADC = 360° - m∠ABC = 360° - x° If the endpoints of an arc lie on a diameter, the arc is a semicircle. The measure of a semicircle is equal to 180°. m ⁀ EFG = 180° E X A M P L E 1 Data Application The circle graph shows the types of music sold during one week at a music store. Find m ⁀ BC. m ⁀ BC = m∠BMC m of arc = m of central ∠. m∠BMC = 0.13 (360°) Central ∠ is 13% = 46.8° of the ⊙. Use the graph to find each of the following. 1a. m∠FMC 1b. m ⁀ AHB 1c. m∠EMD 756 756 Chapter 11 Circles ��
�� Adjacent arcs are arcs of the same circle that intersect at exactly one point. ⁀ RS and ⁀ ST are adjacent arcs. Postulate 11-2-1 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m⁀ABC = m ⁀AB + m ⁀BC E X A M P L E 2 Using the Arc Addition Postulate Find m ⁀CDE m ⁀CD = 90° m∠DFE = 18° m ⁀DE = 18° m ⁀CE = m ⁀CD + m ⁀DE m∠CFD = 90° Vert.  Thm. m∠DFE = 18° Arc Add. Post. = 90° + 18° = 108° Substitute and simplify. Find each measure. 2a. m ⁀JKL 2b. m ⁀LJN Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure, ⁀ ST ≅ ⁀ UV. Theorem 11-2-2 THEOREM HYPOTHESIS CONCLUSION In a circle or congruent circles: (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles. ∠EAD ≅ ∠BAC ̶̶ DE ≅ ̶̶ BC ̶̶ ED ≅ ̶̶ BC ⁀ DE ≅ ⁀ BC ⁀ ED ≅ ⁀ BC ∠DAE ≅ ∠BAC You will prove parts 2 and 3 of Theorem 11-2-2 in Exercises 40 and 41. 11-2 Arcs and Chords 757 757 �� The converses of the parts of Theorem 11-2-2 are also true. For example, with part 1, congruent chords have congruent central angles. PROOF PROOF Theorem 11-2-2 (Part 1) Given: ∠BAC ≅ ∠DAE ̶̶ BC ≅ Prove:
̶̶ DE Proof: Statements Reasons 1. ∠BAC ≅ ∠DAE ̶̶ AB ≅ ̶̶̶ AD, ̶̶ AC ≅ ̶̶ AE 2. 3. △BAC ≅ △DAE ̶̶ DE ̶̶ BC ≅ 4. 1. Given 2. All radii of a ⊙ are ≅. 3. SAS Steps 2, 1 4. CPCTC E X A M P L E 3 Applying Congruent Angles, Arcs, and Chords Find each measure. A ̶̶ RS ≅ ̶̶ TU. Find m ⁀ RS. ⁀ RS ≅ ⁀ TU m ⁀ RS = m ⁀ TU 3x = 2x + 27 x = 27 m ⁀ RS = 3 (27) = 81° ≅ chords have ≅ arcs. Def. of ≅ arcs Substitute the given measures. Subtract 2x from both sides. Substitute 27 for x. Simplify. B ⊙B ≅ ⊙E, and ⁀ AC ≅ ⁀ DF. Find m∠DEF. ∠ABC = ∠DEF m∠ABC = m∠DEF 5y + 5 = 7y - 43 5 = 2y - 43 ≅ arcs have ≅ central . Def. of ≅  Substitute the given measures. Subtract 5y from both sides. 48 = 2y 24 = y m∠DEF = 7 (24) - 43 = 125° Add 43 to both sides. Divide both sides by 2. Substitute 24 for y. Simplify. Find each measure. 3a.  PT bisects ∠RPS. Find RT. 3b. ⊙A ≅ ⊙B, and Find m ⁀ CD. ̶̶ CD ≅ ̶̶ EF. 758 758 Chapter 11 Circles �� Theorems THEOREM HYPOTHESIS CONCLUSION 11-2-3 In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. 11-2-4
In a circle, the perpendicular bisector of a chord is a radius (or diameter). ̶̶ CD bisects ̶̶ EF and ⁀ EF. ̶̶ CD ⊥ ̶̶ EF ̶̶ JK is a diameter of ⊙A. ̶̶ JK is ⊥ bisector of ̶̶̶ GH. You will prove Theorems 11-2-3 and 11-2-4 in Exercises 42 and 43. E X A M P L E 4 Using Radii and Chords Find BD. Step 1 Draw radius ̶̶ AD. AD = 5 Radii of a ⊙ are ≅. Step 2 Use the Pythagorean Theorem. CD 2 + AC 2 = AD 2 CD 2 + 3 2 = 5 2 CD 2 = 16 CD = 4 Step 3 Find BD. BD = 2(4 ) = 8 Substitute 3 for AC and 5 for AD. Subtract 3 2 from both sides. Take the square root of both sides. ̶̶ AE ⊥ ̶̶ BD, so ̶̶ AE bisects ̶̶ BD. 4. Find QR to the nearest tenth. THINK AND DISCUSS 1. What is true about the measure of an arc whose central angle is obtuse? 2. Under what conditions are two arcs the same measure but not congruent? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a definition and draw a sketch. 11-2 Arcs and Chords 759 759 �� 11-2 Exercises Exercises KEYWORD: MG7 11-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An arc that joins the endpoints of a diameter is called a?. (semicircle or ̶̶̶̶ major arc) 2. How do you recognize a central angle of a circle? ABC = 205°. Therefore ⁀ 3. In ⊙P m ⁀ ABC is a?. (major arc or minor arc) ̶̶̶̶ 4. In a circle, an arc that is less than a semicircle is a?. (major arc or minor arc) ̶̶̶̶. 756 Consumer Application Use
the following information for Exercises 5–10. The circle graph shows how a typical household spends money on energy. Find each of the following. 5. m∠PAQ 7. m∠SAQ 9. m ⁀ RQ 6. m∠VAU 8. m ⁀ UT 10. m ⁀ UPT. 757 Find each measure. 11. m ⁀ DF 12. m ⁀ DEB 13. m ⁀ JL 14. m ⁀ HLK 15. ∠QPR ≅ ∠RPS. Find QR. 16. ⊙A ≅ ⊙B, and ⁀ CD ≅ ⁀ EF. Find m∠EBF. p. 758 Multi-Step Find each length to the nearest tenth. p. 759 17. RS 18. EF 760 760 Chapter 11 Circles �� Independent Practice For See Exercises Example 19–24 25–28 29–30 31–32 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Sports Use the following information for Exercises 19–24. The key shows the number of medals won by U.S. athletes at the 2004 Olympics in Athens. Find each of the following to the nearest tenth. �� 19. m∠ADB 21. m ⁀ AB 21. 23. m ⁀ ACB 20. m∠ADC 22. m ⁀ BC 24. m ⁀ CAB Find each measure. 25. m ⁀ MP 26. m ⁀ QNL 29. ⊙A ≅ ⊙B, and ⁀ CD ≅ ⁀ EF. Find m∠CAD. �� 27. m ⁀ WT 28. m ⁀ WTV 30. ̶̶ JK ≅ ̶̶̶ LM. Find m ⁀ JK. Multi-Step Find each length to the nearest tenth. 31. CD 32. RS Determine whether each
statement is true or false. If false, explain why. 33. The central angle of a minor arc is an acute angle. 34. Any two points on a circle determine a minor arc and a major arc. 35. In a circle, the perpendicular bisector of a chord must pass through the center of the circle. 36. Data Collection Use a graphing calculator, a pH probe, and a data-collection device to collect information about the pH levels of ten different liquids. Then create a circle graph with the following sectors: strong basic (9 < pH < 14), weak basic (7 < pH < 9), neutral (pH = 7), weak acidic (5 < pH < 7), and strong acidic (0 < pH < 5). 37. In ⊙E, the measures of ∠AEB, ∠BEC, and ∠CED are in the ratio 3 : 4 : 5. Find m ⁀ AB, m ⁀ BC, and m ⁀ CD. 11-2 Arcs and Chords 761 761 �� Algebra Find the indicated measure. 38. m ⁀ JL 39. m∠SPT 40. Prove ≅ chords have ≅ arcs. ̶̶ Given: ⊙A, BC ≅ Prove: ⁀ BC ≅ ⁀ DE ̶̶ DE 41. Prove ≅ arcs have ≅ central . Given: ⊙A, ⁀ BC ≅ ⁀ DE Prove: ∠BAC ≅ ∠DAE 42. Prove Theorem 11-2-3. Given: ⊙C, Prove: ̶̶ EF ̶̶ EF ̶̶ CD ⊥ ̶̶ CD bisects and ⁀ EF. ̶̶ CE and ̶̶ CF (Hint: Draw and use the HL Theorem.) 43. Prove Theorem 11-2-4. ̶̶ JK ⊥ ̶̶̶ GH Given: ⊙A, bisector of Prove: (Hint: Use the Converse of the ⊥ Bisector Theorem.) ̶̶ JK is a diameter 44. Critical Thinking Roberto folds a circular piece of paper as shown. When he unfolds the paper, how many different-sized
central angles will be formed? One fold Two folds Three folds 45. /////ERROR ANALYSIS///// Below are two solutions to find the value of x. Which solution is incorrect? Explain the error. 46. Write About It According to a school survey, 40% of the students take a bus to school, 35% are driven to school, 15% ride a bike, and the remainder walk. Explain how to use central angles to create a circle graph from this data. 47. This problem will prepare you for the Multi-Step TAKS This problem will prepare you for the Multi-Step TAKS Prep on page 770. Chantal’s bike has wheels with a 27 in. diameter. a. What are AC and AD if DB is 7 in.? b. What is CD to the nearest tenth of an inch? c. What is CE, the length of the top of the bike stand? � � � � � 762 762 Chapter 11 Circles �� 48. Which of these arcs of ⊙Q has the greatest measure? ⁀ WT ⁀ UW ⁀ VR ⁀ TV 49. In ⊙A, CD = 10. Which of these is closest ̶̶ AE? to the length of 3.3 cm 4 cm 5 cm 7.8 cm 50. Gridded Response ⊙P has center P (2, 1) and radius 3. What is the measure, in degrees, of the minor arc with endpoints A (-1, 1) and B (2, -2)? CHALLENGE AND EXTEND ̶̶ 51. In the figure, AB ⊥ ̶̶ CD. Find m ⁀ BD to the nearest tenth of a degree. 52. Two points on a circle determine two distinct arcs. How many arcs are determined by n points on a circle? (Hint: Make a table and look for a pattern.) 53. An angle measure other than degrees is radian measure. 360° converts to 2π radians, or 180°converts to π radians., π _, π
_ a. Convert the following radian angle measures to degrees: π _. 4 3 2 b. Convert the following angle measures to radians: 135°, 270°. SPIRAL REVIEW Simplify each expression. (Previous course) 54. (3x) 3 ( 2y 2 ) ( 3 -2 y 2 ) 55. a 4 b 3 (-2a) -4 2 56. ( -2t 3 s 2 ) ( 3ts 2 ) Find the next term in each pattern. (Lesson 2-1) 57. 1, 3, 7, 13, 21, … 58. C, E, G, I, K,... 59. 1, 6, 15, … In the figure, (Lesson 11-1) ̶̶ QP and ̶̶̶ QM are tangent to ⊙N. Find each measure. 60. m∠NMQ 61. MQ Construction Circle Through Three Noncollinear Points    Draw three noncollinear points. Construct m and n, the ⊥ bisectors of ̶̶ PQ and ̶̶ QR. Label the intersection O. Center the compass at O. Draw a circle through P. 1. Explain why ⊙O with radius ̶̶ OP also contains Q and R. 11-2 Arcs and Chords 763 763 �� 11-3 Sector Area and Arc Length TEKS G.8.B Congruence and the geometry of size: find areas of sectors and arc lengths of circles using proportional reasoning. Objectives Find the area of sectors. Find arc lengths. Vocabulary sector of a circle segment of a circle arc length Who uses this? Farmers use irrigation radii to calculate areas of sectors. (See Example 2.) The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the circle by m°____ 360°. Sector of a Circle TERM NAME DIAGRAM AREA A sector of a circle is a region bounded by two radii of the circle and their intercepted arc. sector ACB Also G.1A, G.1.B, G.9.C A = πr 2 ( m° _ ) 360° E X A M P L E 1 Finding the Area of
a Sector Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. A sector MPN 360° A = π r 2 ( m° _ ) = π (3 ) 2 ( 80° _ ) = 2π in 2 ≈ 6.28 in 2 360° B sector EFG 360° A = π r 2 ( m° _ ) = π (6) 2 ( 120° _ ) = 12π ≈ 37.70 cm 2 360° Use formula for area of a sector. Substitute 3 for r and 80 for m. Simplify. Use formula for area of a sector. Substitute 6 for r and 120 for m. Simplify. Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 1a. sector ACB 1b. sector JKL Write the degree symbol after m in the formula to help you remember to use degree measure not arc length. 764 764 Chapter 11 Circles �� E X A M P L E 2 Agriculture Application A circular plot with a 720 ft diameter is watered by a spray irrigation system. To the nearest square foot, what is the area that is watered as the sprinkler rotates through an angle of 50°? 360° A = π r 2 ( m° _ ) = π (360) 2 ( 50° _ ) ≈ 56,549 ft 2 360° d = 720 ft, r = 360 ft. Simplify. 2. To the nearest square foot, what is the area watered in Example 2 as the sprinkler rotates through a semicircle? A segment of a circle is a region bounded by an arc and its chord. The shaded region in the figure is a segment. Area of a Segment area of segment = area of sector - area of triangle E X A M P L E 3 Finding the Area of a Segment Find the area of segment ACB to the nearest hundredth. Step 1 Find the area of sector ACB. In a 30°-60°-90° triangle, the length of the leg opposite the 60° angle is √  3 times the length of the shorter leg. 360° A = π r 2 ( m° _ ) = π (12) 2 ( 60° _ ) = 24π in 2 360° Use formula for area of a
sector. Substitute 12 for r and 60 for m. Step 2 Find the area of △ACB. ̶̶ AD. Draw altitude bh = 1 _ A = 1 _ (12) (6 √  3 ) 2 2 = 36 √  3 in 2 CD = 6 in., and h = 6 √  3 in. Simplify. Step 3 area of segment = area of sector ACB - area of △ACB = 24π - 36 √  3 ≈ 13.04 in 2 3. Find the area of segment RST to the nearest hundredth. 11-3 Sector Area and Arc Length 765 765 �� In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle. Arc Length TERM DIAGRAM LENGTH Arc length is the distance along an arc measured in linear units. L = 2πr ( m° _ ) 360° E X A M P L E 4 Finding Arc Length Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. A ⁀CD L = 2πr ( m°_ 360°) = 2π(10)( 90°_ 360°) Use formula for arc length. Substitute 10 for r and 90 for m. = 5π ft ≈ 15.71 ft Simplify. B an arc with measure 35° in a circle with radius 3 in. L = 2πr ( m°_ 360°) = 2π(3)( 35°_ 360°) = 7_ 12 in. ≈ 1.83 in. Use formula for arc length. Substitute 3 for r and 35 for m. Simplify. Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 4a. ⁀GH 4b. an arc with measure 135° in a circle with radius 4 cm THINK AND DISCUSS 1. What is the difference between arc measure and arc length? 2. A slice of pizza is a sector of a circle. Explain what measurements you would need to make in order to calculate the area of the slice. 3. GET ORGANIZED Copy and complete the graphic organizer. 766 766 Chapter 11 Circles ��
�� 11-3 Exercises Exercises KEYWORD: MG7 11-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a circle, the region bounded by a chord and an arc is called a?. (sector or segment) ̶̶̶̶. 764 Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 2. sector PQR 3. sector JKL 4. sector ABC. Navigation The beam from a lighthouse is visible for a distance of 3 mi. p. 765 To the nearest square mile, what is the area covered by the beam as it sweeps in an arc of 150°? Multi-Step Find the area of each segment to the nearest hundredth. p. 765 6. 7. 8. 766 Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 9. ⁀ EF 10. ⁀ PQ Independent Practice For See Exercises Example 12–14 15 16–18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 11. an arc with measure 20° in a circle with radius 6 in. PRACTICE AND PROBLEM SOLVING Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 12. sector DEF 13. sector GHJ 14. sector RST 15. Architecture A lunette is a semicircular window that is sometimes placed above a doorway or above a rectangular window. To the nearest square inch, what is the area of the lunette? �� 11-3 Sector Area and Arc Length 767 767 �� Multi-Step Find the area of each segment to the nearest hundredth. 16. 17. 18. Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 19. ⁀ UV 20. ⁀ AB Math History 21. an arc with measure 9° in a circle with diameter 4 ft 22. Math History Greek mathematicians studied the salinon, a figure bounded by four semicircles. What is the perimeter of this salin
on to the nearest tenth of an inch? Hypatia lived 1600 years ago. She is considered one of history’s most important mathematicians. She is credited with contributions to both geometry and astronomy. Tell whether each statement is sometimes, always, or never true. 23. The length of an arc of a circle is greater than the circumference of the circle. 24. Two arcs with the same measure have the same arc length. 25. In a circle, two arcs with the same length have the same measure. Find the radius of each circle. 26. area of sector ABC = 9π 27. arc length of ⁀ EF = 8π 28. Estimation The fraction 22 __ 7 is an approximation for π. a. Use this value to estimate the arc length of ⁀ XY. b. Use the π key on your calculator to find the length of ⁀ XY to 8 decimal places. c. Was your estimate in part a an overestimate or an underestimate? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 770. The pedals of a penny-farthing bicycle are directly connected to the front wheel. a. Suppose a penny-farthing bicycle has a front wheel with a diameter of 5 ft. To the nearest tenth of a foot, how far does the bike move when you turn the pedals through an angle of 90°? b. Through what angle should you turn the pedals in order to move forward by a distance of 4.5 ft? Round to the nearest degree. 768 768 Chapter 11 Circles �� 30. Critical Thinking What is the length of the radius that makes the area of ⊙A = 24 in 2 and the area of sector BAC = 3 in 2? Explain. 31. Write About It Given the length of an arc of a circle and the measure of the arc, explain how to find the radius of the circle. 32. What is the area of sector AOB? 4π 16π 32π 64π 33. What is the length of ⁀ AB? 4π 2π 8π 16π 34. Gridded Response To the nearest hundredth, what is the area of the sector determined by an arc with measure 35° in a circle with radius 12? CHALLENGE AND EXTEND 35. In the diagram, the larger of the two concentric circles has radius 5, and
the smaller circle has radius 2. What is the area of the shaded region in terms of π? 36. A wedge of cheese is a sector of a cylinder. a. To the nearest tenth, what is the volume of the wedge with the dimensions shown? b. What is the surface area of the wedge of cheese to the nearest tenth? 37. Probability The central angles of a target measure 45°. The inner circle has a radius of 1 ft, and the outer circle has a radius of 2 ft. Assuming that all arrows hit the target at random, find the following probabilities. a. hitting a red region b. hitting a blue region c. hitting a red or blue region �� SPIRAL REVIEW Determine whether each line is parallel to y = 4x - 5, perpendicular to y = 4x - 5, or neither. (Previous course) 38. 8x - 2y = 6 39. line passing through the points ( 1 __ 2, 0) and (1 1 __ 2, 2) 40. line with x-intercept 4 and y-intercept 1 Find each measurement. Give your answer in terms of π. (Lesson 10-8) 41. volume of a sphere with radius 3 cm 42. circumference of a great circle of a sphere whose surface area is 4π cm 2 Find the indicated measure. (Lesson 11-2) 43. m∠KLJ 44. m ⁀ KJ 45. m ⁀ JFH 11-3 Sector Area and Arc Length 769 769 �� SECTION 11A Lines and Arcs in Circles As the Wheels Turn The bicycle was invented in the 1790s. The first models didn’t even have pedals—riders moved forward by pushing their feet along the ground! Today the bicycle is a high-tech machine that can include hydraulic brakes and electronic gear changers. 1. A road race bicycle wheel is 28 inches in diameter. A manufacturer makes metal bicycle stands that are 10 in. tall. How long should a stand be to the nearest tenth in order to support a 28 in. wheel? (Hint: Consider the triangle formed by the radii and the top of the stand.) 2. The chain of a bicycle loops around a large gear connected to the bike’s pedals and a small gear attached to the rear wheel. In the diagram, the distance AB between
the centers of the gears the nearest tenth is 15 in. Find CD, the length of the chain between the two gears to the nearest tenth. (Hint: Draw a segment from B to ̶̶ AD that is parallel to ̶̶ CD.) �� 3. By pedaling, you turn the large gear through an angle of 60°. How far does the chain move around the circumference of the gear to the nearest tenth? 4. As the chain moves, it turns the small gear. If you use the distance you calculated in Problem 3, through what angle does the small gear turn to the nearest degree? �� 770 770 Chapter 11 Circles SECTION 11A Quiz for Lessons 11-1 Through 11-3 11-1 Lines That Intersect Circles Identify each line or segment that intersects each circle. 1. 2. 3. The tallest building in Africa is the Carlton Centre in Johannesburg, South Africa. What is the distance from the top of this 732 ft building to the horizon to the nearest mile? (Hint: 5280 ft = 1 mi; radius of Earth = 4000 mi) 11-2 Arcs and Chords Find each measure. 4. ⁀ BC 5. ⁀ BED 6. ⁀ SR 7. ⁀ SQU Find each length to the nearest tenth. 8. JK 9. XY 11-3 Sector Area and Arc Length 10. As part of an art project, Peter buys a circular piece of fabric and then cuts out the sector shown. What is the area of the sector to the nearest square centimeter? Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 11. ⁀ AB 12. ⁀ EF �� 13. an arc with measure 44° in a circle with diameter 10 in. 14. a semicircle in a circle with diameter 92 m Ready to Go On? 771 771 �� 11-4 Inscribed Angles TEKS G.5.B Geometric paterns: use … patterns to make generalizations about geometric properties, including … angle relationships in … circles. Objectives Find the measure of an inscribed angle. Use inscribed angles and their properties to solve problems. Vocabulary inscribed angle intercepted arc subtend Also G.1.A, G
.2.A, G.2.B, G.9.C Why learn this? You can use inscribed angles to find measures of angles in string art. (See Example 2.) String art often begins with pins or nails that are placed around the circumference of a circle. A long piece of string is then wound from one nail to another. The resulting pattern may include hundreds of inscribed angles. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. A chord or arc subtends an angle if its endpoints lie on the sides of the angle. ∠DEF is an inscribed angle. ⁀ DF is the intercepted arc. ⁀ DF subtends ∠DEF. Theorem 11-4-1 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. m∠ABC = 1 _ 2 m ⁀ AC Case 1 Case 2 Case 3 You will prove Cases 2 and 3 of Theorem 11-4-1 in Exercises 30 and 31. PROOF PROOF Inscribed Angle Theorem Given: ∠ABC is inscribed in ⊙X. Prove: m∠ABC = 1 __ 2 m ⁀ AC Proof Case 1: ̶̶ BC. Draw ̶̶ XA. m ⁀ AC = m∠AXC. ̶̶ XA and ̶̶ XB are radii of the circle, ∠ABC is inscribed in ⊙X with X on By the Exterior Angle Theorem m∠AXC = m∠ABX + m∠BAX. Since △AXB is isosceles. Thus m∠ABX = m∠BAX. By the Substitution Property, m ⁀ AC = 2m∠ABX or 2m∠ABC. Thus 1 __ 2 m ⁀ AC = m∠ABC. ̶̶ XA ≅ ̶̶ XB. Then by definition 772 772 Chapter 11 Circles �� E X A M P L E 1 Finding Measures of Arcs and Inscribed Angles Find each measure. A m∠RST m∠RST = 1 _ m ⁀ RT 2 = 1 _ (120°) = 60
° 2 Inscribed ∠ Thm. Substitute 120 for m ⁀ RT. B m ⁀ SU m∠SRU = 1 _ m ⁀ SU 2 40° = 1 _ m ⁀ SU 2 m ⁀ SU = 80° Inscribed ∠ Thm. Substitute 40 for m∠SRU. Mult. both sides by 2. Find each measure. 1a. m ⁀ ADC 1b. m∠DAE Corollary 11-4-2 COROLLARY HYPOTHESIS CONCLUSION If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. ∠ACB ≅ ∠ADB ≅ ∠AEB (and ∠CAE ≅ ∠CBE) ∠ACB, ∠ADB, and ∠AEB intercept ⁀ AB. You will prove Corollary 11-4-2 in Exercise 32. E X A M P L E 2 Hobby Application Find m∠DEC, if m ⁀ AD = 86°. ∠BAC ≅ ∠BDC m∠BAC = m∠BDC m∠BDC = 60° m∠ACD = 1 _ m ⁀ AD 2 = 1 _ (86°) 2 = 43° ∠BAC and ∠BDC intercept ⁀ BC. Def. of ≅ Substitute 60 for m∠BDC. Inscribed ∠ Thm. Substitute 86 for m ⁀ AD. Simplify. m∠DEC + 60 + 43 = 180 △ Sum Theorem m∠DEC = 77° Simplify. 2. Find m∠ABD and m ⁀ BC in the string art. 11-4 Inscribed Angles 773 773 �� Theorem 11-4-3 An inscribed angle subtends a semicircle if and only if the angle is a right angle. You will prove Theorem 11-4-3 in Exercise 43. E X A M P L E 3 Finding Angle Measures in Inscribed Triangles Find each value. A x ∠RQT is a right angle ∠RQT is inscribed in a m∠RQT = 90° 4x + 6 = 90 4x = 84 x
= 21 B m∠ADC semicircle. Def. of rt. ∠ Substitute 4x + 6 for m∠RQT. Subtract 6 from both sides. Divide both sides by 4. m∠ABC = m∠ADC ∠ABC and ∠ADC both 10y - 28 = 7y - 1 3y - 28 = -1 3y = 27 y = 9 intercept ⁀ AC. Substitute the given values. Subtract 7y from both sides. Add 28 to both sides. Divide both sides by 3. m∠ADC = 7 (9 ) -1 = 62° Substitute 9 for y. Find each value. 3a. z 3b. m∠EDF Construction Center of a Circle     Draw a circle and ̶̶ AB. chord Draw chord ̶̶ AC. Construct a line ̶̶ perpendicular to AB at B. Where the line and the circle intersect, label the point C. 774 774 Chapter 11 Circles ̶̶ DF. Repeat steps to draw ̶̶ chords DE and The intersection of ̶̶ and DF is the center of the circle. ̶̶ AC �� Theorem 11-4-4 THEOREM HYPOTHESIS CONCLUSION If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. ∠A and ∠C are supplementary. ∠B and ∠D are supplementary. ABCD is inscribed in ⊙E. You will prove Theorem 11-4-4 in Exercise 44. E X A M P L E 4 Finding Angle Measures in Inscribed Quadrilaterals Find the angle measures of PQRS. Step 1 Find the value of y. m∠P + m∠R = 180° PQRS is inscribed in a ⊙. 6y + 1 + 10y + 19 = 180 16y + 20 = 180 16y = 160 y = 10 Substitute the given values. Simplify. Subtract 20 from both sides. Divide both sides by 16. Step 2 Find the measure of each angle. m∠P = 6 (10) + 1 = 61° m∠R = 10 (10) + 19 = 119° m∠Q = 10 2
+ 48 = 148° m∠Q + m∠S = 180° 148° + m∠S = 180° m∠S = 32° Substitute 10 for y in each expression. ∠Q and ∠S are supp. Substitute 148 for m∠Q. Subtract 148 from both sides. 4. Find the angle measures of JKLM. THINK AND DISCUSS 1. Can ABCD be inscribed in a circle? Why or why not? 2. An inscribed angle intercepts an arc that is 1 __ 4 of the circle. Explain how to find the measure of the inscribed angle. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box write a definition, properties, an example, and a nonexample. 11-4 Inscribed Angles 775 775 �� 11-4 Exercises Exercises KEYWORD: MG7 11-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A, B, and C lie on ⊙P. ∠ABC is an example of an? angle. ̶̶̶̶ (intercepted or inscribed Find each measure. p. 773 2. m∠DEF 3. m ⁀ EG 4. m ⁀ JKL 5. m∠LKM. Crafts A circular loom can be used � p. 773 for knitting. What is the m∠QTR in the knitting loom Find each value. �� p. 774 7. x 8. y 9. m∠XYZ Multi-Step Find the angle measures of each quadrilateral. p. 775 10. PQRS 11. ABCD Independent Practice For See Exercises Example 12–15 16 17–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Find each measure. 12. m ⁀ ML 14. m ⁀ EGH 13. m∠KMN 15. m∠GFH 16. Crafts An artist created a stained 16. glass window. If m∠
BEC = 40° and m ⁀ AB = 44°, what is m∠ADC? 776 776 Chapter 11 Circles ABCDEge07sec11l04004a�� Algebra Find each value. 17. y 18. z 19. m ⁀ AB 20. m∠MPN Multi-Step Find the angle measures of each quadrilateral. 21. BCDE 22. TUVW Tell whether each statement is sometimes, always, or never true. 23. Two inscribed angles that intercept the same arc of a circle are congruent. 24. When a right triangle is inscribed in a circle, one of the legs of the triangle is a diameter of the circle. 25. A trapezoid can be inscribed in a circle. Multi-Step Find each angle measure. 26. m∠ABC if m∠ADC = 112° 27. m∠PQR if m ⁀ PQR = 130° 28. Prove that the measure of a central angle subtended by a chord is twice the measure of the inscribed angle subtended by the chord. Given: In ⊙H Prove: m∠JHK = 2m∠JLK ̶̶ JK subtends ∠JHK and ∠JLK. 29. This problem will prepare you for the Multi-Step TAKS Prep on page 806. A Native American sand painting could be used to indicate the direction of sunrise on the winter and summer solstices. You can make this design by placing six equally spaced points around the circumference of a circle and connecting them as shown. a. Find m∠BAC. b. Find m∠CDE. c. What type of triangle is △FBC? Why? 11-4 Inscribed Angles 777 777 ��ABEDCFge07sec11l04006a 30. Given: ∠ABC is inscribed in ⊙X with X in the interior of ∠ABC. Prove: m∠ABC = 1 _ m �
� AC 2 (Hint: Draw  BX and use Case 1 of the Inscribed Angle Theorem.) History The Winchester Round Table, probably built in the late thirteenth century, is 18 ft across and weighs 1.25 tons. King Arthur’s Round Table of English legend would have been much larger—it could seat 1600 men. 31. Given: ∠ABC is inscribed in ⊙X with X in the exterior of ∠ABC. Prove: m∠ABC = 1 _ m ⁀ AC 2 32. Prove Corollary 11-4-2. Given: ∠ACB and ∠ADB intercept ⁀ AB. Prove: ∠ACB ≅ ∠ADB 33. Multi-Step In the diagram, m ⁀ JKL = 198°, and m ⁀ of quadrilateral JKLM. KLM = 216°. Find the measures of the angles 34. Critical Thinking A rectangle PQRS is inscribed in a circle. What can you conclude about ̶̶ PR? Explain. 35. History The diagram shows the Winchester Round Table with inscribed △ABC. The table may have been made at the request of King Edward III, who created the Order of Garter as a return to the Round Table and an order of chivalry. ̶̶ a. Explain why BC must be a diameter of the circle. b. Find m ⁀ AC. 36. To inscribe an equilateral triangle in a circle, draw a ̶̶ BC. Open the compass to the radius of the circle. diameter Place the point of the compass at C and make arcs on the circle at D and E, as shown. Draw ̶̶ DE. Explain why △BDE is an equilateral triangle. ̶̶ BD, ̶̶ BE, and 37. Write About It A student claimed that if a parallelogram contains a 30° angle, it cannot be inscribed in a circle. Do you agree or disagree? Explain. 38. Construction Circumscribe a circle about a triangle. (Hint: Follow the steps for the construction of a circle through three given noncollinear points.) 39. What is m∠BAC? 38° 43° 66° 81° 40. Equilateral △XCZ is inscribed in a circle. ̶̶ CY bisects ∠C, what is m
⁀ XY? If 15° 30° 60° 120° 41. Quadrilateral ABCD is inscribed in a circle. The ratio of m∠A to m∠C is 4 : 5. What is m∠A? 20° 40° 80° 100° 42. Which of these angles has the greatest measure? ∠STR ∠QPR ∠QSR ∠PQS 778 778 Chapter 11 Circles �� CHALLENGE AND EXTEND 43. Prove that an inscribed angle subtends a semicircle if and only if the angle is a right angle. (Hint: There are two parts.) 44. Prove that if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. (Hint: There are two parts.) 45. Find m ⁀ PQ to the nearest degree. 46. Find m∠ABD. 47. Construction To circumscribe an equilateral triangle ̶̶ AB parallel to the horizontal about a circle, construct diameter of the circle and tangent to the circle. Then use a 30°-60°-90° triangle to draw 60° angles with ̶̶ AB and are tangent to the circle. ̶̶ BC so that they form ̶̶ AC and SPIRAL REVIEW 48. Tickets for a play cost $15.00 for section C, $22.50 for section B, and $30.00 for section A. Amy spent a total of $255.00 for 12 tickets. If she spent the same amount on section C tickets as section A tickets, how many tickets for section B did she purchase? (Previous course) Write a ratio expressing the slope of the line through each pair of points. (Lesson 7-1) 49. (4 1 _, -6) and (8, 1 _ ) 2 2 50. (-9, -8) and (0, -2) 51. (3, -14) and (11, 6) Find each of the following. (Lesson 11-2) 52. m ⁀ ST 53. area of △ABD Construction Tangent to a Circle From an Exterior Point     Draw ⊙C and locate P in the exterior of the circle. ̶̶ CP. Construct M, Draw the midpoint of ̶̶ CP. 1.
Can you draw ̶̶ CR ⊥   RP? Explain. Center the compass at M. Draw a circle through C and P. It will intersect ⊙C at R and S. R and S are the tangent points. Draw   PR and    PS tangent to ⊙C. 11-4 Inscribed Angles 779 779 �� 11-5 Use with Lesson 11-5 Activity 1 Explore Angle Relationships in Circles In Lesson 11-4, you learned that the measure of an angle inscribed in a circle is half the measure of its intercepted arc. Now you will explore other angles formed by pairs of lines that intersect circles. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures …. Also G.1.A, G.2.B, G.3.B, G.4.A, G.5.A, G.5.B, G.9.C 1 Create a circle with center A. Label the point on the circle as B. Create a radius segment from A to a new point C on the circle. 2 Construct a line through C perpendicular to ̶̶ radius AC. Create a new point D on this line, which is tangent to circle A at C. Hide radius ̶̶ AC. 3 Create a new point E on the circle and then ̶̶ CE. construct secant 4 Measure ∠DCE and measure ⁀ CBE. (Hint: To measure an arc in degrees, select the three points and the circle and then choose Arc Angle from the Measure menu.) 5 Drag E around the circle and examine the changes in the measures. Fill in the angle and arc measures in a chart like the one below. Try to create acute, right, and obtuse angles. Can you make a conjecture about the relationship between the angle measure and the arc measure? m∠DCE m ⁀ CBE Angle Type Activity 2 1 Construct a new circle with two secants   CD and   EF that intersect inside the circle at G. 2 Create two new points H and I that are on the circle as shown. These will be used to measure
the arcs. Hide B if desired. (It controls the circle’s size.) 3 Measure ∠DGF formed by the secant lines and measure ⁀ CHE and ⁀ DIF. 4 Drag F around the circle and examine the changes in measures. Be sure to keep H between C and E and I between D and F for accurate arc measurement. Move them if needed. 780 780 Chapter 11 Circles 5 Fill in the angle and arc measures in a chart like the one below. Try to create acute, right, and obtuse angles. Can you make a conjecture about the relationship between the angle measure and the two arc measures? m∠DGF m ⁀ CHE m ⁀ DIF Sum of Arcs Activity 3 1 Use the same figure from Activity 2. Drag points around the circle so that the intersection G is now outside the circle. Move H so it is between E and D and I is between C and F, as shown. 2 Measure ∠FGC formed by the secant lines and measure ⁀ CIF and ⁀ DHE. 3 Drag points around the circle and examine the changes in measures. Fill in the angle and arc measures in a chart like the one below. Can you make a conjecture about the relationship between the angle measure and the two arc measures? m∠FGC m ⁀ CIF m ⁀ DHE Number of Arcs Try This 1. How does the relationship you observed in Activity 1 compare to the relationship between an inscribed angle and its intercepted arc? 2. Why do you think the radius ̶̶ AC is needed in Activity 1 for the construction of the tangent line? What theorem explains this? 3. In Activity 3, try dragging points so that the secants become tangents. What conclusion can you make about the angle and arc measures? 4. Examine the conjectures and theorems about the relationships between angles and arcs in a circle. What is true of an angle with a vertex on the circle? What is true of an angle with a vertex inside the circle? What is true of an angle with a vertex outside the circle? Summarize your findings. 5. Does using geometry software to compare angle and arc measures constitute a formal proof of the relationship observed? 11- 5 Technology Lab 781 781 11-5 Angle Relationships in Circles TEKS G.5.B Geometric paterns: use … patterns to make generalizations about geometric
properties, including … angle relationships in … circles. Also G.1.A, G.2.B, G.5.A, G.9.C Objectives Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems. Who uses this? Circles and angles help optometrists correct vision problems. (See Example 4.) Theorem 11-5-1 connects arc measures and the measures of tangent-secant angles with tangent-chord angles. Theorem 11-5-1 THEOREM HYPOTHESIS CONCLUSION If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. m∠ABC = 1 _ m ⁀ AB 2 Tangent   BC and secant   BA intersect at B. You will prove Theorem 11-5-1 in Exercise 45. E X A M P L E 1 Using Tangent-Secant and Tangent-Chord Angles Find each measure. A m∠BCD m∠BCD = 1 _ m ⁀ BC 2 m∠BCD = 1 _ (142°) 2 = 71° B m ⁀ ABC m∠ACD = 1 _ m ⁀ ABC 2 90° = 1 _ m ⁀ ABC 2 180° = m ⁀ ABC Find each measure. 1a. m∠STU 1b. m ⁀ SR 782 782 Chapter 11 Circles �� Theorem 11-5-2 THEOREM HYPOTHESIS CONCLUSION If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. (m ⁀ AB + m ⁀ CD ) m∠1 = 1 _ 2 ̶ AD and ̶ BC Chords intersect at E. PROOF PROOF Theorem 11-5-2 ̶ ̶ BC intersect at E. AD and Given: (m ⁀ AB + m ⁀ CD ) Prove: m∠1 = 1 _ 2 Proof: Statements Reasons 1. ̶ AD and ̶ BD. 2. Draw ̶ BC intersect at E. 3
. m∠1 = m∠EDB + m∠EBD 4. m∠EDB = 1 __ m ⁀ AB, 2 m∠EBD = 1 __ m ⁀ CD 2 m ⁀ AB + 1 __ 5. m∠1 = 1 __ m ⁀ CD 2 2 (m ⁀ AB + m ⁀ CD ) 6. m∠1 = 1 __ 2 1. Given 2. Two pts. determine a line. 3. Ext. ∠ Thm. 4. Inscribed ∠ Thm. 5. Subst. 6. Distrib. Prop. E X A M P L E 2 Finding Angle Measures Inside a Circle Find each angle measure. m∠SQR (m ⁀ PT + m ⁀ SR ) m∠SQR = 1 _ 2 = 1 _ (32° + 100° ) 2 = 1 _ (132°) 2 = 66° Find each angle measure. 2a. m∠ABD 2b. m∠RNM 11-5 Angle Relationships in Circles 783 783 �� Theorem 11-5-3 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. (m ⁀ AD - m ⁀ BD ) m∠2 = 1 _ m∠1 = 1 _ 2 2 (m ⁀ EHG - m ⁀ EG ) m∠3 = 1 _ 2 (m ⁀ JN - m ⁀ KM ) You will prove Theorem 11-5-3 in Exercises 34–36. E X A M P L E 3 Finding Measures Using Tangents and Secants Find the value of x. A B EHG and ⁀ EG joined ⁀ together make a whole circle. So m ⁀ EHG = 360° - 132° = 228° (m ⁀RS - m ⁀QS ) (174° - 98°) x = 1_ 2 = 1_ 2 = 38° 3. Find the value of x. E X A M P L E 4 Biology Application (m ⁀EHG - m ⁀EG ) (
228° - 132°) x = 1_ 2 = 1_ 2 = 48° When a person is farsighted, light rays enter the eye and are focused behind the retina. In the eye shown, light rays converge at R. If m ⁀PS = 60° and m ⁀QT = 14°, what is m∠PRS? (m ⁀PS - m ⁀QT ) m∠PRS = 1_ 2 = 1_ 2 = 1_ 2 (60° - 14° ) (46°) = 23° 4. Two of the six muscles that control eye movement are attached to the eyeball and intersect behind the eye. If m ⁀AEB = 225°, what is m∠ACB? 784 784 Chapter 11 Circles �� Angle Relationships in Circles VERTEX OF THE ANGLE MEASURE OF ANGLE On a circle Half the measure of its intercepted arc DIAGRAMS m∠1 = 60° m∠2 = 100° (44° + 86° ) m∠1 = 1 _ 2 = 65° Inside a circle Half the sum of the measures of its intercepted arcs Outside a circle Half the difference of the measures of its intercepted arcs (202° - 78°) m∠1 = 1_ 2 = 62° (125° - 45°) m∠2 = 1_ 2 = 40° E X A M P L E 5 Finding Arc Measures Find m ⁀ AF. Step 1 Find m ⁀ ADB. m∠ABC = 1 _ m ⁀ ADB 2 110° = 1 _ m ⁀ ADB 2 m ⁀ ADB = 220° Step 2 Find m ⁀ AD. If a tangent and secant intersect on a ⊙ at the pt. of tangency, then the measure of the ∠ formed is half the measure of its intercepted arc. Substitute 110 for m∠ABC. Mult. both sides by 2. ADB = m ⁀ AD + m ⁀ DB Arc Add. Post. m ⁀ 220° = m ⁀ AD + 160° m ⁀ AD = 60° Substitute. Subtract 160 from both sides. Step 3 Find m ⁀ AF
. m ⁀ AF = 360° - (m ⁀ AD + m ⁀ DB + m ⁀ BF ) = 360° - (60° + 160° + 48° ) = 92° Def. of a ⊙ Substitute. Simplify. 5. Find m ⁀ LP. 11-5 Angle Relationships in Circles 785 785 �� THINK AND DISCUSS 1. Explain how the measure of an angle formed by two chords of a circle is related to the measure of the angle formed by two secants. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box write a theorem and draw a diagram according to where the angle’s vertex is in relationship to the circle. 11-5 Exercises Exercises GUIDED PRACTICE Find each measure. p. 782 1. m∠DAB 2. m ⁀ AC 3. m ⁀ PN 4. m∠MNP KEYWORD: MG7 11-5 KEYWORD: MG7 Parent. m∠STU 6. m∠HFG 7. m∠NPK p. 783 Find the value of x. p. 784 8. 9. 10. 784 11. Science A satellite orbits Mars. When it reaches S it is about 12,000 km above the planet. How many arc degrees of the planet are visible to a camera in the satellite? 786 786 Chapter 11 Circles ��ABCSx°38°ge07se_c11l05004aAB��. 785 Multi-Step Find each measure. 12. m ⁀ DF 13. m ⁀ CD 14. m ⁀ PN 15. m ⁀ KN PRACTICE AND PROBLEM SOLVING Find each measure. 16. m∠BCD 17. m∠ABC 18. m∠XZW 19. m ⁀ XZV Independent Practice For See Exercises Example 16–19 20–22 23–25 26 27–30 1 2 3 4 5 TEKS TEKS TAKS TAKS 20. m∠QPR 21. m∠ABC 22
. m∠MKJ Skills Practice p. S25 Application Practice p. S38 Find the value of x. 23. 24. 25. Archaeology Outside of Hunt, Texas, is a replica of Stonehenge. It is 60 percent as tall as the original and 90 percent as large in circumference. 26. Archaeology Stonehenge is a circular arrangement of massive stones near Salisbury, England. A viewer at V observes the monument from a point where two of the stones A and B are aligned with stones at the endpoints of a diameter of the circular shape. Given that m ⁀ AB = 48°, what is m∠AVB? Multi-Step Find each measure. 27. m ⁀ EG 28. m ⁀ DE 29. m ⁀ PR 30. m ⁀ LP 11-5 Angle Relationships in Circles 787 787 �� In the diagram, m∠ABC = x °. Write an expression in terms of x for each of the following. 31. m ⁀ AB 33. m ⁀ AEB 32. m∠ABD 34. Given: Tangent CD and secant CA (m ⁀ AD - m ⁀ BD ) Prove: m∠ACD = 1 _ 2 Plan: Draw auxiliary line segment ̶ BD. Use the Exterior Angle Theorem to show that m∠ACD = m∠ABD - m∠BDC. Then use the Inscribed Angle Theorem and Theorem 11-5-1. 35. Given: Tangents  FG (m ⁀ Prove: m∠EFG = 1 _ 2  FE and EHG - m ⁀ EG ) 36. Given: Secants ̶ LN (m ⁀ JN - m ⁀ KM ) Prove: m∠JLN = 1 _ 2 ̶ LJ and 37. Critical Thinking Suppose two secants intersect in the exterior of a circle as shown. What is greater, m∠1 or m∠2? Justify your answer
. 38. Write About It The diagrams show the intersection of perpendicular lines on a circle, inside a circle, and outside a circle. Explain how you can use these to help you remember how to calculate the measures of the angles formed. Algebra Find the measures of the three angles of △ABC. 39. 40. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 806. The design was made by placing six equally-spaced points on a circle and connecting them. a. Find m∠BHC. b. Find m∠EGD. c. Classify △EGD by its angle measures and by its side lengths. 788 788 Chapter 11 Circles ��ABEDCFge07sec11l05007aABHG 42. What is m∠DCE? 19° 21° 79° 101° 43. Which expression can be used to calculate m∠ABC? 1 _ (m ⁀ DE - m ⁀ AF ) 2 1 _ (m ⁀ AD - m ⁀ AF ) 2 1 _ (m ⁀ AD + m ⁀ AF ) 2 1 _ (m ⁀ DE + m ⁀ AF ) 2 44. Gridded Response In ⊙Q, m ⁀ MN = 146° and m∠JLK = 45°. Find the degree measure of ⁀ JK. CHALLENGE AND EXTEND 45. Prove Theorem 11-5-1.  BC and secant Given: Tangent Prove: m∠ABC = 1 _ m ⁀ AB 2 (Hint: Consider two cases, one where  BA ̶ AB is a diameter and one where ̶ AB is not a diameter.) ̶ WZ are tangent to ⊙X. m ⁀ WY = 90° 46. Given: ̶ YZ and Prove: WXYZ is a square. 47. Find x. 48. Find m ⁀ GH. SPIRAL REVIEW Determine whether the ordered pair (7, -8) is a solution of the following functions. (Previous course) 49. g (x) = 2 x 2 - 15x - 1
50. f (x) = 29 - 3x 51. y = - 7 _ x 8 Find the volume of each pyramid or cone. Round to the nearest tenth. (Lesson 10-7) 52. regular hexagonal pyramid with a base edge of 4 m and a height of 7 m 53. right cone with a diameter of 12 cm and lateral area of 60π cm 2 54. regular square pyramid with a base edge of 24 in. and a surface area of 1200 in 2 In ⊙P, find each angle measure. (Lesson 11-4) 55. m∠BCA 56. m∠DBC 57. m∠ADC 11-5 Angle Relationships in Circles 789 789 �� 11-6 Use with Lesson 11-6 Activity 1 Explore Segment Relationships in Circles When secants, chords, or tangents of circles intersect, they create several segments. You will measure these segments and investigate their relationships. TEKS G.9.C Congruence and the geometry of size: formulate … conjectures about … circles and the lines that intersect them …. Also G.2.A, G.2.B, G.3.B, G.5.A KEYWORD: MG7 Lab11 1 Construct a circle with center A. Label the point on the circle as B. Construct two secants CD and   EF that intersect outside the circle at G. Hide B if desired. (It controls the circle’s size.) 2 Measure ̶̶ GC, ̶̶̶ GD, ̶̶ GE, and around the circle and examine the changes in the measurements. ̶̶ GF. Drag points 3 Fill in the segment lengths in a chart like the one below. Find the products of the lengths of segments on the same secant. Can you make a conjecture about the relationship of the segments formed by intersecting secants of a circle? GC GD GC ⋅ GD GE GF GE ⋅ GF Try This 1. Make a sketch of the diagram from Activity 1, ̶̶ DE to create △CFG and ̶̶ CF and and create △EDG as shown. 2. Name pairs of congruent angles in the diagram. How are △CFG and △
EDG related? Explain your reasoning. 3. Write a proportion involving sides of the triangles. Cross-multiply and state the result. What do you notice? Activity 2 1 Construct a new circle with center A. Label the point on the circle as B. Create a radius segment from A to a new point C on the circle. 2 Construct a line through C perpendicular to radius point D on this line, which is tangent to circle A at C. Hide radius ̶̶ AC. Create a new ̶̶ AC. 790 790 Chapter 11 Circles 3 Create a secant line through D that intersects the circle at two new points E and F, as shown. 4 Measure ̶̶ DC, ̶̶ DE, and ̶̶ DF. Drag points around the circle and examine the changes in the measurements. Fill in the measurements in a chart like the one below. Can you make a conjecture about the relationship between the segments of a tangent and a secant of a circle? DE DF DE ⋅ DF DC? Try This 4. How are the products for a tangent and a secant similar to the products for secant segments? 5. Try dragging E and F so they overlap (to make the secant segment look like a tangent segment). What do you notice about the segment lengths you measured in Activity 2? Can you state a relationship about two tangent segments from the same exterior point? 6. Challenge Write a formal proof of the relationship you found in Problem 2. Activity 3 1 Construct a new circle with two chords that intersect inside the circle at G. ̶̶ CD and ̶̶ EF 2 Measure ̶̶ GC, ̶̶̶ GD, ̶̶ GE, and the circle and examine the changes in the measurements. ̶̶ GF. Drag points around 3 Fill in the segment lengths in a chart like the ones used in Activities 1 and 2. Find the products of the lengths of segments on the same chord. Can you make a conjecture about the relationship of the segments formed by intersecting chords of a circle? Try This 7. Connect the endpoints of the chords to form two triangles. Name pairs of congruent angles. How are the two triangles that are formed related? Explain your reasoning. 8. Examine the conclusions you made in all three activities about segments formed by secants, chords, and tangents in a circle. Summarize your findings. 11- 6 Technology Lab 791 791 11-6
Segment Relationships in Circles TEKS G.5.A Geometric patterns: use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Also G.1.A, G.2.B Objectives Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems. Vocabulary secant segment external secant segment tangent segment Who uses this? Archaeologists use facts about segments in circles to help them understand ancient objects. (See Example 2.) In 1901, divers near the Greek island of Antikythera discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. The following theorem describes the relationship among the four segments that are formed when two chords intersect in the interior of a circle. Theorem 11-6-1 Chord-Chord Product Theorem THEOREM HYPOTHESIS CONCLUSION If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. AE ⋅ EB = CE ⋅ ED ̶̶ AB and ̶̶ CD Chords intersect at E. You will prove Theorem 11-6-1 in Exercise 28. E X A M P L E 1 Applying the Chord-Chord Product Theorem Find the value of x and the length of each chord. PQ ⋅ QR = SQ ⋅ QT 6 (4) = x ( 8) 24 = 8x 3 = x PR = 6 + 4 = 10 ST = 3 + 8 = 11 1. Find the value of x and the length of each chord. 792 792 Chapter 11 Circles �� E X A M P L E 2 Archaeology Application Archaeologists discovered a fragment of an ancient disk. To calculate its original diameter, they drew a chord ̶̶ PQ. Find the disk’s diameter. bisector ̶̶̶ PQ is the perpendicular bisector of ̶̶ PR is a diameter of the disk. ̶̶ AB and its perpendicular Since a chord, AQ ⋅ QB = PQ ⋅ QR 5 (5) = 3 (QR) 25 = 3QR 8 1 _ 3 in. = QR = 11 1 _ PR = 3 + 8 1 _ 3 3 in. 2. What if…? Suppose the length of chord ̶̶ AB that the archaeologists
drew was 12 in. In this case how much longer is the disk’s diameter compared to the disk in Example 2? A secant segment is a segment of a secant with at least one endpoint on the circle. An external secant segment is a secant segment that lies in the exterior of the circle with one endpoint on the circle. ̶̶̶ NM, ̶̶̶ KM, and ̶̶ ̶̶̶ PM, JM are secant segments of ⊙Q. ̶̶ ̶̶̶ NM and JM are external secant segments. Theorem 11-6-2 Secant-Secant Product Theorem THEOREM HYPOTHESIS CONCLUSION If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole ⋅ outside = whole ⋅ outside) ̶̶ AE and ̶̶ CE Secants intersect at E. AE ⋅ BE = CE ⋅ DE PROOF PROOF Secant-Secant Product Theorem Given: Secant segments Prove: AE ⋅ BE = CE ⋅ DE ̶̶ AE and ̶̶ CE Proof: Draw auxiliary line segments ̶̶ AD and ̶̶ CB. ∠EAD and ∠ECB both intercept ⁀ BD, so ∠EAD ≅ ∠ECB. ∠E ≅ ∠E by the Reflexive Property of ≅. Thus △EAD ∼ △ECB by AA Similarity. Therefore corresponding sides are proportional, and AE ___ = DE ___ BE CE AE ⋅ BE = CE ⋅ DE.. By the Cross Products Property, 11-6 Segment Relationships in Circles 793 793 3 in.5 in.PBARQge07sec11l06002a�� E X A M P L E 3 Applying the Secant-Secant Product Theorem Find the value of x and the length of each secant segment. RT ⋅ RS = RQ ⋅ RP 10 (4) = (x + 5) 5 40 = 5x + 25 15 = 5x 3 = x RT = 4 + 6 = 10 RQ = 5 + 3 = 8 3. Find the value of z and the length of
each secant segment. A tangent segment is a segment of a tangent with one endpoint on the circle. ̶̶ ̶̶ AC are tangent segments. AB and Theorem 11-6-3 Secant-Tangent Product Theorem THEOREM HYPOTHESIS CONCLUSION If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole ⋅ outside = tangent 2 ) AC ⋅ BC = DC 2 ̶̶ AC and tangent ̶̶ DC Secant intersect at C. You will prove Theorem 11-6-3 in Exercise 29. E X A M P L E 4 Applying the Secant-Tangent Product Theorem Find the value of x. SQ ⋅ RQ = PQ 2 9 (4) = x 2 36 = x 2 ±6 = x The value of x must be 6 since it represents a length. 4. Find the value of y. 794 794 Chapter 11 Circles �� THINK AND DISCUSS 1. Does the Chord-Chord Product Theorem apply when both chords are diameters? If so, what does the theorem tell you in this case? 2. Given A in the exterior of a circle, how many different tangent segments can you draw with A as an endpoint? 3. GET ORGANIZED Copy and complete the graphic organizer. 11-6 Exercises Exercises KEYWORD: MG7 11-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary   AB intersects ⊙P at exactly one point. Point A is in the exterior ̶̶ AB is a(n)?. (tangent segment or external ̶̶̶̶ of ⊙P, and point B lies on ⊙P. secant segment Find the value of the variable and the length of each chord. p. 792 2. 3. 4. Engineering A section of an aqueduct p. 793 is based on an arc of a circle as shown. ̶̶̶ ̶̶ EF is the perpendicular bisector of GH. GH = 50 ft, and EF = 20 ft. What is the diameter of the circle Find the value of
the variable and the length of each secant segment. p. 794 6. 7. 8. 11-6 Segment Relationships in Circles 795 795 �� Find the value of the variable. p. 794 9. 10. 11. Independent Practice For See Exercises Example 12–14 15 16–18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Find the value of the variable and the length of each chord. 12. 13. 14. 15. Geology Molokini is a small, crescent- shaped island 2 1 __ 2 miles from the Maui coast. It is all that remains of an extinct volcano. To approximate the diameter of the mouth of the volcano, a geologist can use a diagram like the one shown. What is the approximate diameter of the volcano’s mouth to the nearest foot? Find the value of the variable and the length of each secant segment. 16. 17. 18. Find the value of the variable. 19. 20. 21. Use the diagram for Exercises 22 and 23. 22. M is the midpoint of ̶̶ PQ. RM = 10 cm, and PQ = 24 cm. a. Find MS. b. Find the diameter of ⊙O. ̶̶ PQ.The diameter of ⊙O is 13 in., 23. M is the midpoint of and RM = 4 in. a. Find PM. b. Find PQ. 796 796 Chapter 11 Circles �� Multi-Step Find the value of both variables in each figure. 24. 25. 26. Meteorology A weather satellite S orbits Earth at a distance SE of 6000 mi. Given that the diameter of the earth is approximately 8000 mi, what is the distance from the satellite to P? Round to the nearest mile. 27. /////ERROR ANALYSIS///// The two solutions show how to find the value of x. Which solution is incorrect? Explain the error. Meteorology Satellites are launched to an area above the atmosphere where
there is no friction. The idea is to position them so that when they fall back toward Earth, they fall at the same rate as Earth’s surface falls away from them. 28. Prove Theorem 11-6-1. ̶̶ AB and Given: Chords Prove: AE ⋅ EB = CE ⋅ ED ̶̶ CD intersect at point E. Plan: Draw auxiliary line segments ̶̶ AC and ̶̶ BD. Show that △ECA ∼ △EBD. Then write a proportion comparing the lengths of corresponding sides. 29. Prove Theorem 11-6-3. Given: Secant segment Prove: AC ⋅ BC = DC 2 ̶̶ AC, tangent segment ̶̶ DC 30. Critical Thinking A student drew a circle and two secant segments. By measuring with a ruler, he found ̶̶ PQ ≅ the student’s conclusion? Why or why not? ̶̶ PS. He concluded that ̶̶ ST. Do you agree with ̶̶ QR ≅ 31. Write About It The radius of ⊙A is 4. CD = 4, ̶̶ CB is a tangent segment. Describe two different and methods you can use to find BC. 32. This problem will prepare you for the Multi-Step TAKS Prep on page 806. Some Native American designs are based on eight points that are placed around the circumference of a circle. In ⊙O, BE = 3 cm. AE = 5.2 cm, and EC = 4 cm. a. Find DE to the nearest tenth. b. What is the diameter of the circle to the nearest tenth? c. What is the length of ̶̶ OE to the nearest hundredth? 11-6 Segment Relationships in Circles 797 797 ��ge07se_c11l06005aABSEP�� 33. Which of these is closest to the length of tangent ̶̶ PQ? 6.9 9.2 9.9 10.6 34. What is the length of ̶̶ UT? 5 7 12 14 35. Short Response In ⊙A, ̶̶ AB is the
perpendicular ̶̶ CD. CD = 12, and EB = 3. Find the radius bisector of of ⊙A. Explain your steps. CHALLENGE AND EXTEND 36. Algebra ̶̶ KL is a tangent segment of ⊙N. 37. a. Find the value of x. b. Classify △KLM by its angle measures. Explain. ̶̶ PQ is a tangent segment of a circle with radius 4 in. Q lies on the circle, and PQ = 6 in. Find the distance from P to the circle. Round to the nearest tenth of an inch. 38. The circle in the diagram has radius c. Use this diagram and the Chord-Chord Product Theorem to prove the Pythagorean Theorem. 39. Find the value of y to the nearest hundredth. SPIRAL REVIEW 40. An experiment was conducted to find the probability of rolling two threes in a row on a number cube. The probability was 3.5%. How many trials were performed in this experiment if 14 favorable outcomes occurred? (Previous course) 41. Two coins were flipped together 50 times. In 36 of the flips, at least one coin landed heads up. Based on this experiment, what is the experimental probability that at least one coin will land heads up when two coins are flipped? (Previous course) Name each of the following. (Lesson 1-1) 42. two rays that do not intersect 43. the intersection of  AC and  CD 44. the intersection of  CA and  BD Find each measure. Give your answer in terms of π and rounded to the nearest hundredth. (Lesson 11-3) 45. area of the sector XZW 46. arc length of ⁀ XW 47. m∠YZX if the area of the sector YZW is 40π ft 2 798 798 Chapter 11 Circles �� 11-7 Circles in the Coordinate Plane TEKS G.2.B Geometric structure: make conjectures about … circles … choosing from a variety of approaches such as coordinate …. Also G.1.A, G.4.A, G.5.A Objectives Write equations and graph circles in
the coordinate plane. Use the equation and graph of a circle to solve problems. Who uses this? Meteorologists use circles and coordinates to plan the location of weather stations. (See Example 3.) The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center = √  ( = √  (x - h ) 2 + (y - k ) 2 r 2 = (x - h) 2 + (y - k) 2 Distance Formula Substitute the given values. Square both sides. Theorem 11-7-1 Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h) 2 + (y - k Writing the Equation of a Circle Write the equation of each circle. A ⊙A with center A (4, -2) and radius 3 (x - 4) 2 + (y - (-2) ) (x - h) 2 + (y - kx - 4) 2 + (y + 2) 2 = 9 Equation of a circle Substitute 4 for h, -2 for k, and 3 for r. Simplify. B ⊙B that passes through (-2, 6) and has center B (-6, 3) r =  2 √ ( -2 - (-6) ) + (6 - 3) 2 Distance Formula = √  25 = 5 2 (x - (-6) ) Simplify. + (y - 3) 2 = 5 2 (x + 6) 2 + (y - 3) 2 = 25 Simplify. Substitute -6 for h, 3 for k, and 5 for r. Write the equation of each circle. 1a. ⊙P with center P (0, -3) and radius 8 1b. ⊙Q that passes through (2, 3) and has center Q (2, -1) 11-7 Circles in the Coordinate Plane 799 799 �� If you are given the equation of
a circle, you can graph the circle by making a table or by identifying its center and radius. E X A M P L E 2 Graphing a Circle Graph each equation. x 2 + y 2 = 25 Step 1 Make a table of values. A Since the radius is √  25, or 5, use ±5 and the values between for x-values. x y -5 -4 -3 0 3 4 0 ±3 ±4 ±5 ±4 ±3 5 0 Step 2 Plot the points and connect them to form a circle. Always compare the equation to the form (x - h) 2 + (y - k) 2 = r 2. B (x + 1) 2 + (y - 2) 2 = 9 The equation of the given circle can be written as (x - (-1) ) + (y - 2) 2 = 3 2. So h = -1, k = 2, and r = 3. The center is (-1, 2), and the radius is 3. Plot the point (-1, 2). Then graph a circle having this center and radius 3. 2 Graph each equation. 2a. x 2 + y 2 = 9 2b. (x - 3) 2 + (y + 2) 2 = 4 Graphing Circles I found a way to use my calculator to graph circles. You first need to write the circle’s equation in y = form. For example, to graph x 2 + y 2 = 16, first solve for y. y 2 = 16 - x 2 y = ± √  16 - x 2 Christina Avila Crockett High School Now enter and graph the two equations y 1 = √  16 - x 2 and y 2 = - √  16 - x 2. 800 800 Chapter 11 Circles �� E X A M P L E 3 Meteorology Application Meteorologists are planning the location of a new weather station to cover Osceola, Waco, and Ireland, Texas. To optimize radar coverage, the station must be equidistant from the three cities which are located on a coordinate plane at A (2, 5), B (3, -2), and C (-5, -2). a. What are the coordinates where the station should be built? b.
If each unit of the coordinate plane represents 8.5 miles, what is the diameter of the region covered by the radar? The perpendicular bisectors of a triangle are concurrent at a point equidistant from each vertex. Step 1 Plot the three given points. Step 2 Connect A, B, and C to form a triangle. Step 3 Find a point that is equidistant from the three points by constructing the perpendicular bisectors of two of the sides of △ABC. The perpendicular bisectors of the sides of △ABC intersect at a point that is equidistant from A, B, and C. The intersection of the perpendicular bisectors is P (-1, 1). P is the center of the circle that passes through A, B, and C. The weather station should be built at P (-1, 1), Clifton, Texas. There are approximately 10 units across the circle. So the diameter of the region covered by the radar is approximately 85 miles. 3. What if…? Suppose the coordinates of the three cities in Example 3 are D (6, 2), E (5, -5), and F (-2, -4). What would be the location of the weather station? THINK AND DISCUSS 1. What is the equation of a circle with radius r whose center is at the origin? 2. A circle has a diameter with endpoints (1, 4) and (-3, 4). Explain how you can find the equation of the circle. 3. Can a circle have a radius of -6? Justify your answer. 4. GET ORGANIZED Copy and complete the graphic organizer. First select values for a center and radius. Then use the center and radius you wrote to fill in the other circles. Write the corresponding equation and draw the corresponding graph. 11-7 Circles in the Coordinate Plane 801 801 1800200020001800xyBCAWacoIrelandOsceolaxyBCAWacoIrelandOsceolaHolt, Rinehart & WinstonGeometry © 2007ge07sec11107003a Texas Grid map3rd proof1800200020001800xyPBCAWacoIrelandOsceolaCliftonxyPBCAWacoIrelandOsceolaCliftonHolt, Rinehart & WinstonGeometry © 2007ge07sec11107004a Texas Grid map3rd proof�� 11-7 Exercises Exercises
GUIDED PRACTICE Write the equation of each circle. p. 799 1. ⊙A with center A (3, -5) and radius 12 2. ⊙B with center B (-4, 0) and radius 7 KEYWORD: MG7 11-7 KEYWORD: MG7 Parent 3. ⊙M that passes through (2, 0) and that has center M (4, 0) 4. ⊙N that passes through (2, -2) and that has center N (-1, 2 Multi-Step Graph each equation. p. 800 5. (x - 3) 2 + (y - 3) 2 = 4 7. (x + 3) 2 + (y + 4) 2 = 1 6. (x - 1) 2 + (y + 2) 2 = 9 8. (x - 3) 2 + (y + 4) 2 = 16. 801 9. Communications A radio antenna tower is kept perpendicular to the ground by three wires of equal length. The wires touch the ground at three points on a circle whose center is at the base of the tower. The wires touch the ground at A (2, 6), B (-2, -2), and C (-5, 7). a. What are the coordinates of the base of the tower? b. Each unit of the coordinate plane represents 1 ft. What is the diameter of the circle? PRACTICE AND PROBLEM SOLVING Independent Practice Write the equation of each circle. For See Exercises Example 10. ⊙R with center R (-12, -10) and radius 8 10–13 14–17 18 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 11. ⊙S with center S (1.5, -2.5) and radius √  3 12. ⊙C that passes through (2, 2) and that has center C (1, 1) 13. ⊙D that passes through (-5, 1) and that has center D (1, -2) 13. 15. (x + 1) 2 - y 2 = 16 17. x 2 + (y + 2) 2 = 4 Multi-Step Graph each equation. 14. x 2 + (y - 2) 2 = 9 16. x 2 + y 2 = 100 18.
Anthropology Hundreds of stone circles can be found along the Gambia River in western Africa. The stones are believed to be over 1000 years old. In one of the circles at Ker Batch, three stones have approximate coordinates of A (3, 1), B (4, -2), and C (-6, -2). a. What are the coordinates of the center of the stone circle? b. Each unit of the coordinate plane represents 1 ft. What is the diameter of the stone circle? 802 802 Chapter 11 Circles Entertainment The wooden carousel at Fair Park in Dallas, Texas, was manufactured by Gustav Dentzel and his family around 1914, It has 50 elaborately carved jumping horses, 16 standing horses, and 2 chariots. Algebra Write the equation of each circle. 19. 20. 21. Entertainment In 2004, the world’s largest carousel was located at the House on the Rock, in Spring Green, Wisconsin. Suppose that the center of the carousel is at the origin and that one of the animals on the circumference of the carousel has coordinates (24, 32). a. If one unit of the coordinate plane equals 1 ft, what is the diameter of the carousel? b. As the carousel turns, the animals follow a circular path. Write the equation of this circle. Determine whether each statement is true or false. If false, explain why. 22. The circle x 2 + y 2 = 7 has radius 7. 23. The circle (x - 2) 2 + (y + 3) 2 = 9 passes through the point (-1, -3). 24. The center of the circle (x - 6)2 + (y + 4)2 = 1 lies in the second quadrant. 25. The circle (x + 1) 2 + (y - 4) 2 = 4 intersects the y-axis. 26. The equation of the circle centered at the origin with diameter 6 is x 2 + y 2 = 36. 27. Estimation You can use the graph of a circle to estimate its area. a. Estimate the area of the circle by counting the number of squares of the coordinate plane contained in its interior. Be sure to count partial squares. b. Find the radius of the circle. Then use the area formula to calculate the circle’s area to the nearest tenth. c. Was your estimate in part a an overestimate or an underestimate? 28. Consider the circle whose equation is (x -
4) 2 + (y + 6) 2 = 25. Write, in point-slope form, the equation of the line tangent to the circle at (1, -10). 29. This problem will prepare you for the Multi-Step TAKS Prep on page 806. A hogan is a traditional Navajo home. An artist is using a coordinate plane to draw the symbol for a hogan. The symbol is based on eight equally spaced points placed around the circumference of a circle. a. She positions the symbol at A (-3, 5) and C (0, 2). What are the coordinates of E and G? b. What is the length of a diameter of the symbol? c. Use your answer from part b to write an equation of the circle. 11-7 Circles in the Coordinate Plane 803 803 �� Geology Geology The New Madrid earthquake of 1811 was the largest earthquake known in American history. Large areas sank into the earth, new lakes were formed, forests were destroyed, and the course of the Mississippi River was changed. Find the center and radius of each circle. 30. (x - 2)2 + (y + 3)2 = 81 31. x 2 + (y + 15)2 = 25 32. (x + 1)2 + y 2 = 7 Find the area and circumference of each circle. Express your answer in terms of π. Find the area and circumference of each circle. Express your answer in terms of 33. circle with equation (x + 2) 2 + (y - 7) 2 = 9 33. 34. circle with equation (x - 8)2 + (y + 5)2 = 7 34. 34. 35. circle with center (-1, 3) that passes through (2, -1) 36. Critical Thinking Describe the graph of the equation x 2 + y 2 = r 2 when r = 0. 37. Geology A seismograph measures ground motion during an earthquake. To find the epicenter of an earthquake, scientists take readings in three different locations. Then they draw a circle centered at each location. The radius of each circle is the distance the earthquake is from the seismograph. The intersection of the circles is the epicenter. Use the data below to find the epicenter of the New Madrid earthquake. Seismograph Location Distance to Earthquake A B C (-200, 200) (400, -100)
(100, -500) 300 mi 600 mi 500 mi 38. For what value(s) of the constant k is the circle x 2 + (y - k) 2 = 25 tangent to the x-axis? 39. ⊙A has a diameter with endpoints (-3, -2) and (5, -2). Write the equation of ⊙A. 40. Recall that a locus is the set of points that satisfy a given condition. Draw and describe the locus of points that are 3 units from (2, 2). 41. Write About It The equation of ⊙P is (x - 2) 2 + (y - 1) 2 = 9. Without graphing, explain how you can determine whether the point (3, -1) lies on ⊙P, in the interior of ⊙P, or in the exterior of ⊙P. 42. Which of these circles intersects the x-axis? (x - 3) 2 + (y + 3) 2 = 4 (x + 1) 2 + (y - 4) 2 = 9 (x + 2) 2 + (y + 1) 2 = 1 (x + 1) 2 + (y + 4) 2 = 9 43. What is the equation of a circle with center (-3, 5) that passes through the point (1, 5)? (x + 3) 2 + (y - 5) 2 = 4 (x - 3) 2 + (y + 5) 2 = 4 (x + 3) 2 + (y - 5) 2 = 16 (x - 3) 2 + (y + 5) 2 = 16 44. On a map of a park, statues are located at (4, -2), (-1, 3), and (-5, -5). A circular path connects the three statues, and the circle has a fountain at its center. Find the coordinates of the fountain. (-1, -2) (2, 1) (-2, 1) (1, -2) 804 804 Chapter 11 Circles CharlestonDetroitMinneapolisEpicenterHolt, Rinehart & WinstonGeometry © 2007ge07sec11107009a Epicenter map2nd proof CHALLENGE AND EXTEND 45. In three dimensions, the equation of a sphere is similar to that of a circle. The equation of a sphere with center (h, j,
k) and radius r is (x - h) 2 + (y - j) 2 + (z - k) 2 = r 2. a. Write the equation of a sphere with center (2, -4, 3) that contains the point (1, -2, -5). b.   AC and   BC are tangents from the same exterior point. If AC = 15 m, what is BC? Explain. 46. Algebra Find the point(s) of intersection of the line x + y = 5 and the circle x 2 + y 2 = 25 by solving the system of equations. Check your result by graphing the line and the circle. 47. Find the equation of the circle with center (3, 4) that is tangent to the line whose equation is y = 2x + 3. (Hint: First find the point of tangency.) SPIRAL REVIEW Simplify each expression. (Previous course) 18a + 4 (9a + 3) __ 6 2 x 2 - 2 (4 x 2 + 1) __ 2 49. 48. 50. 3 (x + 3y) - 4 (3x + 2y) - (x - 2y) In isosceles △DEF, EF = 4y - 1. Find the value of each variable. (Lesson 4-8) ̶̶ DE ≅ ̶̶ EF. m∠E = 60°, and m∠D = (7x + 4) °. DE = 2y + 10, and 51. x 52. y Find each measure. (Lesson 11-5) 53. m ⁀ LNQ 54. m∠NMP KEYWORD: MG7 Career Q: What math classes did you take in high school? A: I took Algebra 1 and Geometry. I also took Drafting and Woodworking. Those classes aren’t considered math classes, but for me they were since math was used in them. Q: What type of furniture do you make? A: I mainly design and make household furniture, such as end tables, bedroom furniture, and entertainment centers. Q: How do you use math? A: Taking appropriate and precise measurements is very important. If wood is not measured correctly, the end result doesn’t turn out as expected. Understanding angle measures is also important. Some
of the furniture I build has 30° or 40° angles at the edges. Q: What are your future plans? A: Someday I would love to design all the furniture in my own home. It would be incredibly satisfying to know that all my furniture was made with quality and attention to detail. 11-7 Circles in the Coordinate Plane 805 805 Bryan Moreno Furniture Maker �� SECTION 11B Angles and Segments in Circles Native American Design The members of a Native American cultural center are painting a circle of colors on their gallery floor. They start by laying out the circle and chords shown. Before they apply their paint to the design, they measure angles and lengths to check for accuracy. 1. The circle design is based on twelve equally spaced points placed around the circumference of the circle. As the group lays out the design, what should be m∠AGB? 2. What should be m∠KAE? Why? 3. What should be m∠KMJ? Why? 4. The diameter of the circle is 22 ft. KM ≈ 4.8 ft, and JM ≈ 6.4 ft. What should be the length of ̶̶̶ MB? 5. The group members use a coordinate plane to help them position the design. Each square of a grid represents one square foot, and the center of the circle is at (20, 14). What is the equation of the circle? 6. What are the coordinates of points L, C, F, and I? 806 806 Chapter 11 Circles FEGDHCIBJAKMLge07sec11ac2003a SECTION 11B Quiz for Lessons 11-4 Through 11-7 11-4 Inscribed Angles Find each measure. 1. m∠BAC 2. m ⁀ CD 3. m∠FGH 4. m ⁀ JFG 11-5 Angle Relationships in Circles Find each measure. 5. m∠ RST 6. m∠AEC 7. A manufacturing company is creating a plastic stand for DVDs. They want to make the stand with m ⁀ MN = 102°. What should be the measure of ∠MPN? 11-6 Segment Relationships in Circles Find the value of the variable and the length of each chord or secant segment. 8. 9. 10. An archaeologist discovers a portion of a circular stone wall, shown by �
�� ST in the figure. ST = 12.2 m, and UR = 3.9 m. What was the diameter of the original circular wall? Round to the nearest hundredth. 11-7 Circles in the Coordinate Plane Write the equation of each circle. 11. ⊙A with center A (-2, -3) and radius 3 12. ⊙B that passes through (1, 1) and that has center B (4, 5) 13. A television station serves residents of three cities located at J (5, 2), K (-7, 2), and L (-5, -8). The station wants to build a new broadcast facility that is equidistant from the three cities. What are the coordinates of the location where the facility should be built? Ready to Go On? 807 807 �� EXTENSION EXTENSION Polar Coordinates TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Objectives Convert between polar and rectangular coordinates. Plot points using polar coordinates. Vocabulary polar coordinate system pole polar axis In a Cartesian coordinate system, a point is represented by the two coordinates x and y. In a polar coordinate system, a point A is represented by its distance from the origin r, and an angle θ. θ is measured counterclockwise from  OA. The ordered pair (r, θ ) the horizontal axis to represents the polar coordinates of point A. In a polar coordinate system, the origin is called the pole. The horizontal axis is called the polar axis. y You can use the equation of a circle r 2 = x 2 + y 2 and the tangent ratio θ = convert rectangular coordinates to polar coordinates. __ x to E X A M P L E 1 Converting Rectangular Coordinates to Polar Coordinates Convert (3, 4) to polar coordinates = 25 r = 5 tan θ = 4 _ 3 θ = tan -1(4_ 3) ≈ 53° The polar coordinates are (5, 53°). 1. Convert (4, 1) to polar coordinates. You can use the relationships x = r cos θ and y = r sin θ to convert polar coordinates to rectangular coordinates. E X A M P L E
2 Converting Polar Coordinates to Rectangular Coordinates Convert (2, 130°) to rectangular coordinates. x = r cos θ x = 2 cos 130° ≈ -1.29 y = r sinθ y = 2 sin 130° ≈ 1.53 The rectangular coordinates are (-1.29, 1.53). 2. Convert (4, 60°) to rectangular coordinates. 808 808 Chapter 11 Circles �� E X A M P L E 3 Plotting Polar Coordinates Plot the point (4, 225°). Step 1 Measure 225° counterclockwise from the polar axis. Step 2 Locate the point on the ray that is 4 units from the pole. 3. Plot the point (4, 300°). E X A M P L E 4 Graphing Polar Equations Graph r = 4. Make a table of values and plot the points. θ r 0° 4 45° 135° 270° 300° 4 4 4 4 4. Graph r = 2. EXTENSION Exercises Exercises Convert to polar coordinates. 1. (2, 2) 2. (1, 0) Convert to rectangular coordinates. 6. (5, 214°) 5. (3, 150°) 3. (3, 7) 4. (0, 15) 7. (4, 303°) 8. (4.5, 90°) Plot each point. 9. (4, 45°) 10. (3, 165°) 11. (1, 240°) 12. (3.5, 315°) 13. Critical Thinking Graph the equation r = 5. What can you say about the graph of an equation of the form r = a, where a is a positive real number? Technology Graph each equation. 14. r = -5 sin θ 17. r = 5 cos 3θ 15. r = 3 sin 4θ 18. r = 3 cos 2θ 16. r = -4 cos θ 19. r = 2 + 4 sin θ Chapter 11 Extension 809 809 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary adjacent arcs............... 757 exterior of a circle.....

the height of the original cone. a. Find the surface area of the original cone. b. Find the lateral area of the top of the cone. c. Find the area of the top base of the frustum. d. Use your results from parts a, b, and c to find the �� surface area of the frustum of the cone. �� 42. A frustum of a pyramid is a part of the pyramid with two parallel bases. The lateral faces of the frustum are trapezoids. Use the area formula for a trapezoid to derive a formula for the lateral area of a frustum of a regular square pyramid with base edge lengths b 1 and b 2 and slant height ℓ. 43. Use the net to derive the formula for the lateral area of a right cone with radius r and slant height ℓ. a. The length of the curved edge of the lateral surface must equal the circumference of the base. Find the circumference c of the base in terms of r. b. The lateral surface is part of a larger circle. Find the circumference C of the larger circle. c. The lateral surface area is c __ times the area of C the larger circle. Use your results from parts a and b to find c __. C �� d. Find the area of the larger circle. Use your result and the result from part c to find the lateral area L. SPIRAL REVIEW State whether the following can be described by a linear function. (Previous course) 44. the surface area of a right circular cone with height h and radius r 45. the perimeter of a rectangle with a height h that is twice as large as its width w 46. the area of a circle with radius r A point is chosen randomly in ACEF. Find the probability of each event. Round to the nearest hundredth. (Lesson 9-6) 47. The point is in △BDG. 48. The point is in ⊙H. 49. The point is in the shaded region. Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth. (Lesson 10-4) � � � � �� 50. 51. �� 52. �� 696 696 Chapter 10 Spatial Reasoning 10-6 Volume of Prisms and Cylinders TEKS G

.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the volume of a prism. Learn and apply the formula for the volume of a cylinder. Vocabulary volume Also G.1.B, G.5.A, G.5.B, G.11.D Who uses this? Marine biologists must ensure that aquariums are large enough to accommodate the number of fish inside them. (See Example 2.) The volume of a threedimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill the interior. A cube built out of 27 unit cubes has a volume of 27 cubic units. Cavalieri’s principle says that if two three-dimensional figures have the same height and have the same cross-sectional area at every level, they have the same volume. A right prism and an oblique prism with the same base and height have the same volume. Volume of a Prism The volume of a prism with base area B and height h is V = Bh. The volume of a right rectangular prism with length ℓ, width w, and height h is V = ℓwh. The volume of a cube with edge length s is Finding Volumes of Prisms Find the volume of each prism. Round to the nearest tenth, if necessary. A V = ℓwh = (10) (12) (8) = 960 cm 3 Volume of a right rectangular prism Substitute 10 for ℓ, 12 for w, and 8 for h. B a cube with edge length 10 cm V = s 3 = 10 3 = 1000 cm 3 Volume of a cube Substitute 10 for s. 10-6 Volume of Prisms and Cylinders 697 697 �� To review the area of a regular polygon, see page 601. To review tangent ratios, see page 525. Find the volume of each prism. Round to the nearest tenth, if necessary. C a right regular pentagonal prism with base edge length 5 m and height 7 m Step 1 Find the apothem a of the base. First draw a right triangle on one base as shown. The measure of the angle with its vertex at = 36°. 10 the center is 360° _ tan 36° = 2.5 _ a a = 2.5 _ tan 36° The leg of

the triangle is half the side length, or 2.5 m. Solve for a. Step 2 Use the value of a to find the base area. aP =.5 _ tan 36° ) (25) = 31.25 _ tan 36° P = 5 (5) = 25 m Step 3 Use the base area to find the volume. V = Bh = 31.25 _ · 7 ≈ 301.1 m 3 tan 36° 1. Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with legs 7 yd and 5 yd long. E X A M P L E 2 Marine Biology Application The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds. (Hint: 1 gallon ≈ 0.134 ft 3 ) Step 1 Find the volume of the aquarium in cubic feet. V = ℓwh = (120) (60) (8) = 57,600 ft 3 Step 2 Use the conversion factor to estimate the volume in gallons. 57, 600 ft 3 · 1 gallon _ 0.134 ft 3 ≈ 429,851 gallons 1 gallon _ = 1 0.134 ft 3 1 gallon _ 0.134 ft 3 Step 3 Use the conversion factor of the water. 8.33 pounds __ 1 gallon to estimate the weight 429,851 gallons · ≈ 3,580,659 pounds 8.33 pounds __ 1 gallon 8.33 pounds __ = 1 1 gallon Marine Biology The Islands of Steel habitat at the Texas State Aquarium in Corpus Christi has a volume of 132,000 gallons. Over 150 animals live in the habitat, including a sand tiger shark and nurse sharks. The aquarium holds about 429,851 gallons. The water in the aquarium weighs about 3,580,659 pounds. 2. What if…? Estimate the volume in gallons and the weight of the water in the aquarium above if the height were doubled. 698 698 Chapter 10 Spatial Reasoning 120 ft60 ft8 ftge07se_c10l06003aAB �� Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume. Volume of a Cylinder The volume of a cylinder with base area B, radius r, and height h is V = Bh,

or V = π r 2h. E X A M P L E 3 Finding Volumes of Cylinders Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. A V = π r 2h = π (8)2(12) Volume of a cylinder Substitute 8 for r and 12 for h. = 768π cm 3 ≈ 2412.7 cm 3 B a cylinder with a base area of 36π in 2 and a height equal to twice the radius Step 1 Use the base area to find the radius. π r 2 = 36π r = 6 Substitute 36π for the base area. Solve for r. Step 2 Use the radius to find the height. The height is equal to twice the radius. h = 2r = 2 (6) = 12 cm Step 3 Use the radius and height to find the volume. Volume of a cylinder V = π r 2h = π (6)2(12)= 432π in 3 ≈ 1357.2 in 3 Substitute 6 for r and 12 for h. 3. Find the volume of a cylinder with a diameter of 16 in. and a height of 17 in. Give your answer both in terms of π and rounded to the nearest tenth. 10-6 Volume of Prisms and Cylinders 699 699 �� E X A M P L E 4 Exploring Effects of Changing Dimensions The radius and height of the cylinder are multiplied by 1 __. Describe the effect on the volume. 2 original dimensions6) 2 (12) = 432π m 3 radius and height multiplied by 1 __ 3) 2 (6) = 54π m 3 Notice that 54π = 1 __ 8 (432π). If the radius and height are multiplied by 1 __ 2, the volume is multiplied by ( 1 __ 2 ), or 1 __ 8. 3 4. The length, width, and height of the prism are doubled. Describe the effect on the volume. E X A M P L E 5 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The base area of the prism is B = 1 __ 2 (6) (8) = 24 m 2. The volume of the prism is V = Bh = 24 (9) = 216 m 3. �� The cylinder’s diameter equals the hypotenuse of

the prism’s base, 10 m. So the radius is 5 m. The volume of the cylinder is V = π r 2 h = π (5) 2 (5) = 125π m 3. �� The total volume of the figure is the sum of the volumes. V = 216 + 125π ≈ 608.7 m 3 �� 5. Find the volume of the composite figure. Round to the nearest tenth. THINK AND DISCUSS 1. Compare the formula for the volume of a prism with the formula for the volume of a cylinder. 2. Explain how Cavalieri’s principle relates to the formula for the volume of an oblique prism. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the formula for the volume. 700 700 Chapter 10 Spatial Reasoning �� 10-6 Exercises Exercises KEYWORD: MG7 10-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a right cylinder, the altitude is? the axis. (longer than, shorter ̶̶̶̶ than, or the same length as Find the volume of each prism. p. 697 2. 3. 4. a cube with edge length 8 ft. Food The world’s largest ice cream cake, built in p. 698 New York City on May 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream cake in gallons. If the density of the ice cream was 4.73 pounds per gallon, estimate the weight of the cake. (Hint: 1 gallon ≈ 0.134 cubic feet. 699 Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. 6. 7. 8. a cylinder with base area 25π cm 2 and height 3 cm more than the radius. 700 Describe the effect of each change on the volume of the given figure. 9. The dimensions are multiplied by 1 _. 4 10. The dimensions are tripled Find the volume of each composite figure. Round to the nearest tenth. p. 700 11. 12. 10-6 Volume of Prisms and Cylinders 701 701 �� PRACTICE AND

PROBLEM SOLVING Find the volume of each prism. 13. 14. Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS 15. a square prism with a base area of 49 ft 2 and a height 2 ft less than the base Skills Practice p. S23 Application Practice p. S37 edge length 16. Landscaping Colin is buying dirt to fill a garden bed that is a 9 ft by 16 ft rectangle. If he wants to fill it to a depth of 4 in., how many cubic yards of dirt does he need? If dirt costs $25 per yd 3, how much will the project cost? (Hint: 1 yd 3 = 27 ft 3 ) Find the volume of each cylinder. Give your answers both in terms of π and rounded to the nearest tenth. 17. 18. 19. a cylinder with base area 24π cm 2 and height 16 cm Describe the effect of each change on the volume of the given figure. 20. The dimensions are multiplied by 5. 21. The dimensions are multiplied by 3 _. 5 Find the volume of each composite figure. 22. 23. 24. One cup is equal to 14.4375 in 3. If a 1 c cylindrical measuring cup has a radius of 2 in., what is its height? If the radius is 1.5 in., what is its height? 25. Food A cake is a cylinder with a diameter of 10 in. and a height of 3 in. For a party, a coin has been mixed into the batter and baked inside the cake. The person who gets the piece with the coin wins a prize. a. Find the volume of the cake. Round to the nearest tenth. b. Probability Keka gets a piece of cake that is a right rectangular prism with a 3 in. by 1 in. base. What is the probability that the coin is in her piece? Round to the nearest tenth. 702 702 Chapter 10 Spatial Reasoning �� 26. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A cylindrical juice container with a 3 in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw

can be inserted. a. Find the height h of the container to the nearest tenth. b. Find the volume of the container to the nearest tenth. c. How many ounces of juice does the container hold? (Hint: 1 in 3 ≈ 0.55 oz) Math History 27. Find the height of a rectangular prism with length 5 ft, width 9 ft, and volume 495 ft 3. 28. Find the area of the base of a rectangular prism with volume 360 in 3 and height 9 in. 29. Find the volume of a cylinder with surface area 210π m 2 and height 8 m. 30. Find the volume of a rectangular prism with vertices (0, 0, 0), (0, 3, 0), (7, 0, 0), (7, 3, 0), (0, 0, 6), (0, 3, 6), (7, 0, 6), and (7, 3, 6). 31. You can use displacement to find the volume of an irregular object, such as a stone. Suppose the tank shown is filled with water to a depth of 8 in. A stone is placed in the tank so that it is completely covered, causing the water level to rise by 2 in. Find the volume of the stone. �� Archimedes (287–212 B.C.E.) used displacement to find the volume of a gold crown. He discovered that the goldsmith had cheated the king by substituting an equal weight of silver for part of the gold. 32. Food A 1 in. cube of cheese is one serving. How many servings are in a 4 in. by 4 in. by 1 __ 4 in. slice? 33. History In 1919, a cylindrical tank containing molasses burst and flooded the city of Boston, Massachusetts. The tank had a 90 ft diameter and a height of 52 ft. How many gallons of molasses were in the tank? (Hint: 1 gal ≈ 0.134 ft 3 ) 34. Meteorology If 3 in. of rain fall on the property shown, what is the volume in cubic feet? In gallons? The density of water is 8.33 pounds per gallon. What is the weight of the rain in pounds? (Hint: 1 gal ≈ 0.134 ft 3 ) �� 35. Critical Thinking The dimensions of a prism with volume V and surface area S are multiplied by a scale factor of k to form a

similar prism. Make a conjecture about the ratio of the surface area of the new prism to its volume. Test your conjecture using a cube with an edge length of 1 and a scale factor of 2. 36. Write About It How can you change the edge length of a cube so that its volume is doubled? 37. Abigail has a cylindrical candle mold with the dimensions shown. If Abigail has a rectangular block of wax measuring 15 cm by 12 cm by 18 cm, about how many candles can she make after melting the block of wax? 14 31 35 76 10-6 Volume of Prisms and Cylinders 703 703 �� 38. A 96-inch piece of wire was cut into equal segments that were then connected to form the edges of a cube. What is the volume of the cube? 576 in 3 512 in 3 729 in 3 1728 in 3 39. One juice container is a rectangular prism with a height of 9 in. and a 3 in. by 3 in. square base. Another juice container is a cylinder with a radius of 1.75 in. and a height of 9 in. Which best describes the relationship between the two containers? The prism has the greater volume. The cylinder has the greater volume. The volumes are equivalent. The volumes cannot be determined. 40. What is the volume of the three-dimensional object with the dimensions shown in the three views below? �� 160 cm 3 240 cm 3 840 cm 3 1000 cm 3 CHALLENGE AND EXTEND Algebra Find the volume of each three-dimensional figure in terms of x. 41. 42. �� 43. � �� 44. The volume in cubic units of a cylinder is equal to its surface area in square units. Prove that the radius and height must both be greater than 2. SPIRAL REVIEW 45. Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled? (Previous course) 46. Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his

term paper? (Previous course) ABCD is a parallelogram. Find each measure. (Lesson 6-2) 47. m∠ABC 48. BC 49. AB Find the surface area of each figure. Round to the nearest tenth. (Lesson 10-5) 50. a square pyramid with slant height 10 in. and base edge length 8 in. �� 51. a regular pentagonal pyramid with slant height 8 cm and base edge length 6 cm 52. a right cone with slant height 2 ft and a base with circumference of π ft 704 704 Chapter 10 Spatial Reasoning 10-7 Volume of Pyramids and Cones TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of..., pyramids,..., cones,.... Objectives Learn and apply the formula for the volume of a pyramid. Learn and apply the formula for the volume of a cone. Also G.5.A, G.5.B, G.11.D Who uses this? The builders of the Rainforest Pyramid in Galveston, Texas, needed to calculate the volume of the pyramid to plan the climate control system. (See Example 2.) The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube. Volume of a Pyramid The volume of a pyramid with base area B and height h is V = 1 _ 3 Bh. E X A M P L E 1 Finding Volumes of Pyramids Find the volume of each pyramid. A a rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft Bh = 1 _ V = 1 _ (7 · 9) (12) = 252 ft 3 3 3 B the square pyramid The base is a square with a side length of 4 in., and the height is 6 in. Bh = 6) = 32 in 3 3 3 10-7 Volume of Pyramids and Cones 705 705 �� Find the volume of the pyramid. C the trapezoidal pyramid with base ABCD, where and ̶̶ AE ⊥ plane ABC ̶̶

AB ǁ ̶̶ CD Step 1 Find the area of the base9 + 18) 6 2 = 81 m 2 Area of a trapezoid Substitute 9 for b 1, 18 for b 2, and 6 for h. Simplify. Step 2 Use the base area and the height to find the volume. ̶̶ AE is the altitude, so the height Because ̶̶ AE ⊥ plane ABC, Bh is equal to AE81) (10) 3 = 270 m 3 Volume of a pyramid Substitute 81 for B and 10 for h. 1. Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of the base. E X A M P L E 2 Architecture Application The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd 2, 1 story ≈ 10 ft) The base is a square with an area of about 4840 yd 2. The base edge length is √  is about 10 (10) = 100 ft, or about 33 yd. 4840 ≈ 70 yd. The height First find the volume in cubic yards. Bh V = 1 _ 3 = 1 _ ( 70 2 ) (33) = 53,900 yd 3 3 Volume of a regular pyramid Substitute 70 2 for B and 33 for h. Then convert your answer to find the volume in cubic feet. The volume of one cubic yard is (3 ft) (3 ft) (3 ft) = 27 ft 3. Use the conversion factor 27 ft 3 ____ 1 yd 3 to find the volume in cubic feet. 53,900 yd 3 · 27 ft 3 _ 1 yd 3 ≈ 1,455,300 ft 3 2. What if…? What would be the volume of the Rainforest Pyramid if the height were doubled? 706 706 Chapter 10 Spatial Reasoning �� Volume of Cones The volume of a cone with base area B, radius r, and height h is V = 1 _ 3 or V = 1 _ π r 2 h. 3 Bh, E X A M P L E 3 Finding Volumes of Cones Find the volume of each cone. Give your answers both in terms of π and rounded

to the nearest tenth. A a cone with radius 5 cm and height 12 cm 5) 2 (12) 3 = 100π cm 3 ≈ 314.2 cm 3 Volume of a cone Substitute 5 for r and 12 for h. Simplify. B a cone with a base circumference of 21π cm and a height 3 cm less than twice the radius Step 1 Use the circumference to find the radius. 2πr = 21π Substitute 21π for C. r = 10.5 cm Divide both sides by 2π. Step 2 Use the radius to find the height. 2 (10.5) - 3 = 18 cm The height is 3 cm less than twice the radius. Step 3 Use the radius and height to find the volume10.5) 2 (18) 3 Volume of a cone Substitute 10.5 for r and 18 for h. = 661.5π cm 3 ≈ 2078.2 cm 3 Simplify. C Step 1 Use the Pythagorean Theorem to find the height. 7 2 + h 2 = 25 2 h 2 = 576 h = 24 Pythagorean Theorem Subtract 7 2 from both sides. Take the square root of both sides. Step 2 Use the radius and height to find the volume7) 2 (24) 3 Volume of a cone Substitute 7 for r and 24 for h. = 392π ft 3 ≈ 1231.5 ft 3 Simplify. 3. Find the volume of the cone. 10-7 Volume of Pyramids and Cones 707 707 �� E X A M P L E 4 Exploring Effects of Changing Dimensions The length, width, and height of the rectangular pyramid are multiplied by 1 __. 4 Describe the effect on the volume. original dimensions: Bh V = 1 _ 3 = 1 _ (24 · 20) (20) 3 = 3200 ft 3 length, width, and height multiplied by 1 _ : 4 Bh V = 1 _ 3 = 1 _ (6 · 5) (5) 3 = 50 ft 3 Notice that 50 = 1 __ 64 (3200). If the length, width, and height are multiplied by 1 __ 4, the volume is multiplied by ( 1 __ 4 ), or 1 __ 64. 3 4. The radius and height of the cone are doubled. Describe the effect on the volume. E X A M P L E 5 Finding Volumes of Composite Three-Dimensional Figures Find the volume of the

composite figure. Round to the nearest tenth. The volume of the cylinder is V = π r 2 h = π (2) 2 (2) = 8π in 3. The volume of the cone is 2) 2 (3) = 4π in 3. 3 3 The volume of the composite figure is the sum of the volumes. V = 8π + 4π = 12π in 3 ≈ 37.7 in 3 5. Find the volume of the composite figure. THINK AND DISCUSS 1. Explain how the volume of a pyramid is related to the volume of a prism with the same base and height. 2. GET ORGANIZED Copy and complete the graphic organizer. 708 708 Chapter 10 Spatial Reasoning �� 10-7 Exercises Exercises KEYWORD: MG7 10-7 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary The altitude of a pyramid is or oblique)? to the base. (perpendicular, parallel, ̶̶̶̶ Find the volume of each pyramid. Round to the nearest tenth, if necessary. p. 705 2. 3. 4. a hexagonal pyramid with a base area of 25 ft 2 and a height of 9 ft. 706 5. Geology A crystal is cut into the shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of 3 mm. What is the volume to the nearest cubic millimeter of the crystal. 707 Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. 6. 7. 8. a cone with radius 12 m and height 20 Describe the effect of each change on the volume of the given figure. p. 708 9. The dimensions are tripled. 10. The dimensions are multiplied by Find the volume of each composite figure. Round to the nearest tenth, if necessary. p. 708 11. 12. 10-7 Volume of Pyramids and Cones 709 709 ge07sec10l07003aAB3 mm5.7 mm�� Independent Practice For See

Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the volume of each pyramid. Round to the nearest tenth, if necessary. 13. 14. 15. 15. a regular square pyramid with base edge length 12 ft and slant height 10 ft 16. Carpentry A roof that encloses an attic is a square pyramid with a base edge length of 45 feet and a height of 5 yards. What is the volume of the attic in cubic feet? In cubic yards? �� Find the volume of each cone. Give your answers both in terms of π and rounded to the nearest tenth. 17. 18. 19. a cone with base area 36π ft 2 and a height equal to twice the radius �� Describe the effect of each change on the volume of the given figure. 20. The dimensions are multiplied by 1 _. 3 21. The dimensions are multiplied by 6. Find the volume of each composite figure. Round to the nearest tenth, if necessary. 22. 23. Find the volume of each right cone with the given dimensions. Give your answers in terms of π. 24. radius 3 in. height 7 in. 25. diameter 5 m height 2 m 26. radius 28 ft 27. diameter 24 cm slant height 53 ft slant height 13 cm 710 710 Chapter 10 Spatial Reasoning �� Find the volume of each regular pyramid with the given dimensions. Round to the nearest tenth, if necessary. Number of sides of base Base edge length Height Volume 28. 29. 30. 31. 3 4 5 6 10 ft 15 m 9 in. 8 cm 6 ft 18 m 12 in. 3 cm 32. Find the height of a rectangular pyramid with length 3 m, width 8 m, and volume 112 m 3. 33. Find the base circumference of a cone with height 5 cm and volume 125π cm 3. 34. Find the volume of a cone with slant height 10 ft and height 8 ft. 35. Find the volume of a square pyramid with slant height 17 in. and surface area 800 in 2. 36. Find the surface area of a cone with height 20 yd and volume 1500π yd 3.

37. Find the volume of a triangular pyramid with vertices (0, 0, 0), (5, 0, 0), (0, 3, 0), and (0, 0, 7). 38. /////ERROR ANALYSIS///// Which volume is incorrect? Explain the error. 39. Critical Thinking Write a ratio comparing the volume of the prism to the volume of the composite figure. Explain your answer. 40. Write About It Explain how you would find the volume of a cone, given the radius and the surface area. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice stand sells smoothies in cone-shaped cups that are 8 in. tall. The regular size has a 4 in. diameter. The jumbo size has an 8 in. diameter. a. Find the volume of the regular size to the nearest tenth. b. Find the volume of the jumbo size to the nearest tenth. c. The regular size costs $1.25. What would be a reasonable price for the jumbo size? Explain your reasoning. �� 10-7 Volume of Pyramids and Cones 711 711 �� 42. Find the volume of the cone. 432π cm 3 720π cm 3 1296π cm 3 2160π cm 3 43. A square pyramid has a slant height of 25 m and a lateral area of 350 m 2. Which is closest to the volume? 392 m 3 1176 m 3 404 m 3 1225 m 3 44. A cone has a volume of 18π in 3. Which are possible dimensions of the cone? Diameter 3 in., height 6 in. Diameter 1 in., height 18 in. Diameter 6 in., height 6 in. Diameter 6 in., height 3 in. 45. Gridded Response Find the height in centimeters of a square pyramid with a volume of 243 cm 3 and a base edge length equal to the height. CHALLENGE AND EXTEND Each cone is inscribed in a regular pyramid with a base edge length of 2 ft and a height of 2 ft. Find the volume of each cone. 46. 47. 48. 49. A regular octahedron has 8 faces that are equilateral triangles. Find the volume of a regular octahedron with a side length of 10 cm. 50. A cylinder has a radius

of 5 in. and a height of 3 in. Without calculating the volumes, find the height of a cone with the same base and the same volume as the cylinder. Explain your reasoning. SPIRAL REVIEW Find the unknown numbers. (Previous course) 51. The difference of two numbers is 24. The larger number is 4 less than 3 times the smaller number. 52. Three times the first number plus the second number is 88. The first number times 10 is equal to 4 times the second. 53. The sum of two numbers is 197. The first number is 20 more than 1 __ 2 of the second number. Explain why the triangles are similar, then find each length. (Lesson 7-3) 54. AB 55. PQ Find AB and the coordinates of the midpoint of if necessary. (Lesson 10-3) 56. A (1, 1, 2), B (8, 9, 10) 57. A (-4, -1, 0), B (5, 1, -4) ̶̶ AB. Round to the nearest tenth, 58. A (2, -2, 4), B (-2, 2, -4) 59. A (-3, -1, 2), B (-1, 5, 5) 712 712 Chapter 10 Spatial Reasoning �� Functional Relationships in Formulas Algebra You have studied formulas for several solid figures. Here you will see how a change in one dimension affects the measurements of the other dimensions. See Skills Bank page S63 Example A square prism has a volume of 21 cubic units. Write an equation that describes the base edge length s in terms of the height h. Graph the relationship in a coordinate plane with h on the horizontal axis and s on the vertical axis. What happens to the base edge length as the height increases? First use the volume formula to write an equation. V = Bh 100 = s 2h Volume of a prism Substitute 100 for V and s 2 for B. Then solve for s to get an equation for s in terms of h. s 2 = 100_ h s = √100_ h s = 10_ √  h Divide both sides by h. Take the square root of both sides. √  100 = 10 Graph the equation. First make a table of h- and s-values. Then plot the points and draw a

smooth curve through the points. Notice that the function is not defined for h = 0. h 1 4 9 16 25 s 10 5 ̶ 3 3. 2.5 2 As the height of the prism increases, the base edge length decreases. Try This TAKS Grades 9–11 Obj. 8 1. A right cone has a radius of 10 units. Write an equation that describes the slant height ℓ in terms of the surface area S. Graph the relationship in a coordinate plane with S on the horizontal axis and ℓ on the vertical axis. What happens to the slant height as the surface area increases? 2. A cylinder has a height of 5 units. Write an equation that describes the radius r in terms of the volume V. Graph the relationship in a coordinate plane with V on the horizontal axis and r on the vertical axis. What happens to the radius as the volume increases? On Track for TAKS 713 713 �� 10-8 Spheres TEKS G.8.D Congruence and the geometry of size: find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites …. Objectives Learn and apply the formula for the volume of a sphere. Learn and apply the formula for the surface area of a sphere. Vocabulary sphere center of a sphere radius of a sphere hemisphere great circle Who uses this? Biologists study the eyes of deep-sea predators such as the giant squid to learn about their behavior. (See Example 2.) A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres. Also G.5.A, G.5.B, G.11.D The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalieri’s Principle. You will prove that the cross sections have equal areas in Exercise 39. V (hemisphere) = V (cylinder) - V (cone = 2_ 3 = 2_ 3 = 2_ 3 π r 2(r) π r 2h π r 3 The height of the hemisphere is equal to the radius. The volume of a sphere with

radius r is twice the volume of the hemisphere, or V = 4__ 3 π r 3. Volume of a Sphere The volume of a sphere with radius r is Finding Volumes of Spheres Find each measurement. Give your answer in terms of π. A the volume of the sphere 9) 3 3 Substitute 9 for r. = 972π cm 3 Simplify. 714 714 Chapter 10 Spatial Reasoning �� Find each measurement. Give your answer in terms of π. B the diameter of a sphere with volume 972π in 3 972π = 4 _ π r 3 3 729 = r 3 r = 9 d = 18 in. Substitute 972π for V. Divide both sides by 4 _ π. 3 Take the cube root of both sides. d = 2r C the volume of the hemisphere 4) 3 = 128π _ = 2 _ 3 3 m 3 Volume of a hemisphere Substitute 4 for r. 1. Find the radius of a sphere with volume 2304π ft Biology Application Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximately a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximately a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball? human eyeball: giant squid eyeball1.25) 3 ≈ 8.18 cm 12.5) 3 ≈ 8181.23 cm 3 3 A giant squid eyeball is about 1000 times as great in volume as a human eyeball. 2. A hummingbird eyeball has a diameter of approximately 0.6 cm. How many times as great is the volume of a human eyeball as the volume of a hummingbird eyeball? In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximately the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r. Br + 1 _ V (sphere) ≈ 1 _ Br + … + Br) 4 _ 3 4π r 2 ≈ nB 3 Br The sphere’s volume is close to the sum of the volumes of the pyramids. Divide both sides by 1 _ 3 πr. If the py

ramids fill the sphere, the total area of the bases is approximately equal to the surface area of the sphere S, so 4π r 2 ≈ S. As the number of pyramids increases, the approximation gets closer to the actual surface area. 10 - 8 Spheres 715 715 �� Surface Area of a Sphere The surface area of a sphere with radius r is S = 4π Finding Surface Area of Spheres Find each measurement. Give your answers in terms of π. A the surface area of a sphere with diameter 10 ft S = 4π r 2 S = 4π (5)2 = 200π ft 2 Substitute 5 for r. B the volume of a sphere with surface area 144π m 2 S = 4π r 2 144π = 4π 6) 3 = 288π m 3 3 Substitute 144π for S. Solve for r. Substitute 6 for r. The volume of the sphere is 288π m 3. C the surface area of a sphere with a great circle that has an area of 4π in 2 π r 2 = 4π Substitute 4π for A in the formula for the area of a circle. Solve for r. r = 2 S = 4π r 2 = 4π (2) 2 = 16π in 2 Substitute 2 for r in the surface area formula. 3. Find the surface area of the sphere. E X A M P L E 4 Exploring Effects of Changing Dimensions The radius of the sphere is tripled. Describe the effect on the volume. original dimensions3) 3 3 = 36π m 3 radius tripled9) 3 3 = 972π m 3 Notice that 972π = 27 (36π). If the radius is tripled, the volume is multiplied by 27. 4. The radius of the sphere above is divided by 3. Describe the effect on the surface area. 716 716 Chapter 10 Spatial Reasoning �� E X A M P L E 5 Finding Surface Areas and Volumes of Composite Figures Find the surface area and volume of the composite figure. Give your answers in terms of π. Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone. S (hemisphere) = 1 _ (4π r 2 ) = 2π (7) 2 = 98π cm 2 2 L (cone) = πr�

�� = π (7) (25) = 175π cm 2 The surface area of the composite figure is 98π + 175π = 273π cm 2. Step 2 Find the volume of the composite figure. First find the height of the cone. h = √  25 2 - 7 2 Pythagorean Theorem = √  576 = 24 cm Simplify. The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the conehemispherecone) = 1 _ π (7) 2 (24) = 392π cm 3 3 3 π (7) 3 = 686π _ 3 3 cm 3 The volume of the composite figure is 686π _ + 392π = 1862π _ cm 3. 3 3 5. Find the surface area and volume of the composite figure. THINK AND DISCUSS 1. Explain how to find the surface area of a sphere when you know the area of a great circle. 2. Compare the volume of the sphere with the volume of the composite figure. 3. GET ORGANIZED Copy and complete the graphic organizer. 10 - 8 Spheres 717 717 �� 10-8 Exercises Exercises KEYWORD: MG7 10-8 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe the endpoints of a radius of a sphere Find each measurement. Give your answers in terms of π. p. 714 2. the volume of the hemisphere 3. the volume of the sphere 4. the radius of a sphere with volume 288π cm. Food Approximately how many times as great is the volume of the grapefruit p. 715 as the volume of the lime? �� Find each measurement. Give your answers in terms of π. p. 716 6. the surface area of the sphere 7. the surface area of the sphere 8. the volume of a sphere with surface area 6724π ft Describe the effect of each change on the given measurement of the figure. p. 716 9. surface area 10. volume The dimensions are doubled. The dimensions are multiplied by Find the surface area and volume of each composite figure. p. 717 11

. 12. 718 718 Chapter 10 Spatial Reasoning �� Independent Practice For See Exercises Example 13–15 16 17–19 20–21 22–23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S23 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find each measurement. Give your answers in terms of π. 13. the volume of the sphere 14. the volume of the hemisphere 15. the diameter of a sphere with volume 7776π in 3 16. Jewelry The size of a cultured pearl is typically indicated by its diameter in mm. How many times as great is the volume of the 9 mm pearl as the volume of the 6 mm pearl? �� Find each measurement. Give your answers in terms of π. �� 17. the surface area of the sphere 18. the surface area of the sphere 19. the volume of a sphere with surface area 625π m 2 Describe the effect of each change on the given measurement of the figure. 20. surface area The dimensions are multiplied by 1 _. 5 21. volume The dimensions are multiplied by 6. Find the surface area and volume of each composite figure. 22. 23. 24. Find the radius of a hemisphere with a volume of 144π cm 3. 25. Find the circumference of a sphere with a surface area of 60π in 2. 26. Find the volume of a sphere with a circumference of 36π ft. 27. Find the surface area and volume of a sphere centered at (0, 0, 0) that passes through the point (2, 3, 6). 28. Estimation A bead is formed by drilling a cylindrical hole with a 2 mm diameter through a sphere with an 8 mm diameter. Estimate the surface area and volume of the bead. 10 - 8 Spheres 719 719 �� Marine Biology In 1934, the bathysphere reached a record depth of 3028 feet. The pressure on the hull was about half a ton per square inch. Sports Find the unknown dimensions of the ball for each sport. Sport Ball Diameter Circumference Surface Area Volume 29. Golf 30. Cricket 31. Tennis 32. Petanque 9 in. 1.68 in. 2.5 in. 74 mm 33. Marine Biology The bathysphere was

an early version of a submarine, invented in the 1930s. The inside diameter of the bathysphere was 54 inches, and the steel used to make the sphere was 1.5 inches thick. It had three 8-inch diameter windows. Estimate the volume of steel used to make the bathysphere. 34. Geography Earth’s radius is approximately 4000 mi. About two-thirds of Earth’s surface is covered by water. Estimate the land area on Earth. Astronomy Use the table for Exercises 35–38. 35. How many times as great is the volume of Jupiter as the volume of Earth? 36. The sum of the volumes of Venus and Mars is about equal to the volume of which planet? 37. Which is greater, the sum of the surface areas of Uranus and Neptune or the surface area of Saturn? 38. How many times as great is the surface area of Mercury as the surface area of Pluto? Planet Diameter (mi) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto 3,032 7,521 7,926 4,222 88,846 74,898 31,763 30,775 1,485 39. Critical Thinking In the figure, the hemisphere and the cylinder both have radius and height r. Prove that the shaded cross sections have equal areas. 40. Write About It Suppose a sphere and a cube have equal surface areas. Using r for the radius of the sphere and s for the side of a cube, write an equation to show the relationship between r and s. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A company sells orange juice in spherical containers that look like oranges. Each container has a surface area of approximately 50.3 in 2. a. What is the volume of the container? Round to the nearest tenth. b. The company decides to increase the radius of the container by 10%. What is the volume of the new container? 720 720 Chapter 10 Spatial Reasoning �� 42. A sphere with radius 8 cm is inscribed in a cube. Find the ratio of the volume of the cube to the volume of the sphere. 2 : 1 _ π 3 2 : 3π 43. What is the surface area of a sphere with volume 10 2 _ π in 3? 3 8π in 2 10 2 _ π in 2 3 16π in 2 32π in 2 44. Which expression represents the volume of the

composite figure formed by a hemisphere with radius r and a cube with side length 2r? 3 π + 82π + 12 CHALLENGE AND EXTEND 45. Food The top of a gumball machine is an 18 in. sphere. The machine holds a maximum of 3300 gumballs, which leaves about 43% of the space in the machine empty. Estimate the diameter of each gumball. 46. The surface area of a sphere can be used to determine its volume. a. Solve the surface area formula of a sphere to get an expression for r in terms of S. b. Substitute your result from part a into the volume formula to find the volume V of a sphere in terms of its surface area S. c. Graph the relationship between volume and surface area with S on the horizontal axis and V on the vertical axis. What shape is the graph? Use the diagram of a sphere inscribed in a cylinder for Exercises 47 and 48. 47. What is the relationship between the volume of the sphere and the volume of the cylinder? 48. What is the relationship between the surface area of the sphere and the lateral area of the cylinder? SPIRAL REVIEW Write an equation that describes the functional relationship for each set of ordered pairs. (Previous course)   (0, 1), (1, 2), (-1, 2), (2, 5), (-2, 5) 49. ⎬ ⎨     (-1, 9), (0, 10), (1, 11), (2, 12), (3, 13) 50. ⎬ ⎨   Find the shaded area. Round to the nearest tenth, if necessary. (Lesson 9-3) 51. 52. Describe the effect on the volume that results from the given change. (Lesson 10-6) 53. The side lengths of a cube are multiplied by 3 __ 4. 54. The height and the base area of a prism are multiplied by 5. 10 - 8 Spheres 721 721 �� 10-8 Use with Lesson 10-8 Activity 1 Compare Surface Areas and Volumes In some situations you may need to find the minimum surface area for a given volume. In others you may need to find the maximum volume for a given surface area. Spreadsheet software can help you analyze these

problems. TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed …. 1 Create a spreadsheet to compare surface areas and volumes of rectangular prisms. Create columns for length L, width W, height H, surface area SA, volume V, and ratio of surface area to volume SA/V. In the column for SA, use the formula shown. 2 Create a formula for the V column and a formula for the SA/V column. 3 Fill in the measurements L = 8, W = 2, and H = 4 for the first rectangular prism. 4 Choose several values for L, W, and H to create rectangular prisms that each have the same volume as the first one. Which has the least surface area? Sketch the prism and describe its shape in words. (Is it tall or short, skinny or wide, flat or cubical?) Make a conjecture about what type of shape has the minimum surface area for a given volume. 722 722 Chapter 10 Spatial Reasoning �� Try This 1. Repeat Activity 1 for cylinders. Create columns for radius R, height H, surface area SA, volume V, and ratio of surface area to volume SA/V. What shape cylinder has the minimum surface area for a given volume? (Hint: To use π in a formula, input “PI( )” into your spreadsheet.) 2. Investigate packages such as cereal boxes and soda cans. Do the manufacturers appear to be using shapes with the minimum surface areas for their volume? What other factors might influence a company’s choice of packaging? Activity 2 1 Create a new spreadsheet with the same column headings used in Activity 1. Fill in the measurements L = 8, W = 2, and H = 4 for the first rectangular prism. To create a new prism with the same surface area, choose new values for L and W, and use the formula shown to calculate H. 2 Choose several more values for L and W, and calculate H so that SA = 112. Examine the V and SA/V columns. Which prism has the greatest volume? Sketch the prism and describe it in words. Make a conjecture about what type of shape has the maximum volume for a given surface area. Try This 3. Repeat Activity 2 for cylinders. Create columns for radius R, height H, surface area SA, volume V, and the ratio of surface area to volume SA/

V. What shape cylinder has the maximum volume for a given surface area? 4. Solve the formula SA = 2LW + 2LH + 2WH for H. Use your result to explain the formula that was used to find H in Activity 2. 5. If a rectangular prism, a pyramid, a cylinder, a cone, and a sphere all had the same volume, which do you think would have the least surface area? Which would have the greatest surface area? Explain. 6. Use a spreadsheet to analyze what happens to the ratio of surface area to volume of a rectangular prism when the dimensions are doubled. Explain how you set up the spreadsheet and describe your results. 10-8 Technology Lab 723 723 �� SECTION 10B Surface Area and Volume Juice for Fun You are in charge of designing containers for a new brand of juice. Your company wants you to compare several different container shapes. The container must be able to hold a 6-inch straw so that exactly 1 inch remains outside the container when the straw is inserted as far as possible. 1. One possible container is a cylinder with a base diameter of 4 in., as shown. How much material is needed to make this container? Round to the nearest tenth. 2. Estimate the volume of juice in ounces that the cylinder will hold. Round to the nearest tenth. (Hint: 1 in 3 ≈ 0.55 oz) 3. Another option is a square prism with a 3 in. by 3 in. base, as shown. How much material is needed to make this container? 4. Estimate the volume of juice in ounces that the prism will hold. (Hint: 1 in 3 ≈ 0.55 oz) 5. Which container would you recommend to your company? Justify your answer. �� 724 724 Chapter 10 Spatial Reasoning SECTION 10B Quiz for Lessons 10-4 Through 10-8 10-4 Surface Area of Prisms and Cylinders Find the surface area of each figure. Round to the nearest tenth, if necessary. 1. 2. 3. 4. The dimensions of a 12 mm by 8 mm by 24 mm right rectangular prism are multiplied by 3 __ 4. Describe the effect on the surface area. 10-5 Surface Area of Pyramids and Cones Find the surface area of each figure. Round to the nearest tenth, if necessary. 5. a regular pentagonal pyramid with base edge length 18 yd and sl

ant height 20 yd 6. a right cone with diameter 30 in. and height 8 in. 7. the composite figure formed by two cones 10-6 Volume of Prisms and Cylinders Find the volume of each figure. Round to the nearest tenth, if necessary. 8. a regular hexagonal prism with base area 23 in 2 and height 9 in. 9. a cylinder with radius 8 yd and height 14 yd 10. A brick patio measures 10 ft by 12 ft by 4 in. Find the volume of the bricks. If the density of brick is 130 pounds per cubic foot, what is the weight of the patio in pounds? 11. The dimensions of a cylinder with diameter 2 ft and height 1 ft are doubled. Describe the effect on the volume. 10-7 Volume of Pyramids and Cones Find the volume of each figure. Round to the nearest tenth, if necessary. 12. 13. 14. 10-8 Spheres Find the surface area and volume of each figure. 15. a sphere with diameter 20 in. 16. a hemisphere with radius 12 in. 17. A baseball has a diameter of approximately 3 in., and a softball has a diameter of approximately 5 in. About how many times as great is the volume of a softball as the volume of a baseball? Ready to Go On? 725 725 �� EXTENSION Spherical Geometry EXTENSION TEKS G.1.C Geometric structure: compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. Objective Understand spherical geometry as an example of non-Euclidean geometry. Vocabulary non-Euclidean geometry spherical geometry Euclidean geometry is based on figures in a plane. Non-Euclidean geometry is based on figures in a curved surface. In a non-Euclidean geometry system, the Parallel Postulate is not true. One type of non-Euclidean geometry is spherical geometry, which is the study of figures on the surface of a sphere. A line in Euclidean geometry is the shortest path between two points. On a sphere, the shortest path between two points is along a great circle, so “lines” in spherical geometry are defined as great circles. In spherical geometry, there are no parallel lines. Any two lines intersect at two points. Pil

ots usually fly along great circles because a great circle is the shortest route between two points on Earth. Spherical Geometry Parallel Postulate Through a point not on a line, there is no line parallel to a given line. E X A M P L E 1 Classifying Figures in Spherical Geometry Name a line, a segment, and a triangle on the sphere. The two points used to name a line cannot be exactly opposite each other on the sphere. In Example 1,   AB could refer to more than one line.  AC is a line. ̶̶ AC is a segment. △ACD is a triangle. 1. Name another line, segment, and triangle on the sphere above. In Example 1, the lines   AC and   AD are both perpendicular to   CD. This means that △ACD has two right angles. So the sum of its angle measures must be greater than 180°. Imagine cutting an orange in half and then cutting each half in quarters using two perpendicular cuts. Each of the resulting triangles has three right angles. Spherical Triangle Sum Theorem The sum of the angle measures of a spherical triangle is greater than 180°. 726 726 Chapter 10 Spatial Reasoning �� E X A M P L E 2 Classifying Spherical Triangles Classify each spherical triangle by its angle measures and by its side lengths. A △ABC △ABC is an obtuse scalene triangle. B △NPQ on Earth has vertex N at the North Pole and vertices P and Q on the equator. PQ is equal to 1 __ the circumference of Earth. 3 NP and NQ are both equal to 1 __ 4 the circumference of Earth. The equator is perpendicular to both of the other two sides of the triangle. Thus △NPQ is an isosceles right triangle. 2. Classify △VWX by its angle measures and by its side lengths. The area of a spherical triangle is part of the surface area of the sphere. For the piece of orange on page 726, the area is 1 __ 8 of the surface area of the orange, or 1 __ 8 (4π r 2 ) = π r 2 ___ 2. If you know the radius of a sphere and the measure of each angle, you can

find the area of the triangle. Area of a Spherical Triangle The area of spherical △ABC on a sphere with radius r is A = π r 2 _ 180° (m∠A + m∠B + m∠C - 180°). E X A M P L E 3 Finding the Area of Spherical Triangles Find the area of each spherical triangle. Round to the nearest tenth, if necessary. A △ABC (m∠A + m∠B + m∠C - 180°) (100 + 106 + 114 - 180) ≈ 152.4 cm 2 B △DEF on Earth’s surface with m∠D = 75°, m∠E = 80°, and m∠F = 30°. (Hint: average radius of Earth = 3959 miles) (m∠D + m∠E + m∠F - 180°) (75 + 80 + 30 - 180) ≈ 1,367,786.7 mi 2 A = π r 2 _ 180° π (14) 2 _ 180° A = A = π r 2 _ 180° π (3959) 2 _ 180° = 3. Find the area of △KLM on a sphere with diameter 20 ft, where m∠K = 90°, m∠L = 90°, and m∠M = 30°. Round to the nearest tenth. Chapter 10 Extension 727 727 �� EXTENSION Exercises Exercises Use the figure for Exercises 1–3. 1. Name all lines on the sphere. 2. Name three segments on the sphere. 3. Name a triangle on the sphere. Determine whether each figure is a line in spherical geometry. 4. m 5. n 6. p Classify each spherical triangle by its angle measures and by its side lengths. 7. 9. Find the area of each spherical triangle. 11. 13. 8. 10. 12. 14. 15. △ABC on the Moon’s surface with m∠A = 35°, m∠B = 48°, and m∠C = 100° (Hint: average radius of the Moon ≈ 1079 miles) 16. △RST on a scale model of Earth with radius 6 m, m∠R = 80°, m∠S = 130°,

and m∠T = 150° 728 728 Chapter 10 Spatial Reasoning �� 17. △ABC is an acute triangle. a. Write an inequality for the sum of the angle measures of △ABC, based on the fact that △ABC is acute. b. Use your result from part a to write an inequality for the area of △ABC. c. Use your result from part b to compare the area of an acute spherical triangle to the total surface area of the sphere. 18. Draw a quadrilateral on a sphere. Include one diagonal in your drawing. Use the sum of the angle measures of the quadrilateral to write an inequality. Geography Compare each length to the length of a great circle on Earth. 19. the distance between the North 20. the distance between the North Pole and the South Pole Pole and any point on the equator 21. Geography If the area of a triangle on Earth’s surface is 100,000 mi 2, what is the sum of its angle measures? (Hint: average radius of Earth ≈ 3959 miles) 22. Sports Describe the curves on the basketball that are lines in spherical geometry. 23. Navigation Pilots navigating long distances often travel along the lines of spherical geometry. Using a globe and string, determine the shortest route for a plane traveling from Washington, D.C., to London, England. What do you notice? 24. Write About It Can a spherical triangle be right and obtuse at the same time? Explain. 25. Write About It A 2-gon is a polygon with two edges. Draw two lines on a sphere. How many 2-gons are formed? What can you say about the positions of the vertices of the 2-gons on the sphere? 26. Challenge Another type of non-Euclidean geometry, called hyperbolic geometry, is defined on a surface that is curved like the bell of a trumpet. What do you think is true about the sum of the angle measures of the triangle shown at right? Compare the sum of the angle measures of a triangle in Euclidean, spherical, and hyperbolic geometry. Chapter 10 Extension 729 729 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary altitude................

.... 680 isometric drawing.......... 662 right cone.................. 690 altitude of a cone........... 690 lateral edge................. 680 right cylinder............... 681 altitude of a pyramid........ 689 lateral face................. 680 right prism................. 680 axis of a cone............... 690 lateral surface.............. 681 slant height of a axis of a cylinder............ 681 net........................ 655 center of a sphere........... 714 oblique cone............... 690 cone....................... 654 oblique cylinder............ 681 cross section............... 656 oblique prism.............. 680 cube....................... 654 orthographic drawing....... 661 cylinder.................... 654 perspective drawing........ 662 edge....................... 654 polyhedron................. 670 face........................ 654 prism......

................ 654 great circle................. 714 pyramid................... 654 hemisphere................ 714 radius of a sphere........... 714 horizon.................... 662 regular pyramid............ 689 regular pyramid.......... 689 slant height of a right cone............... 690 space...................... 671 sphere..................... 714 surface area................ 680 vanishing point............. 662 vertex...................... 654 vertex of a cone............. 690 vertex of a pyramid......... 689 volume.................... 697 Complete the sentences below with vocabulary words from the list above. 1. A(n)? has at least one nonrectangular lateral face. ̶̶̶̶ 2. A name given to the intersection of a three-dimensional figure and a plane is?. ̶̶̶̶ 10-1 Solid Geometry (pp. 654–660) TEKS G.2.B, G.6.A, G.6.B, G.9.D E X A M P L E S ■ Classify the figure. Name the vertices, edges, and bases. pentagonal prism vertices: A, B, C, D, E, F, EXERCISES Classify each figure. Name the vertices, edges, and bases. 3. 4. G, H, J,

K ̶̶ AB, ̶̶ EK, edges: ̶̶ ̶̶ AF, KF, ̶̶ BC, ̶̶ DJ, ̶̶ CD, ̶̶ CH, ̶̶ DE, ̶̶ BG ̶̶ AE, ̶̶ FG, ̶̶̶ GH, ̶̶ HJ, ̶̶ JK, bases: ABCDE, FGHJK ■ Describe the three-dimensional figure that can be made from the given net. The net forms a rectangular prism. 730 730 Chapter 10 Spatial Reasoning Describe the three-dimensional figure that can be made from the given net. 5. 6. �� 10-2 Representations of Three-Dimensional Figures (pp. 661–668) TEKS G.6.C, G.9.D E X A M P L E S ■ Draw all six orthographic views of the given object. Assume there are no hidden cubes. EXERCISES Use the figure made of unit cubes for Exercises 7–10. Assume there are no hidden cubes. 7. Draw all six orthographic views. Top: Bottom: 8. Draw an isometric view. Front: Back: Left: Right: ■ Draw an isometric view of the given object. Assume there are no hidden cubes. 9. Draw the object in one-point perspective. 10. Draw the object in two-point perspective. Determine whether each drawing represents the given object. Assume there are no hidden cubes. 11. 12. 10-3 Formulas in Three Dimensions (pp. 670–677) TEKS G.5.A, G.7.C, G.8.C, G.9.D E X A M P L E S ■ Find the number of vertices, edges, and faces of the given polyhedron. Use your results to verify Euler’s formula. V = 12, E = 18, F = 8 12 - 18 + 8 = 2 ■ Find the distance between the points (6, 3, 4) and (2, 7, 9). Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. distance: d = √ �

� (2 - 6) 2 + (7 - 3) 2 + (9 - 4) 2 = √  57 ≈ 7.5 midpoint4, 5, 6.5) EXERCISES Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 13. 14. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 15. (2, 6, 4) and (7, 1, 1) 16. (0, 3, 0) and (5, 7, 8) 17. (7, 2, 6) and (9, 1, 5) 18. (6, 2, 8) and (2, 7, 4) Study Guide: Review 731 731 10-4 Surface Area of Prisms and Cylinders (pp. 680–687) E X A M P L E S EXERCISES TEKS G.5.A, G.5.B, G.6.B, G.8.D, G.11.D Find the lateral area and surface area of each right prism or cylinder. ■ Find the lateral area and surface area of each right prism or cylinder. Round to the nearest tenth, if necessary. 19. L = Ph = 28 (10) = 280 in 2 S = Ph + 2B = 280 + 2 (49) = 378 in 2 20. a cube with side length 5 ft ■ a cylinder with radius 8 m and height 12 m L = 2πrh = 2π (8)(12) = 192π ≈ 603.2 m 2 S = L + 2B = 192π + 2π (8)2 = 320π ≈ 1005.3 m 2 21. an equilateral triangular prism with height 7 m and base edge lengths 6 m 22. a regular pentagonal prism with height 8 cm and base edge length 4 cm 10-5 Surface Area of Pyramids and Cones (pp. 689–696) E X A M P L E S EXERCISES TEKS G.5.A, G.5.B, G.6.B, G.8.D, G.11.D Find the lateral area and surface area of each right pyramid or cone. ■ Find the lateral area and surface area of each right pyramid or

cone. 23. a square pyramid with side length 15 ft and slant height 21 ft 24. a cone with radius 7 m and height 24 m The radius is 8 m, so the slant height is 25. a cone with diameter 20 in. and slant height 15 in. √ 8 2 + 15 2 = 17 m. L = πrℓ = π (8)(17) = 136π m 2 S = πrℓ + π r 2 = 136π + (8)2π = 200π m 2 ■ a regular hexagonal pyramid with base edge length 8 in. and slant height 20 in. L = 1 _ Pℓ = 1 _ (48) (20) = 480 in 2 2 2 S = L + B = 480 + 1 _ (4 √  3 ) (48) ≈ 646.3 in 2 2 Find the surface area of each composite figure. 26. 27. 10-6 Volume of Prisms and Cylinders (pp. 697–704) E X A M P L E S ■ Find the volume of the prism. 2 V = Bh = ( 1 _ aP) h = 1 _ (4 √  3 ) (48) (12) 2 = 1152 √  3 ≈ 1995.3 cm 3 732 732 Chapter 10 Spatial Reasoning EXERCISES Find the volume of each prism. 28. 29. TEKS G.1.B, G.5.A, G.5.B, G.8.D, G.11.D �� ■ Find the volume of the cylinder6) 2 (14) = 504π ≈ 1583.4 ft 3 Find the volume of each cylinder. 30. 31. 10-7 Volume of Pyramids and Cones (pp. 705–712) TEKS G.5.A, G.5.B, G.8.D, G.11.D E X A M P L E S ■ Find the volume of the pyramid. Bh = 1 _ V = 1 _ (8 · 3) (14) 3 3 = 112 in 3 ■ Find the volume of the cone9) 2 (16) 3 3 = 432π

ft 3 ≈ 1357.2 ft 3 EXERCISES Find the volume of each pyramid or cone. 32. a hexagonal pyramid with base area 42 m 2 and height 8 m 33. an equilateral triangular pyramid with base edge 3 cm and height 8 cm 34. a cone with diameter 12 cm and height 10 cm 35. a cone with base area 16π ft 2 and height 9 ft Find the volume of each composite figure. 36. 37. 10-8 Spheres (pp. 714–721) TEKS G.5.A, G.5.B, G.8.D, G.11.D E X A M P L E EXERCISES ■ Find the volume and surface area of the sphere. Give your answers in terms of π9) 3 = 972π m 2 3 3 S = 4π r 2 = 4π (9) 2 = 324π m 2 Find each measurement. Give your answers in terms of π. 38. the volume of a sphere with surface area 100π m 2 39. the surface area of a sphere with volume 288π in 3 40. the diameter of a sphere with surface area 256π ft 2 Find the surface area and volume of each composite figure. 41. 42. Study Guide: Review 733 733 �� Use the diagram for Items 1–3. 1. Classify the figure. Name the vertices, edges, and bases. 2. Describe a cross section made by a plane parallel to the base. 3. Find the number of vertices, edges, and faces of the polyhedron. Use your results to verify Euler’s formula. Use the figure made of unit cubes for Items 4–6. Assume there are no hidden cubes. 4. Draw all six orthographic views. 5. Draw an isometric view. 6. Draw the object in one-point perspective. Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 7. (0, 0, 0) and (5, 5, 5) 8. (6, 0, 9) and (7, 1, 4) 9. (-1, 4, 3) and (2, -5, 7) Find the surface area of each figure. Round to

the nearest tenth, if necessary. 10. 13. 11. 14. 12. 15. Find the volume of each figure. Round to the nearest tenth, if necessary. 16. 19. 17. 20. 18. 21. 22. Earth’s diameter is approximately 7930 miles. The Moon’s diameter is approximately 2160 miles. About how many times as great is the volume of Earth as the volume of the Moon? 734 734 Chapter 10 Spatial Reasoning �� FOCUS ON SAT MATHEMATICS SUBJECT TEST SAT Mathematics Subject Test results include scaled scores and percentiles. Your scaled score is a number from 200 to 800, calculated using a formula that varies from year to year. The percentile indicates the percentage of people who took the same test and scored lower than you did. You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete. The questions are written so that you should not need to do any lengthy calculations. If you find yourself getting involved in a long calculation, think again about all of the information in the problem to see if you might have missed something helpful. 1. A line intersects a cube at two points, A and B. If each edge of the cube is 4 cm, what is the greatest possible distance between A and B? 4. If triangle ABC is rotated about the x-axis, what is the volume of the resulting cone? (A) 2 √  3 cm (B) 4 cm (C) 4 √  2 cm (D) 4 √  3 cm (E) 16 √  3 cm 2. The lateral area of a right cylinder is 3 times the area of its base. What is the height h of the cylinder in terms of its radius r? (A) 1 _ r 2 (B) 2 _ r 3 (C) 3 _ r 2 (D) 3r (E) 3 r 2 3. What is the lateral area of a right cone with radius 6 ft and height 8 ft? (A) 30π ft 2 (B) 48π ft 2 (C) 60π ft 2 (D) 180π ft 2 (E) 360π ft 2 (A) 100π cubic units (B) 144π cubic units (

C) 240π cubic units (D) 300π cubic units (E) 720π cubic units in 3 5. An oxygen tank is the shape of a cylinder with a hemisphere at each end. If the radius of the tank is 5 inches and the overall length is 32 inches, what is the volume of the tank? (A) 500 _ 3π (B) 2275 _ 12 (C) 1900 _ 3 (D) 2150 _ 3 (E) 2900 _ 3 π in 3 π in 3 π in 3 π in 3 College Entrance Exam Practice 735 735 �� Any Question Type: Measure to Solve Problems On some tests, you may have to measure a figure in order to answer a question. Pay close attention to the units of measure asked for in the question. Some questions ask you to measure to the nearest centimeter, and some ask you to measure to the nearest inch. Multiple Choice: The net of a square pyramid is shown below. Use a ruler to measure the dimensions of the pyramid to the nearest centimeter. Which of the following best represents the total surface area of the square pyramid? 9 square centimeters 21 square centimeters 33 square centimeters 36 square centimeters Use a centimeter ruler to measure one side of the square base. The measurement to the nearest centimeter is 3 cm. The base is a square, so all four side lengths are 3 cm. Measure the altitude of a triangular face, which is the slant height of the pyramid. The altitude is 2 cm. Label the drawing with the measurements. To find the total surface area of the pyramid, find the base area and the lateral area. The base of the pyramid is a square. The base area of the pyramid is A = s 2 = (3) 2 = 9 cm 2. The area of one triangular face is A = 1 _ 2 bh = 1 _ (3) (2) = 3 cm 2. 2 The pyramid has 4 faces, so the lateral area is 4 (3) = 12 cm 2. The total surface area is 9 + 12 = 21 cm 2. The correct answer choice is B. 736 736 Chapter 10 Spatial Reasoning �� Read each test item and answer the questions that follow. �� Measure carefully and make sure you are using the correct units to

measure the figure. Item A The net of a cube is shown below. Use a ruler to measure the dimensions of the cube to the nearest 1 __ Which best represents the volume of the cube to the nearest cubic inch? inch. 4 1 cubic inch 2 cubic inches 5 cubic inches 9 cubic inches 1. Measure one edge of the net for the cube. What is the length to the nearest 1__ 4 inch? 2. How would you use the measurement to find the volume of the cube? Item B The net of a cylinder is shown below. Use a ruler to measure the dimensions of the cylinder to the nearest tenth of a centimeter. Which best represents the total surface area of the cylinder to the nearest square centimeter? 6 square centimeters 16 square centimeters 19 square centimeters 42 square centimeters 3. Which part of the net do you need to measure in order to find the height of the cylinder? Find the height of the cylinder to the nearest tenth of a centimeter. 4. What other measurement(s) do you need in order to find the surface area of the cylinder? Find the measurement(s) to the nearest tenth of a centimeter. 5. How would you use the measurements to find the surface area of the cylinder? TAKS Tackler 737 737 KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–10 Multiple Choice 1. If a point (x, y) is chosen at random in the coordinate plane such that -1 ≤ x ≤ 1 and -5 ≤ y ≤ 3, what is the probability that x ≥ 0 and y ≥ 0? 0.1875 0.25 0.375 0.8125 2. △ABC ∼ △DEF, and △DEF ∼ △GHI. If the similarity ratio of △ABC to △DEF is 1 __ and the 2 similarity ratio of △DEF to △GHI is 3 __, what is the 4 similarity ratio of △ABC to △GHI. Which expression represents the number of faces of a prism with bases that are n-gons? n + 1 n + 2 2n 3n 4. Parallelogram ABCD has a diagonal ̶̶ AC with endpoints A (-1, 3) and C (-3, -3). If B has coordinates (x, y), which of the following represents the coordinates for D? D (-3x, -y) D (-x, -y) D (-

x - 4, -y) D (x - 2, y) 5. Right △ABC with legs AB = 9 millimeters and BC = 12 millimeters is the base of a right prism that has a surface area of 450 square millimeters. What is the height of the prism? 4.75 millimeters 9.5 millimeters 6 millimeters 11 millimeters 6. The radius of a sphere is doubled. What happens to the ratio of the volume of the sphere to the surface area of the sphere? It remains the same. It is doubled. It is increased by a factor of 4. It is increased by a factor of 8. 738 738 Chapter 10 Spatial Reasoning 7. ̶̶ AB has endpoints A (x, y, z) and B (-2, 6, 13) and midpoint M (2, -6, 3). What are the coordinates of A? A (-6, 18, 23) A (0, 0, 8) A (2, -6, 19) A (6, -18, -7) 8. If ̶̶ DE bisects ∠CEF, which of the following additional statements would allow you to conclude that △DEF ≅ △ABC? ∠DEF ≅ ∠BAC ∠DEF ≅ ∠CDE ̶̶ ̶̶ CD EF ≅ ̶̶ ̶̶ EC EF ≅ 9. To the nearest tenth of a cubic centimeter, what is the volume of a right regular octagonal prism with base edge length 4 centimeters and height 7 centimeters? 180.3 cubic centimeters 224.0 cubic centimeters 270.4 cubic centimeters 540.8 cubic centimeters 10. Which of the following must be true about a conditional statement? If the inverse is false, then the converse is false. If the conditional is true, then the contrapositive is false. If the conditional is true, then the converse is false. If the hypothesis of the conditional is true, then the conditional is true. �� It may be helpful to include units in your calculations of measures of geometric figures. If your answer includes the proper units, you are less likely to have made an error. 11. A right cylinder has a height of 10 inches. The area of the base is 63.6 square inches. To the nearest tenth of a square inch, what is the lateral area for this cylinder? 53.6 square inches 282.7 square inches 409

.9 square inches 634.6 square inches 12. The volume of the smaller sphere is 288 cubic centimeters. Find the volume of the larger sphere. 864 cubic centimeters 2,592 cubic centimeters 7,776 cubic centimeters 23,328 cubic centimeters Gridded Response 13.  u = 〈3, -7〉, and  v = 〈-6, 5〉. What is the magnitude of the resultant vector to the nearest tenth of  u and  v? STANDARDIZED TEST PREP Short Response 17. The area of trapezoid GHIJ is 103.5 square centimeters. Find each of the following. Round answers to the nearest tenth. Show your work or explain in words how you found your answers. a. the height of trapezoid GHIJ b. m∠J 18. The figure shows the top view of a stack of cubes. The number on each cube represents the number of stacked cubes. Draw the bottom, back, and right views of the object. 19. △ABC has vertices A (1, -2), B (-2, -3), and C (-2, 2). a. Graph △A'B'C', the image of △ABC, after a dilation with a scale factor of 3 __. 2 b. Show that   AB ǁ   A'B',   BC ǁ   B'C', and   CA ǁ   C'A'. Use slope to justify your answer. 14. If a polyhedron has 12 vertices and 8 faces, how many edges does the polyhedron have? Extended Response 15. If Y is the circumcenter of △PQR, what is the value of x? 20. A right cone has a lateral area of 30π square inches and a slant height of 6 inches. 16. How many cubes with edge length 3 centimeters will fit in a box that is a rectangular prism with length 12 centimeters, width 15 centimeters, and height 24 centimeters? a. Find the height of the cone. Show your work or explain in words how you determined your answer. Round

your answer to the nearest tenth. b. Find the volume of this cone. Round your answer to the nearest tenth. c. Given a right cone with a lateral area of L and a slant height of ℓ, find an equation for the volume in terms of L and ℓ. Show your work. Cumulative Assessment, Chapters 1–10 739 739 �� T E X A S TAKS Grades 9–11 Obj. 10 Reliant Stadium When Houston’s Reliant Stadium opened in 2002, it was the first NFL stadium to have a retractable roof. In addition to football games, Reliant Stadium hosts rodeos, concerts, and other events. When configured for football, up to 72,000 fans can sit around 97,000 square feet of playing field. Choose one or more strategies to solve each problem. 1. Inside the playing field, the football field is 160 ft by 360 ft. Approximately how many acres of land surround the football field? (Hint: 1 acre = 43,560 ft 2 ) �� For 2 and 3, use the table. 2. There are two scoreboards in Reliant Stadium. Suppose the ratio of each scoreboard’s length to its width were 46 : 7. What would the dimensions of each scoreboard be? 3. Each scoreboard is equipped with a large video screen. A screen’s width is 72.5 ft greater than its height. What are the dimensions of the video screens on the scoreboards? Approximate Scoreboard Measurements Perimeter (ft) Scoreboard Video Screen 636 241 Area ( ft 2 ) 11,592 2,316 4. Reliant Stadium can be divided into 12 seating sections of equal size. When the roof is opened at a certain time of day, 4 of these sections are in the sunlight. Suppose you choose a seat in the arena at random when the roof is retracted. What is the probability that you are sitting in sunlight? What is the probability that you are sitting in the shade? 740 740 Chapter 10 Spatial Reasoning �� Texas Coins Although Texas does not have its own mint, Texas has a rich history in coins. Starting in the mid-1800s, many Texas businesses issued tokens that were redeemable for merchandise in place of U.S. money. In 1935, the U.S. Mint issued a coin to celebrate explorer Al

var Nunez Cabeza de Vaca’s travels across southern America and Texas. In 2004, the U.S. Mint issued the Texas state quarter. Choose one or more strategies and use the table to solve each problem. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List 1. Quarters are stamped out of a rectangular metal strip that is 13 in. wide by 1500 ft long. Given that the diameter of a quarter is just under an inch (0.955 in.), what is the minimum number of strips needed for 700,000 Texas state quarters? 2. A Spanish Trail memorial half dollar contains a small amount of copper, but most of the metal in the coin is silver. The volume of copper in a Spanish Trail memorial half dollar is about 6.58 mm 3. What percent of the half dollar is copper? 3. Many Texas trade tokens were made from aluminum. About how many tokens could be made from a block of aluminum with a volume of 1 m 3? Texas State Quarter Spanish Trail Memorial Half Dollar Texas Trade Token Coin Specifications Diameter (mm) Thickness (mm) 24.26 1.75 30.60 2.15 44.00 1.42 4. 4. The regular octagonal token shown was issued by a 4. barber shop in Fort Worth. If the distance from the midpoint of one side to the midpoint of the opposite side is 25 mm, what is the area of the face of the coin? Problem Solving on Location 741 741 Circles 11A Lines and Arcs in Circles 11-1 Lines That Intersect Circles 11-2 Arcs and Chords 11-3 Sector Area and Arc Length 11B Angles and Segments in Circles 11-4 Inscribed Angles Lab Explore Angle Relationships in Circles 11-5 Angle Relationships in Circles Lab Explore Segment Relationships in Circles 11-6 Segment Relationships in Circles 11-7 Circles in the Coordinate Plane Ext Polar Coordinates KEYWORD: MG7 ChProj On the floor of the Capitol rotunda are six seals, commemorating the flags that have flown over Texas. 742 742 Chapter 11 Vocabulary Match each term on the left with a definition on the right. A. the distance around a circle 1. radius 2. pi

3. circle 4. circumference B. the locus of points in a plane that are a fixed distance from a given point C. a segment with one endpoint on a circle and one endpoint at the center of the circle D. the point at the center of a circle E. the ratio of a circle’s circumference to its diameter Tables and Charts The table shows the number of students in each grade level at Middletown High School. Find each of the following. 5. the percentage of students who are freshman 6. the percentage of students who are juniors 7. the percentage of students who are sophomores or juniors Year Freshman Sophomore Junior Senior Number of Students 192 208 216 184 Circle Graphs The circle graph shows the age distribution of residents of Mesa, Arizona, according to the 2000 census. The population of the city is 400,000. 8. How many residents are between the ages of 18 and 24? 9. How many residents are under the age of 18? 10. What percentage of the residents are over the age of 45? 11. How many residents are over the age of 45? Solve Equations with Variables on Both Sides Solve each equation. 12. 11y - 8 = 8y + 1 14. z + 30 = 10z - 15 16. -2x - 16 = x + 6 13. 12x + 32 = 10 + x 15. 4y + 18 = 10y + 15 17. -2x - 11 = -3x - 1 Solve Quadratic Equations Solve each equation. 18. 17 = x 2 - 32 20. 4 x 2 + 12 = 7 x 2 19. 2 + y 2 = 18 21. 188 - 6 x 2 = 38 Circles 743 743 �� Key Vocabulary/Vocabulario arc arco arc length longitud de arco central angle ángulo central chord secant cuerda secante sector of a circle sector de un círculo segment of a circle segmento de un círculo semicircle semicírculo tangent of a circle tangente de un círculo Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, answer the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word semicircle begins with

the prefix semi-. List some other words that begin with semi-. What do all of these words have in common? 2. The word central means “located at the center.” How can you use this definition to understand the term central angle of a circle? 3. The word tangent comes from the Latin word tangere, which means “to touch.” What does this tell you about a line that is a tangent to a circle? Geometry TEKS Les. 11-1 Les. 11-2 Les. 11-3 Les. 11-4 G.1.A Geometric structure* develop an awareness of the ★ ★ ★ ★ 11-5 Tech. Lab 11-6 Tech. Lab Les. 11-5 ★ Les. 11-6 Les. 11-7 Ext ★ ★ ★ structure of a mathematical system,... G.1.B Geometric structure* recognize the historical development of geometric systems... ★ G.2.A Geometric structure* use constructions to explore ★ ★ ★ ★ ★ attributes of geometric figures and to make conjectures... G.2.B Geometric structure* make conjectures about... circles,... and determine the validity of the conjectures,... ★ ★ ★ ★ ★ ★ ★ ★ G.3.B Geometric structure* construct and justify statements ★ about geometric figures... G.5.A Geometric patterns* use... patterns to develop algebraic ★ ★ ★ ★ ★ expressions representing geometric properties G.5.B Geometric patterns* use numeric and geometric ★ ★ ★ patterns to make generalizations about geometric properties, including... angle relationships in... circles G.8.B Congruence and the geometry of size* find areas of ★ sectors and arc lengths of circles using proportional reasoning G.8.C Congruence and the geometry of size* … use the ★ Pythagorean Theorem G.9.C Congruence and the geometry of size* formulate and test conjectures about the properties and attributes of circles and the lines that intersect them... ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 744 744 Chapter 11 Reading Strategy: Read to Solve Problems A word problem may be overwhelming at first. Once you identify the important parts of the problem and translate the words into math language, you will find that the problem is similar to others you have solved. Reading Tips: ✔ Read each phrase slowly. Write down ✔ Translate the words or

phrases what the words mean as you read them. into math language. ✔ Draw a diagram. Label the diagram so it ✔ Highlight what is being asked. makes sense to you. ✔ Read the problem again before finding your solution. From Lesson 10-3: Use the Reading Tips to help you understand this problem. 14. After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The elevation of the camp is 0.6 km higher than the starting point. What is the distance from the camp to the starting point? After a day hike, a group of hikers set up a camp 3 km east and 7 km north of the starting point. The starting point can be represented by the ordered triple (0, 0, 0). The elevation of the camp is 0.6 km higher than the starting point. The camp can be represented by the ordered triple (3, 7, 0.6). What is the distance from the camp to the starting point? Distance can be found using the Distance Formula. Use the Distance Formula to find the distance between the camp and the starting point = √  = √  (3 - 0) 2 + (7 - 0) 2 + (0.6 - 0) 2 ≈ 7.6 km Try This For the following problem, apply the following reading tips. Do not solve. • Identify key words. • Translate each phrase into math language. • Draw a diagram to represent the problem. 1. The height of a cylinder is 4 ft, and the diameter is 9 ft. What effect does doubling each measure have on the volume? Circles 745 745 �� 11-1 Lines That Intersect Circles TEKS G.9.C Congruence and the geometry of size: … test conjectures about the properties and attributes of circles and the lines that intersect …. Also G.1.A, Objectives Identify tangents, secants, and chords. Use properties of tangents to solve problems. Vocabulary interior of a circle exterior of a circle chord sec

ant tangent of a circle point of tangency congruent circles concentric circles tangent circles common tangent Why learn this? You can use circle theorems to solve problems about Earth. (See Example 3.) This photograph was taken 216 miles above Earth. From this altitude, it is easy to see the curvature of the horizon. Facts about circles can help us understand details about Earth. Recall that a circle is the set of all points in a plane that are equidistant from a given point, called the center of the circle. A circle with center C is called circle C, or ⊙C. The interior of a circle is the set of all points inside the circle. The exterior of a circle is the set of all points outside the circle. �� Lines and Segments That Intersect Circles TERM DIAGRAM A chord is a segment whose endpoints lie on a circle. Also G.2.A, G.2.B A secant is a line that intersects a circle at two points. A tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency. E X A M P L E 1 Identifying Lines and Segments That Intersect Circles Identify each line or segment that intersects ⊙A. chords: ̶̶ EF and ̶̶ BC tangent: ℓ ̶̶ AC and radii: ̶̶ AB secant:   EF diameter: ̶̶ BC 746 746 Chapter 11 Circles �� 1. Identify each line or segment that intersects ⊙P. Remember that the terms radius and diameter may refer to line segments, or to the lengths of segments. Pairs of Circles TERM DIAGRAM Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles. ̶̶ ⊙A ≅ ⊙B if BD. ̶̶ ̶̶ BD if ⊙A ≅ ⊙B. AC ≅ ̶̶ AC ≅ Internally tangent circles Externally tangent circles E X A M P L E

2 Identifying Tangents of Circles Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. radius of ⊙A : 4 radius of ⊙B : 2 Center is (-1, 0). Pt. on ⊙ is (3, 0). Dist. between the 2 pts. is 4. Center is (1, 0). Pt. on ⊙ is (3, 0). Dist. between the 2 pts. is 2. point of tangency: (3, 0) Pt. where the ⊙s and tangent line intersect equation of tangent line: x = 3 Vert. line through (3, 0) 2. Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 11- 1 Lines That Intersect Circles 747 747 �� A common tangent is a line that is tangent to two circles. Lines ℓ and m are common external tangents to ⊙A and ⊙B. Lines p and q are common internal tangents to ⊙A and ⊙B. Construction Tangent to a Circle at a Point    Draw ⊙P. Locate a point on the circle and label it Q. Draw  PQ. Construct the perpendicular ℓ to  PQ at Q. This line is tangent to ⊙P at Q. Notice that in the construction, the tangent line is perpendicular to the radius at the point of tangency. This fact is the basis for the following theorems. Theorems THEOREM HYPOTHESIS CONCLUSION 11-1-1 Theorem 11-1-2 is the converse of Theorem 11-1-1. 11-1-2 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. (line tangent to ⊙ → line ⊥ to radius) If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. (line ⊥ to radius → line tangent to ⊙) ℓ is tangent

to ⊙A m is ⊥ to ̶̶ CD at D ℓ ⊥ ̶̶ AB m is tangent to ⊙C. You will prove Theorems 11-1-1 and 11-1-2 in Exercises 28 and 29. 748 748 Chapter 11 Circles �� E X A M P L E 3 Problem Solving Application The summit of Mount Everest is approximately 29,000 ft above sea level. What is the distance from the summit to the horizon to the nearest mile? Understand the Problem The answer will be the length of an imaginary segment from the summit of Mount Everest to Earth’s horizon. Make a Plan Draw a sketch. Let C be the center of Earth, E be the summit of Mount Everest, and H be a point on the horizon. You need to find the length of tangent to ⊙C at H. By Theorem 11-1-1, △CHE is a right triangle. ̶̶ EH, which is ̶̶ EH ⊥ ̶̶ CH. So 5280 ft = 1 mi Earth’s radius ≈ 4000 mi Solve ED = 29,000 ft Given = 29,000 _ 5280 ≈ 5.49 mi EC = CD + ED Change ft to mi. Seg. Add. Post. = 4000 + 5.49 = 4005.49 mi Substitute 4000 for CD and 5.49 for ED. EC 2 = EH 2 + CH 2 4005.49 2 = EH 2 + 4000 2 43,950.14 ≈ EH 2 210 mi ≈ EH Look Back Pyth. Thm. Substitute the given values. Subtract 4000 2 from both sides. Take the square root of both sides. The problem asks for the distance to the nearest mile. Check if your answer is reasonable by using the Pythagorean Theorem. Is 210 2 + 4000 2 ≈ 4005 2? Yes, 16,044,100 ≈ 16,040,025. 3. Kilimanjaro, the tallest mountain in Africa, is 19,340 ft tall. What is the distance from the summit of Kilimanjaro to the horizon to the nearest mile? Theorem 11-1-3 THEOREM HYPOTHESIS CONCLUSION If two segments are tangent to a circle from the same external point, then the segments are congruent

. (2 segs. tangent to ⊙ from same ext. pt. → segs. ≅) ̶̶ AB ≅ ̶̶ AC ̶̶ AB and ̶̶ AC are tangent to ⊙P. You will prove Theorem 11-1-3 in Exercise 30. 11- 1 Lines That Intersect Circles 749 749 12��34�� You can use Theorem 11-1-3 to find the length of segments drawn tangent to a circle from an exterior point. E X A M P L E 4 Using Properties of Tangents ̶̶ DE and ̶̶ DF are tangent to ⊙C. Find DF. DE = DF 2 segs. tangent to ⊙ from same ext. pt. → segs. ≅. 5y - 28 = 3y Substitute 5y - 28 for DE and 3y for DF. 2y - 28 = 0 Subtract 3y from both sides. 2y = 28 y = 14 DF = 3 (14) = 42 Add 28 to both sides. Divide both sides by 2. Substitute 14 for y. Simplify. ̶̶ RS and 4a. ̶̶ RT are tangent to ⊙Q. Find RS. 4b. THINK AND DISCUSS 1. Consider ⊙A and ⊙B. How many different lines are common tangents to both circles? Copy the circles and sketch the common external and common internal tangent lines. 2. Is it possible for a line to be tangent to two concentric circles? Explain your answer. 3. Given ⊙P, is the center P a part of the circle? Explain your answer. 4. In the figure, ̶̶ RQ is tangent to ⊙P at Q. Explain how you can find m∠PRQ. 5. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a definition and draw a sketch. 750 750 Chapter 11 Circles �� 11-1 Exercises Exercises KEYWORD: MG7 11-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A? is a line in the plane of a

circle that intersects the circle at two points. ̶̶̶̶ (secant or tangent) 2. Coplanar circles that have the same center are called?. ̶̶̶̶ (concentric or congruent) 3. ⊙Q and ⊙R both have a radius of 3 cm. Therefore the circles are?. ̶̶̶̶ (concentric or congruent Identify each line or segment that intersects each circle. p. 746 4. 5. 747 Multi-Step Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 6. 7. 749 8. Space Exploration The International Space Station orbits Earth at an altitude of 240 mi. What is the distance from the space station to Earth’s horizon to the nearest mile The segments in each figure are tangent to the circle. Find each length. p. 750 9. JK 10. ST 11- 1 Lines That Intersect Circles 751 751 �� PRACTICE AND PROBLEM SOLVING Identify each line or segment that intersects each circle. 11. 12. Multi-Step Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point. 13. 14. Independent Practice For See Exercises Example 11–12 13–14 15 16–17 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 Astronomy 15. Astronomy Olympus Mons’s peak rises 25 km above the surface of the planet Mars. The diameter of Mars is approximately 6794 km. What is the distance from the peak of Olympus Mons to the horizon to the nearest kilometer? Olympus Mons, located on Mars, is the tallest known volcano in the solar system. The segments in each figure are tangent to the circle. Find each length. 16. AB 17. RT Tell whether each statement is sometimes, always, or never true. 18. Two circles with the same center are congruent. 19. A tangent to a circle intersects the circle at two points. 20. Tangent circles have the same center. 21. A tangent to a circle will form a right angle with a radius that is drawn to the point of tangency. 22. A chord of

a circle is a diameter. Graphic Design Use the following diagram for Exercises 23–25. The blue topaz was adopted as the Texas state gemstone in 1969. Identify the following. � 23. diameter 24. radii 25. chord � � � � 752 752 Chapter 11 Circles �� In each diagram, ̶̶ PR and 26. m∠Q ̶̶ PS are tangent to ⊙Q. Find each angle measure. 27. m∠P 28. Complete this indirect proof of Theorem 11-1-1. Given: ℓ is tangent to ⊙A at point B. Prove: ℓ ⊥ ̶̶ AB ̶̶ AC such that ̶̶ AB. Then it is possible to ̶̶ AC ⊥ ℓ. If this is true, then △ACB is?. Since ℓ is a ̶̶̶̶ Proof: Assume that ℓ is not ⊥ draw a right triangle. AC < AB because a. tangent line, it can only intersect ⊙A at b.?, and C must be in the ̶̶̶̶ exterior of ⊙A. That means that AC > AB since contradicts the fact that AC < AB. Thus the assumption is false, and d.?. This ̶̶̶̶ ̶̶ AB is a c.?. ̶̶̶̶ 29. Prove Theorem 11-1-2. ̶̶ Given: m ⊥ CD Prove: m is tangent to ⊙C. (Hint: Choose a point on m. Then use the Pythagorean Theorem to prove that if the point is not D, then it is not on the circle.) 30. Prove Theorem 11-1-3. ̶̶ AB and ̶̶ ̶̶ AB ≅ AC Given: Prove: ̶̶ AC are tangent to ⊙P. Plan: Draw auxiliary segments the triangles formed are congruent. Then use CPCTC. ̶̶ PA, ̶̶ PB, and ̶̶ PC. Show that Algebra Assume the segments that appear to be tangent are tangent. Find each length. 31. ST 32. DE 33. JL 34. ⊙M has center M (2

, 2) and radius 3. ⊙N has center N (-3, 2) and is tangent to ⊙M. Find the coordinates of the possible points of tangency of the two circles. 35. This problem will prepare you for the Multi-Step TAKS Prep on page 770. The diagram shows the gears of a bicycle. AD = 5 in., and BC = 3 in. CD, the length of the chain between the gears, is 17 in. a. What type of quadrilateral is BCDE? Why? b. Find BE and AE. c. What is AB to the nearest tenth of an inch? � � � � � 11- 1 Lines That Intersect Circles 753 753 �� 36. Critical Thinking Given a circle with diameter ̶̶ BC, is it possible to draw tangents to B and C from an external point X? If so, make a sketch. If not, explain why it is not possible. 37. Write About It ̶̶ PR and Explain why ∠P and ∠Q are supplementary. ̶̶ PS are tangent to ⊙Q. 38. ̶̶ AB and is closest to AD? ̶̶ AC are tangent to ⊙D. Which of these 9.5 cm 10 cm 10.4 cm 13 cm 39. ⊙P has center P (3, -2) and radius 2. Which of these lines is tangent to ⊙P? x = 4 y = -4 y = -2 x = 0 40. ⊙A has radius 5. ⊙B has radius 6. What is the ratio of the area of ⊙A to that of ⊙B? 125 _ 216 25 _ 36 5 _ 6 36 _ 25 CHALLENGE AND EXTEND ̶̶̶ 41. Given: ⊙G with GH ⊥ ̶̶ KH ̶̶ JH ≅ Prove: ̶̶ JK 42. Multi-Step ⊙A has radius 5, ⊙B has radius 2, ̶̶ CD is a common tangent. What is AB? and (Hint: Draw a perpendicular segment from B to E, a point on ̶̶ AC.) 43. Manufacturing A company builds metal stands for bicycle wheels. A new design calls for a V-shaped

stand that will hold wheels with a 13 in. radius. The sides of the stand form a 70° angle. To the nearest tenth of an inch, what should be the length XY of a side so that it is tangent to the wheel? SPIRAL REVIEW 44. Andrea and Carlos both mow lawns. Andrea charges $14.00 plus $6.25 per hour. Carlos charges $12.50 plus $6.50 per hour. If they both mow h hours and Andrea earns more money than Carlos, what is the range of values of h? (Previous course) ̶̶ LR. A point is chosen randomly on Use the diagram to find the probability of each event. (Lesson 9-6) ̶̶̶ MP. 45. The point is not on ̶̶̶ MN or 47. The point is on ̶̶ PR. 754 754 Chapter 11 Circles 46. The point is on 48. The point is on ̶̶ LP. ̶̶ QR. �� Circle Graphs Data Analysis A circle graph compares data that are parts of a whole unit. When you make a circle graph, you find the measure of each central angle. A central angle is an angle whose vertex is the center of the circle. See Skills Bank page S80 Example Make a circle graph to represent the following data. Step 1 Add all the amounts. 110 + 40 + 300 + 150 = 600 Step 2 Write each part as a fraction of the whole. fiction: 110 _ ; nonfiction: 40 _ ; children’s: 300 _ ; audio books: 150 _ 600 600 600 600 Books in the Bookmobile Fiction Nonfiction Children’s 110 40 300 150 Step 3 Multiply each fraction by 360° to calculate the central Audio books angle measure. 110 _ 600 (360°) = 66°; 40 _ 600 (360°) = 24°; 300 _ (360°) = 180°; 150 _ (360°) = 90° 600 600 Step 4 Make a circle graph. Then color each section of the circle to match the data. The section with a central angle of 66° is green, 24° is orange, 180° is purple, and 90° is yellow. Try This TAKS Grades 9–11 Obj. 8, 9 Choose the circle graph that best represents the data. Show each step. 1. Books in Linda’s Library Novels

Reference Textbooks 18 10 8 2. Vacation Expenses ($) 3. Puppy Expenses ($) Travel Meals Lodging Other 450 120 900 330 Food Health Training Other 190 375 120 50 On Track for TAKS 755 755 �� 11-2 Arcs and Chords TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system …. Also G.2.A, G.2.B, G.8.C, G.9.C Objectives Apply properties of arcs. Apply properties of chords. Who uses this? Market analysts use circle graphs to compare sales of different products. Vocabulary central angle arc minor arc major arc semicircle adjacent arcs congruent arcs Minor arcs may be named by two points. Major arcs and semicircles must be named by three points. A central angle is an angle whose vertex is the center of a circle. An arc is an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them. Arcs and Their Measure ARC MEASURE DIAGRAM A minor arc is an arc whose points are on or in the interior of a central angle. The measure of a minor arc is equal to the measure of its central angle. m ⁀ AC = m∠ABC = x° A major arc is an arc whose points are on or in the exterior of a central angle. The measure of a major arc is equal to 360° minus the measure of its central angle. m ⁀ ADC = 360° - m∠ABC = 360° - x° If the endpoints of an arc lie on a diameter, the arc is a semicircle. The measure of a semicircle is equal to 180°. m ⁀ EFG = 180° E X A M P L E 1 Data Application The circle graph shows the types of music sold during one week at a music store. Find m ⁀ BC. m ⁀ BC = m∠BMC m of arc = m of central ∠. m∠BMC = 0.13 (360°) Central ∠ is 13% = 46.8° of the ⊙. Use the graph to find each of the following. 1a. m∠FMC 1b. m ⁀ AHB 1c. m∠EMD 756 756 Chapter 11 Circles ��

�� Adjacent arcs are arcs of the same circle that intersect at exactly one point. ⁀ RS and ⁀ ST are adjacent arcs. Postulate 11-2-1 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m⁀ABC = m ⁀AB + m ⁀BC E X A M P L E 2 Using the Arc Addition Postulate Find m ⁀CDE m ⁀CD = 90° m∠DFE = 18° m ⁀DE = 18° m ⁀CE = m ⁀CD + m ⁀DE m∠CFD = 90° Vert.  Thm. m∠DFE = 18° Arc Add. Post. = 90° + 18° = 108° Substitute and simplify. Find each measure. 2a. m ⁀JKL 2b. m ⁀LJN Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure, ⁀ ST ≅ ⁀ UV. Theorem 11-2-2 THEOREM HYPOTHESIS CONCLUSION In a circle or congruent circles: (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles. ∠EAD ≅ ∠BAC ̶̶ DE ≅ ̶̶ BC ̶̶ ED ≅ ̶̶ BC ⁀ DE ≅ ⁀ BC ⁀ ED ≅ ⁀ BC ∠DAE ≅ ∠BAC You will prove parts 2 and 3 of Theorem 11-2-2 in Exercises 40 and 41. 11-2 Arcs and Chords 757 757 �� The converses of the parts of Theorem 11-2-2 are also true. For example, with part 1, congruent chords have congruent central angles. PROOF PROOF Theorem 11-2-2 (Part 1) Given: ∠BAC ≅ ∠DAE ̶̶ BC ≅ Prove:

̶̶ DE Proof: Statements Reasons 1. ∠BAC ≅ ∠DAE ̶̶ AB ≅ ̶̶̶ AD, ̶̶ AC ≅ ̶̶ AE 2. 3. △BAC ≅ △DAE ̶̶ DE ̶̶ BC ≅ 4. 1. Given 2. All radii of a ⊙ are ≅. 3. SAS Steps 2, 1 4. CPCTC E X A M P L E 3 Applying Congruent Angles, Arcs, and Chords Find each measure. A ̶̶ RS ≅ ̶̶ TU. Find m ⁀ RS. ⁀ RS ≅ ⁀ TU m ⁀ RS = m ⁀ TU 3x = 2x + 27 x = 27 m ⁀ RS = 3 (27) = 81° ≅ chords have ≅ arcs. Def. of ≅ arcs Substitute the given measures. Subtract 2x from both sides. Substitute 27 for x. Simplify. B ⊙B ≅ ⊙E, and ⁀ AC ≅ ⁀ DF. Find m∠DEF. ∠ABC = ∠DEF m∠ABC = m∠DEF 5y + 5 = 7y - 43 5 = 2y - 43 ≅ arcs have ≅ central . Def. of ≅  Substitute the given measures. Subtract 5y from both sides. 48 = 2y 24 = y m∠DEF = 7 (24) - 43 = 125° Add 43 to both sides. Divide both sides by 2. Substitute 24 for y. Simplify. Find each measure. 3a.  PT bisects ∠RPS. Find RT. 3b. ⊙A ≅ ⊙B, and Find m ⁀ CD. ̶̶ CD ≅ ̶̶ EF. 758 758 Chapter 11 Circles �� Theorems THEOREM HYPOTHESIS CONCLUSION 11-2-3 In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. 11-2-4

In a circle, the perpendicular bisector of a chord is a radius (or diameter). ̶̶ CD bisects ̶̶ EF and ⁀ EF. ̶̶ CD ⊥ ̶̶ EF ̶̶ JK is a diameter of ⊙A. ̶̶ JK is ⊥ bisector of ̶̶̶ GH. You will prove Theorems 11-2-3 and 11-2-4 in Exercises 42 and 43. E X A M P L E 4 Using Radii and Chords Find BD. Step 1 Draw radius ̶̶ AD. AD = 5 Radii of a ⊙ are ≅. Step 2 Use the Pythagorean Theorem. CD 2 + AC 2 = AD 2 CD 2 + 3 2 = 5 2 CD 2 = 16 CD = 4 Step 3 Find BD. BD = 2(4 ) = 8 Substitute 3 for AC and 5 for AD. Subtract 3 2 from both sides. Take the square root of both sides. ̶̶ AE ⊥ ̶̶ BD, so ̶̶ AE bisects ̶̶ BD. 4. Find QR to the nearest tenth. THINK AND DISCUSS 1. What is true about the measure of an arc whose central angle is obtuse? 2. Under what conditions are two arcs the same measure but not congruent? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a definition and draw a sketch. 11-2 Arcs and Chords 759 759 �� 11-2 Exercises Exercises KEYWORD: MG7 11-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An arc that joins the endpoints of a diameter is called a?. (semicircle or ̶̶̶̶ major arc) 2. How do you recognize a central angle of a circle? ABC = 205°. Therefore ⁀ 3. In ⊙P m ⁀ ABC is a?. (major arc or minor arc) ̶̶̶̶ 4. In a circle, an arc that is less than a semicircle is a?. (major arc or minor arc) ̶̶̶̶. 756 Consumer Application Use

the following information for Exercises 5–10. The circle graph shows how a typical household spends money on energy. Find each of the following. 5. m∠PAQ 7. m∠SAQ 9. m ⁀ RQ 6. m∠VAU 8. m ⁀ UT 10. m ⁀ UPT. 757 Find each measure. 11. m ⁀ DF 12. m ⁀ DEB 13. m ⁀ JL 14. m ⁀ HLK 15. ∠QPR ≅ ∠RPS. Find QR. 16. ⊙A ≅ ⊙B, and ⁀ CD ≅ ⁀ EF. Find m∠EBF. p. 758 Multi-Step Find each length to the nearest tenth. p. 759 17. RS 18. EF 760 760 Chapter 11 Circles �� Independent Practice For See Exercises Example 19–24 25–28 29–30 31–32 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Sports Use the following information for Exercises 19–24. The key shows the number of medals won by U.S. athletes at the 2004 Olympics in Athens. Find each of the following to the nearest tenth. �� 19. m∠ADB 21. m ⁀ AB 21. 23. m ⁀ ACB 20. m∠ADC 22. m ⁀ BC 24. m ⁀ CAB Find each measure. 25. m ⁀ MP 26. m ⁀ QNL 29. ⊙A ≅ ⊙B, and ⁀ CD ≅ ⁀ EF. Find m∠CAD. �� 27. m ⁀ WT 28. m ⁀ WTV 30. ̶̶ JK ≅ ̶̶̶ LM. Find m ⁀ JK. Multi-Step Find each length to the nearest tenth. 31. CD 32. RS Determine whether each

statement is true or false. If false, explain why. 33. The central angle of a minor arc is an acute angle. 34. Any two points on a circle determine a minor arc and a major arc. 35. In a circle, the perpendicular bisector of a chord must pass through the center of the circle. 36. Data Collection Use a graphing calculator, a pH probe, and a data-collection device to collect information about the pH levels of ten different liquids. Then create a circle graph with the following sectors: strong basic (9 < pH < 14), weak basic (7 < pH < 9), neutral (pH = 7), weak acidic (5 < pH < 7), and strong acidic (0 < pH < 5). 37. In ⊙E, the measures of ∠AEB, ∠BEC, and ∠CED are in the ratio 3 : 4 : 5. Find m ⁀ AB, m ⁀ BC, and m ⁀ CD. 11-2 Arcs and Chords 761 761 �� Algebra Find the indicated measure. 38. m ⁀ JL 39. m∠SPT 40. Prove ≅ chords have ≅ arcs. ̶̶ Given: ⊙A, BC ≅ Prove: ⁀ BC ≅ ⁀ DE ̶̶ DE 41. Prove ≅ arcs have ≅ central . Given: ⊙A, ⁀ BC ≅ ⁀ DE Prove: ∠BAC ≅ ∠DAE 42. Prove Theorem 11-2-3. Given: ⊙C, Prove: ̶̶ EF ̶̶ EF ̶̶ CD ⊥ ̶̶ CD bisects and ⁀ EF. ̶̶ CE and ̶̶ CF (Hint: Draw and use the HL Theorem.) 43. Prove Theorem 11-2-4. ̶̶ JK ⊥ ̶̶̶ GH Given: ⊙A, bisector of Prove: (Hint: Use the Converse of the ⊥ Bisector Theorem.) ̶̶ JK is a diameter 44. Critical Thinking Roberto folds a circular piece of paper as shown. When he unfolds the paper, how many different-sized

central angles will be formed? One fold Two folds Three folds 45. /////ERROR ANALYSIS///// Below are two solutions to find the value of x. Which solution is incorrect? Explain the error. 46. Write About It According to a school survey, 40% of the students take a bus to school, 35% are driven to school, 15% ride a bike, and the remainder walk. Explain how to use central angles to create a circle graph from this data. 47. This problem will prepare you for the Multi-Step TAKS This problem will prepare you for the Multi-Step TAKS Prep on page 770. Chantal’s bike has wheels with a 27 in. diameter. a. What are AC and AD if DB is 7 in.? b. What is CD to the nearest tenth of an inch? c. What is CE, the length of the top of the bike stand? � � � � � 762 762 Chapter 11 Circles �� 48. Which of these arcs of ⊙Q has the greatest measure? ⁀ WT ⁀ UW ⁀ VR ⁀ TV 49. In ⊙A, CD = 10. Which of these is closest ̶̶ AE? to the length of 3.3 cm 4 cm 5 cm 7.8 cm 50. Gridded Response ⊙P has center P (2, 1) and radius 3. What is the measure, in degrees, of the minor arc with endpoints A (-1, 1) and B (2, -2)? CHALLENGE AND EXTEND ̶̶ 51. In the figure, AB ⊥ ̶̶ CD. Find m ⁀ BD to the nearest tenth of a degree. 52. Two points on a circle determine two distinct arcs. How many arcs are determined by n points on a circle? (Hint: Make a table and look for a pattern.) 53. An angle measure other than degrees is radian measure. 360° converts to 2π radians, or 180°converts to π radians., π _, π

_ a. Convert the following radian angle measures to degrees: π _. 4 3 2 b. Convert the following angle measures to radians: 135°, 270°. SPIRAL REVIEW Simplify each expression. (Previous course) 54. (3x) 3 ( 2y 2 ) ( 3 -2 y 2 ) 55. a 4 b 3 (-2a) -4 2 56. ( -2t 3 s 2 ) ( 3ts 2 ) Find the next term in each pattern. (Lesson 2-1) 57. 1, 3, 7, 13, 21, … 58. C, E, G, I, K,... 59. 1, 6, 15, … In the figure, (Lesson 11-1) ̶̶ QP and ̶̶̶ QM are tangent to ⊙N. Find each measure. 60. m∠NMQ 61. MQ Construction Circle Through Three Noncollinear Points    Draw three noncollinear points. Construct m and n, the ⊥ bisectors of ̶̶ PQ and ̶̶ QR. Label the intersection O. Center the compass at O. Draw a circle through P. 1. Explain why ⊙O with radius ̶̶ OP also contains Q and R. 11-2 Arcs and Chords 763 763 �� 11-3 Sector Area and Arc Length TEKS G.8.B Congruence and the geometry of size: find areas of sectors and arc lengths of circles using proportional reasoning. Objectives Find the area of sectors. Find arc lengths. Vocabulary sector of a circle segment of a circle arc length Who uses this? Farmers use irrigation radii to calculate areas of sectors. (See Example 2.) The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the circle by m°____ 360°. Sector of a Circle TERM NAME DIAGRAM AREA A sector of a circle is a region bounded by two radii of the circle and their intercepted arc. sector ACB Also G.1A, G.1.B, G.9.C A = πr 2 ( m° _ ) 360° E X A M P L E 1 Finding the Area of

a Sector Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. A sector MPN 360° A = π r 2 ( m° _ ) = π (3 ) 2 ( 80° _ ) = 2π in 2 ≈ 6.28 in 2 360° B sector EFG 360° A = π r 2 ( m° _ ) = π (6) 2 ( 120° _ ) = 12π ≈ 37.70 cm 2 360° Use formula for area of a sector. Substitute 3 for r and 80 for m. Simplify. Use formula for area of a sector. Substitute 6 for r and 120 for m. Simplify. Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 1a. sector ACB 1b. sector JKL Write the degree symbol after m in the formula to help you remember to use degree measure not arc length. 764 764 Chapter 11 Circles �� E X A M P L E 2 Agriculture Application A circular plot with a 720 ft diameter is watered by a spray irrigation system. To the nearest square foot, what is the area that is watered as the sprinkler rotates through an angle of 50°? 360° A = π r 2 ( m° _ ) = π (360) 2 ( 50° _ ) ≈ 56,549 ft 2 360° d = 720 ft, r = 360 ft. Simplify. 2. To the nearest square foot, what is the area watered in Example 2 as the sprinkler rotates through a semicircle? A segment of a circle is a region bounded by an arc and its chord. The shaded region in the figure is a segment. Area of a Segment area of segment = area of sector - area of triangle E X A M P L E 3 Finding the Area of a Segment Find the area of segment ACB to the nearest hundredth. Step 1 Find the area of sector ACB. In a 30°-60°-90° triangle, the length of the leg opposite the 60° angle is √  3 times the length of the shorter leg. 360° A = π r 2 ( m° _ ) = π (12) 2 ( 60° _ ) = 24π in 2 360° Use formula for area of a

sector. Substitute 12 for r and 60 for m. Step 2 Find the area of △ACB. ̶̶ AD. Draw altitude bh = 1 _ A = 1 _ (12) (6 √  3 ) 2 2 = 36 √  3 in 2 CD = 6 in., and h = 6 √  3 in. Simplify. Step 3 area of segment = area of sector ACB - area of △ACB = 24π - 36 √  3 ≈ 13.04 in 2 3. Find the area of segment RST to the nearest hundredth. 11-3 Sector Area and Arc Length 765 765 �� In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle. Arc Length TERM DIAGRAM LENGTH Arc length is the distance along an arc measured in linear units. L = 2πr ( m° _ ) 360° E X A M P L E 4 Finding Arc Length Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. A ⁀CD L = 2πr ( m°_ 360°) = 2π(10)( 90°_ 360°) Use formula for arc length. Substitute 10 for r and 90 for m. = 5π ft ≈ 15.71 ft Simplify. B an arc with measure 35° in a circle with radius 3 in. L = 2πr ( m°_ 360°) = 2π(3)( 35°_ 360°) = 7_ 12 in. ≈ 1.83 in. Use formula for arc length. Substitute 3 for r and 35 for m. Simplify. Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 4a. ⁀GH 4b. an arc with measure 135° in a circle with radius 4 cm THINK AND DISCUSS 1. What is the difference between arc measure and arc length? 2. A slice of pizza is a sector of a circle. Explain what measurements you would need to make in order to calculate the area of the slice. 3. GET ORGANIZED Copy and complete the graphic organizer. 766 766 Chapter 11 Circles ��

�� 11-3 Exercises Exercises KEYWORD: MG7 11-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a circle, the region bounded by a chord and an arc is called a?. (sector or segment) ̶̶̶̶. 764 Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 2. sector PQR 3. sector JKL 4. sector ABC. Navigation The beam from a lighthouse is visible for a distance of 3 mi. p. 765 To the nearest square mile, what is the area covered by the beam as it sweeps in an arc of 150°? Multi-Step Find the area of each segment to the nearest hundredth. p. 765 6. 7. 8. 766 Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 9. ⁀ EF 10. ⁀ PQ Independent Practice For See Exercises Example 12–14 15 16–18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S24 Application Practice p. S38 11. an arc with measure 20° in a circle with radius 6 in. PRACTICE AND PROBLEM SOLVING Find the area of each sector. Give your answer in terms of π and rounded to the nearest hundredth. 12. sector DEF 13. sector GHJ 14. sector RST 15. Architecture A lunette is a semicircular window that is sometimes placed above a doorway or above a rectangular window. To the nearest square inch, what is the area of the lunette? �� 11-3 Sector Area and Arc Length 767 767 �� Multi-Step Find the area of each segment to the nearest hundredth. 16. 17. 18. Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 19. ⁀ UV 20. ⁀ AB Math History 21. an arc with measure 9° in a circle with diameter 4 ft 22. Math History Greek mathematicians studied the salinon, a figure bounded by four semicircles. What is the perimeter of this salin

on to the nearest tenth of an inch? Hypatia lived 1600 years ago. She is considered one of history’s most important mathematicians. She is credited with contributions to both geometry and astronomy. Tell whether each statement is sometimes, always, or never true. 23. The length of an arc of a circle is greater than the circumference of the circle. 24. Two arcs with the same measure have the same arc length. 25. In a circle, two arcs with the same length have the same measure. Find the radius of each circle. 26. area of sector ABC = 9π 27. arc length of ⁀ EF = 8π 28. Estimation The fraction 22 __ 7 is an approximation for π. a. Use this value to estimate the arc length of ⁀ XY. b. Use the π key on your calculator to find the length of ⁀ XY to 8 decimal places. c. Was your estimate in part a an overestimate or an underestimate? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 770. The pedals of a penny-farthing bicycle are directly connected to the front wheel. a. Suppose a penny-farthing bicycle has a front wheel with a diameter of 5 ft. To the nearest tenth of a foot, how far does the bike move when you turn the pedals through an angle of 90°? b. Through what angle should you turn the pedals in order to move forward by a distance of 4.5 ft? Round to the nearest degree. 768 768 Chapter 11 Circles �� 30. Critical Thinking What is the length of the radius that makes the area of ⊙A = 24 in 2 and the area of sector BAC = 3 in 2? Explain. 31. Write About It Given the length of an arc of a circle and the measure of the arc, explain how to find the radius of the circle. 32. What is the area of sector AOB? 4π 16π 32π 64π 33. What is the length of ⁀ AB? 4π 2π 8π 16π 34. Gridded Response To the nearest hundredth, what is the area of the sector determined by an arc with measure 35° in a circle with radius 12? CHALLENGE AND EXTEND 35. In the diagram, the larger of the two concentric circles has radius 5, and

the smaller circle has radius 2. What is the area of the shaded region in terms of π? 36. A wedge of cheese is a sector of a cylinder. a. To the nearest tenth, what is the volume of the wedge with the dimensions shown? b. What is the surface area of the wedge of cheese to the nearest tenth? 37. Probability The central angles of a target measure 45°. The inner circle has a radius of 1 ft, and the outer circle has a radius of 2 ft. Assuming that all arrows hit the target at random, find the following probabilities. a. hitting a red region b. hitting a blue region c. hitting a red or blue region �� SPIRAL REVIEW Determine whether each line is parallel to y = 4x - 5, perpendicular to y = 4x - 5, or neither. (Previous course) 38. 8x - 2y = 6 39. line passing through the points ( 1 __ 2, 0) and (1 1 __ 2, 2) 40. line with x-intercept 4 and y-intercept 1 Find each measurement. Give your answer in terms of π. (Lesson 10-8) 41. volume of a sphere with radius 3 cm 42. circumference of a great circle of a sphere whose surface area is 4π cm 2 Find the indicated measure. (Lesson 11-2) 43. m∠KLJ 44. m ⁀ KJ 45. m ⁀ JFH 11-3 Sector Area and Arc Length 769 769 �� SECTION 11A Lines and Arcs in Circles As the Wheels Turn The bicycle was invented in the 1790s. The first models didn’t even have pedals—riders moved forward by pushing their feet along the ground! Today the bicycle is a high-tech machine that can include hydraulic brakes and electronic gear changers. 1. A road race bicycle wheel is 28 inches in diameter. A manufacturer makes metal bicycle stands that are 10 in. tall. How long should a stand be to the nearest tenth in order to support a 28 in. wheel? (Hint: Consider the triangle formed by the radii and the top of the stand.) 2. The chain of a bicycle loops around a large gear connected to the bike’s pedals and a small gear attached to the rear wheel. In the diagram, the distance AB between

the centers of the gears the nearest tenth is 15 in. Find CD, the length of the chain between the two gears to the nearest tenth. (Hint: Draw a segment from B to ̶̶ AD that is parallel to ̶̶ CD.) �� 3. By pedaling, you turn the large gear through an angle of 60°. How far does the chain move around the circumference of the gear to the nearest tenth? 4. As the chain moves, it turns the small gear. If you use the distance you calculated in Problem 3, through what angle does the small gear turn to the nearest degree? �� 770 770 Chapter 11 Circles SECTION 11A Quiz for Lessons 11-1 Through 11-3 11-1 Lines That Intersect Circles Identify each line or segment that intersects each circle. 1. 2. 3. The tallest building in Africa is the Carlton Centre in Johannesburg, South Africa. What is the distance from the top of this 732 ft building to the horizon to the nearest mile? (Hint: 5280 ft = 1 mi; radius of Earth = 4000 mi) 11-2 Arcs and Chords Find each measure. 4. ⁀ BC 5. ⁀ BED 6. ⁀ SR 7. ⁀ SQU Find each length to the nearest tenth. 8. JK 9. XY 11-3 Sector Area and Arc Length 10. As part of an art project, Peter buys a circular piece of fabric and then cuts out the sector shown. What is the area of the sector to the nearest square centimeter? Find each arc length. Give your answer in terms of π and rounded to the nearest hundredth. 11. ⁀ AB 12. ⁀ EF �� 13. an arc with measure 44° in a circle with diameter 10 in. 14. a semicircle in a circle with diameter 92 m Ready to Go On? 771 771 �� 11-4 Inscribed Angles TEKS G.5.B Geometric paterns: use … patterns to make generalizations about geometric properties, including … angle relationships in … circles. Objectives Find the measure of an inscribed angle. Use inscribed angles and their properties to solve problems. Vocabulary inscribed angle intercepted arc subtend Also G.1.A, G

.2.A, G.2.B, G.9.C Why learn this? You can use inscribed angles to find measures of angles in string art. (See Example 2.) String art often begins with pins or nails that are placed around the circumference of a circle. A long piece of string is then wound from one nail to another. The resulting pattern may include hundreds of inscribed angles. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. A chord or arc subtends an angle if its endpoints lie on the sides of the angle. ∠DEF is an inscribed angle. ⁀ DF is the intercepted arc. ⁀ DF subtends ∠DEF. Theorem 11-4-1 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. m∠ABC = 1 _ 2 m ⁀ AC Case 1 Case 2 Case 3 You will prove Cases 2 and 3 of Theorem 11-4-1 in Exercises 30 and 31. PROOF PROOF Inscribed Angle Theorem Given: ∠ABC is inscribed in ⊙X. Prove: m∠ABC = 1 __ 2 m ⁀ AC Proof Case 1: ̶̶ BC. Draw ̶̶ XA. m ⁀ AC = m∠AXC. ̶̶ XA and ̶̶ XB are radii of the circle, ∠ABC is inscribed in ⊙X with X on By the Exterior Angle Theorem m∠AXC = m∠ABX + m∠BAX. Since △AXB is isosceles. Thus m∠ABX = m∠BAX. By the Substitution Property, m ⁀ AC = 2m∠ABX or 2m∠ABC. Thus 1 __ 2 m ⁀ AC = m∠ABC. ̶̶ XA ≅ ̶̶ XB. Then by definition 772 772 Chapter 11 Circles �� E X A M P L E 1 Finding Measures of Arcs and Inscribed Angles Find each measure. A m∠RST m∠RST = 1 _ m ⁀ RT 2 = 1 _ (120°) = 60

° 2 Inscribed ∠ Thm. Substitute 120 for m ⁀ RT. B m ⁀ SU m∠SRU = 1 _ m ⁀ SU 2 40° = 1 _ m ⁀ SU 2 m ⁀ SU = 80° Inscribed ∠ Thm. Substitute 40 for m∠SRU. Mult. both sides by 2. Find each measure. 1a. m ⁀ ADC 1b. m∠DAE Corollary 11-4-2 COROLLARY HYPOTHESIS CONCLUSION If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. ∠ACB ≅ ∠ADB ≅ ∠AEB (and ∠CAE ≅ ∠CBE) ∠ACB, ∠ADB, and ∠AEB intercept ⁀ AB. You will prove Corollary 11-4-2 in Exercise 32. E X A M P L E 2 Hobby Application Find m∠DEC, if m ⁀ AD = 86°. ∠BAC ≅ ∠BDC m∠BAC = m∠BDC m∠BDC = 60° m∠ACD = 1 _ m ⁀ AD 2 = 1 _ (86°) 2 = 43° ∠BAC and ∠BDC intercept ⁀ BC. Def. of ≅ Substitute 60 for m∠BDC. Inscribed ∠ Thm. Substitute 86 for m ⁀ AD. Simplify. m∠DEC + 60 + 43 = 180 △ Sum Theorem m∠DEC = 77° Simplify. 2. Find m∠ABD and m ⁀ BC in the string art. 11-4 Inscribed Angles 773 773 �� Theorem 11-4-3 An inscribed angle subtends a semicircle if and only if the angle is a right angle. You will prove Theorem 11-4-3 in Exercise 43. E X A M P L E 3 Finding Angle Measures in Inscribed Triangles Find each value. A x ∠RQT is a right angle ∠RQT is inscribed in a m∠RQT = 90° 4x + 6 = 90 4x = 84 x

= 21 B m∠ADC semicircle. Def. of rt. ∠ Substitute 4x + 6 for m∠RQT. Subtract 6 from both sides. Divide both sides by 4. m∠ABC = m∠ADC ∠ABC and ∠ADC both 10y - 28 = 7y - 1 3y - 28 = -1 3y = 27 y = 9 intercept ⁀ AC. Substitute the given values. Subtract 7y from both sides. Add 28 to both sides. Divide both sides by 3. m∠ADC = 7 (9 ) -1 = 62° Substitute 9 for y. Find each value. 3a. z 3b. m∠EDF Construction Center of a Circle     Draw a circle and ̶̶ AB. chord Draw chord ̶̶ AC. Construct a line ̶̶ perpendicular to AB at B. Where the line and the circle intersect, label the point C. 774 774 Chapter 11 Circles ̶̶ DF. Repeat steps to draw ̶̶ chords DE and The intersection of ̶̶ and DF is the center of the circle. ̶̶ AC �� Theorem 11-4-4 THEOREM HYPOTHESIS CONCLUSION If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. ∠A and ∠C are supplementary. ∠B and ∠D are supplementary. ABCD is inscribed in ⊙E. You will prove Theorem 11-4-4 in Exercise 44. E X A M P L E 4 Finding Angle Measures in Inscribed Quadrilaterals Find the angle measures of PQRS. Step 1 Find the value of y. m∠P + m∠R = 180° PQRS is inscribed in a ⊙. 6y + 1 + 10y + 19 = 180 16y + 20 = 180 16y = 160 y = 10 Substitute the given values. Simplify. Subtract 20 from both sides. Divide both sides by 16. Step 2 Find the measure of each angle. m∠P = 6 (10) + 1 = 61° m∠R = 10 (10) + 19 = 119° m∠Q = 10 2

+ 48 = 148° m∠Q + m∠S = 180° 148° + m∠S = 180° m∠S = 32° Substitute 10 for y in each expression. ∠Q and ∠S are supp. Substitute 148 for m∠Q. Subtract 148 from both sides. 4. Find the angle measures of JKLM. THINK AND DISCUSS 1. Can ABCD be inscribed in a circle? Why or why not? 2. An inscribed angle intercepts an arc that is 1 __ 4 of the circle. Explain how to find the measure of the inscribed angle. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box write a definition, properties, an example, and a nonexample. 11-4 Inscribed Angles 775 775 �� 11-4 Exercises Exercises KEYWORD: MG7 11-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A, B, and C lie on ⊙P. ∠ABC is an example of an? angle. ̶̶̶̶ (intercepted or inscribed Find each measure. p. 773 2. m∠DEF 3. m ⁀ EG 4. m ⁀ JKL 5. m∠LKM. Crafts A circular loom can be used � p. 773 for knitting. What is the m∠QTR in the knitting loom Find each value. �� p. 774 7. x 8. y 9. m∠XYZ Multi-Step Find the angle measures of each quadrilateral. p. 775 10. PQRS 11. ABCD Independent Practice For See Exercises Example 12–15 16 17–20 21–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Find each measure. 12. m ⁀ ML 14. m ⁀ EGH 13. m∠KMN 15. m∠GFH 16. Crafts An artist created a stained 16. glass window. If m∠

BEC = 40° and m ⁀ AB = 44°, what is m∠ADC? 776 776 Chapter 11 Circles ABCDEge07sec11l04004a�� Algebra Find each value. 17. y 18. z 19. m ⁀ AB 20. m∠MPN Multi-Step Find the angle measures of each quadrilateral. 21. BCDE 22. TUVW Tell whether each statement is sometimes, always, or never true. 23. Two inscribed angles that intercept the same arc of a circle are congruent. 24. When a right triangle is inscribed in a circle, one of the legs of the triangle is a diameter of the circle. 25. A trapezoid can be inscribed in a circle. Multi-Step Find each angle measure. 26. m∠ABC if m∠ADC = 112° 27. m∠PQR if m ⁀ PQR = 130° 28. Prove that the measure of a central angle subtended by a chord is twice the measure of the inscribed angle subtended by the chord. Given: In ⊙H Prove: m∠JHK = 2m∠JLK ̶̶ JK subtends ∠JHK and ∠JLK. 29. This problem will prepare you for the Multi-Step TAKS Prep on page 806. A Native American sand painting could be used to indicate the direction of sunrise on the winter and summer solstices. You can make this design by placing six equally spaced points around the circumference of a circle and connecting them as shown. a. Find m∠BAC. b. Find m∠CDE. c. What type of triangle is △FBC? Why? 11-4 Inscribed Angles 777 777 ��ABEDCFge07sec11l04006a 30. Given: ∠ABC is inscribed in ⊙X with X in the interior of ∠ABC. Prove: m∠ABC = 1 _ m �

� AC 2 (Hint: Draw  BX and use Case 1 of the Inscribed Angle Theorem.) History The Winchester Round Table, probably built in the late thirteenth century, is 18 ft across and weighs 1.25 tons. King Arthur’s Round Table of English legend would have been much larger—it could seat 1600 men. 31. Given: ∠ABC is inscribed in ⊙X with X in the exterior of ∠ABC. Prove: m∠ABC = 1 _ m ⁀ AC 2 32. Prove Corollary 11-4-2. Given: ∠ACB and ∠ADB intercept ⁀ AB. Prove: ∠ACB ≅ ∠ADB 33. Multi-Step In the diagram, m ⁀ JKL = 198°, and m ⁀ of quadrilateral JKLM. KLM = 216°. Find the measures of the angles 34. Critical Thinking A rectangle PQRS is inscribed in a circle. What can you conclude about ̶̶ PR? Explain. 35. History The diagram shows the Winchester Round Table with inscribed △ABC. The table may have been made at the request of King Edward III, who created the Order of Garter as a return to the Round Table and an order of chivalry. ̶̶ a. Explain why BC must be a diameter of the circle. b. Find m ⁀ AC. 36. To inscribe an equilateral triangle in a circle, draw a ̶̶ BC. Open the compass to the radius of the circle. diameter Place the point of the compass at C and make arcs on the circle at D and E, as shown. Draw ̶̶ DE. Explain why △BDE is an equilateral triangle. ̶̶ BD, ̶̶ BE, and 37. Write About It A student claimed that if a parallelogram contains a 30° angle, it cannot be inscribed in a circle. Do you agree or disagree? Explain. 38. Construction Circumscribe a circle about a triangle. (Hint: Follow the steps for the construction of a circle through three given noncollinear points.) 39. What is m∠BAC? 38° 43° 66° 81° 40. Equilateral △XCZ is inscribed in a circle. ̶̶ CY bisects ∠C, what is m

⁀ XY? If 15° 30° 60° 120° 41. Quadrilateral ABCD is inscribed in a circle. The ratio of m∠A to m∠C is 4 : 5. What is m∠A? 20° 40° 80° 100° 42. Which of these angles has the greatest measure? ∠STR ∠QPR ∠QSR ∠PQS 778 778 Chapter 11 Circles �� CHALLENGE AND EXTEND 43. Prove that an inscribed angle subtends a semicircle if and only if the angle is a right angle. (Hint: There are two parts.) 44. Prove that if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. (Hint: There are two parts.) 45. Find m ⁀ PQ to the nearest degree. 46. Find m∠ABD. 47. Construction To circumscribe an equilateral triangle ̶̶ AB parallel to the horizontal about a circle, construct diameter of the circle and tangent to the circle. Then use a 30°-60°-90° triangle to draw 60° angles with ̶̶ AB and are tangent to the circle. ̶̶ BC so that they form ̶̶ AC and SPIRAL REVIEW 48. Tickets for a play cost $15.00 for section C, $22.50 for section B, and $30.00 for section A. Amy spent a total of $255.00 for 12 tickets. If she spent the same amount on section C tickets as section A tickets, how many tickets for section B did she purchase? (Previous course) Write a ratio expressing the slope of the line through each pair of points. (Lesson 7-1) 49. (4 1 _, -6) and (8, 1 _ ) 2 2 50. (-9, -8) and (0, -2) 51. (3, -14) and (11, 6) Find each of the following. (Lesson 11-2) 52. m ⁀ ST 53. area of △ABD Construction Tangent to a Circle From an Exterior Point     Draw ⊙C and locate P in the exterior of the circle. ̶̶ CP. Construct M, Draw the midpoint of ̶̶ CP. 1.

Can you draw ̶̶ CR ⊥   RP? Explain. Center the compass at M. Draw a circle through C and P. It will intersect ⊙C at R and S. R and S are the tangent points. Draw   PR and    PS tangent to ⊙C. 11-4 Inscribed Angles 779 779 �� 11-5 Use with Lesson 11-5 Activity 1 Explore Angle Relationships in Circles In Lesson 11-4, you learned that the measure of an angle inscribed in a circle is half the measure of its intercepted arc. Now you will explore other angles formed by pairs of lines that intersect circles. TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures …. Also G.1.A, G.2.B, G.3.B, G.4.A, G.5.A, G.5.B, G.9.C 1 Create a circle with center A. Label the point on the circle as B. Create a radius segment from A to a new point C on the circle. 2 Construct a line through C perpendicular to ̶̶ radius AC. Create a new point D on this line, which is tangent to circle A at C. Hide radius ̶̶ AC. 3 Create a new point E on the circle and then ̶̶ CE. construct secant 4 Measure ∠DCE and measure ⁀ CBE. (Hint: To measure an arc in degrees, select the three points and the circle and then choose Arc Angle from the Measure menu.) 5 Drag E around the circle and examine the changes in the measures. Fill in the angle and arc measures in a chart like the one below. Try to create acute, right, and obtuse angles. Can you make a conjecture about the relationship between the angle measure and the arc measure? m∠DCE m ⁀ CBE Angle Type Activity 2 1 Construct a new circle with two secants   CD and   EF that intersect inside the circle at G. 2 Create two new points H and I that are on the circle as shown. These will be used to measure

the arcs. Hide B if desired. (It controls the circle’s size.) 3 Measure ∠DGF formed by the secant lines and measure ⁀ CHE and ⁀ DIF. 4 Drag F around the circle and examine the changes in measures. Be sure to keep H between C and E and I between D and F for accurate arc measurement. Move them if needed. 780 780 Chapter 11 Circles 5 Fill in the angle and arc measures in a chart like the one below. Try to create acute, right, and obtuse angles. Can you make a conjecture about the relationship between the angle measure and the two arc measures? m∠DGF m ⁀ CHE m ⁀ DIF Sum of Arcs Activity 3 1 Use the same figure from Activity 2. Drag points around the circle so that the intersection G is now outside the circle. Move H so it is between E and D and I is between C and F, as shown. 2 Measure ∠FGC formed by the secant lines and measure ⁀ CIF and ⁀ DHE. 3 Drag points around the circle and examine the changes in measures. Fill in the angle and arc measures in a chart like the one below. Can you make a conjecture about the relationship between the angle measure and the two arc measures? m∠FGC m ⁀ CIF m ⁀ DHE Number of Arcs Try This 1. How does the relationship you observed in Activity 1 compare to the relationship between an inscribed angle and its intercepted arc? 2. Why do you think the radius ̶̶ AC is needed in Activity 1 for the construction of the tangent line? What theorem explains this? 3. In Activity 3, try dragging points so that the secants become tangents. What conclusion can you make about the angle and arc measures? 4. Examine the conjectures and theorems about the relationships between angles and arcs in a circle. What is true of an angle with a vertex on the circle? What is true of an angle with a vertex inside the circle? What is true of an angle with a vertex outside the circle? Summarize your findings. 5. Does using geometry software to compare angle and arc measures constitute a formal proof of the relationship observed? 11- 5 Technology Lab 781 781 11-5 Angle Relationships in Circles TEKS G.5.B Geometric paterns: use … patterns to make generalizations about geometric

properties, including … angle relationships in … circles. Also G.1.A, G.2.B, G.5.A, G.9.C Objectives Find the measures of angles formed by lines that intersect circles. Use angle measures to solve problems. Who uses this? Circles and angles help optometrists correct vision problems. (See Example 4.) Theorem 11-5-1 connects arc measures and the measures of tangent-secant angles with tangent-chord angles. Theorem 11-5-1 THEOREM HYPOTHESIS CONCLUSION If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. m∠ABC = 1 _ m ⁀ AB 2 Tangent   BC and secant   BA intersect at B. You will prove Theorem 11-5-1 in Exercise 45. E X A M P L E 1 Using Tangent-Secant and Tangent-Chord Angles Find each measure. A m∠BCD m∠BCD = 1 _ m ⁀ BC 2 m∠BCD = 1 _ (142°) 2 = 71° B m ⁀ ABC m∠ACD = 1 _ m ⁀ ABC 2 90° = 1 _ m ⁀ ABC 2 180° = m ⁀ ABC Find each measure. 1a. m∠STU 1b. m ⁀ SR 782 782 Chapter 11 Circles �� Theorem 11-5-2 THEOREM HYPOTHESIS CONCLUSION If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. (m ⁀ AB + m ⁀ CD ) m∠1 = 1 _ 2 ̶ AD and ̶ BC Chords intersect at E. PROOF PROOF Theorem 11-5-2 ̶ ̶ BC intersect at E. AD and Given: (m ⁀ AB + m ⁀ CD ) Prove: m∠1 = 1 _ 2 Proof: Statements Reasons 1. ̶ AD and ̶ BD. 2. Draw ̶ BC intersect at E. 3

. m∠1 = m∠EDB + m∠EBD 4. m∠EDB = 1 __ m ⁀ AB, 2 m∠EBD = 1 __ m ⁀ CD 2 m ⁀ AB + 1 __ 5. m∠1 = 1 __ m ⁀ CD 2 2 (m ⁀ AB + m ⁀ CD ) 6. m∠1 = 1 __ 2 1. Given 2. Two pts. determine a line. 3. Ext. ∠ Thm. 4. Inscribed ∠ Thm. 5. Subst. 6. Distrib. Prop. E X A M P L E 2 Finding Angle Measures Inside a Circle Find each angle measure. m∠SQR (m ⁀ PT + m ⁀ SR ) m∠SQR = 1 _ 2 = 1 _ (32° + 100° ) 2 = 1 _ (132°) 2 = 66° Find each angle measure. 2a. m∠ABD 2b. m∠RNM 11-5 Angle Relationships in Circles 783 783 �� Theorem 11-5-3 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs. (m ⁀ AD - m ⁀ BD ) m∠2 = 1 _ m∠1 = 1 _ 2 2 (m ⁀ EHG - m ⁀ EG ) m∠3 = 1 _ 2 (m ⁀ JN - m ⁀ KM ) You will prove Theorem 11-5-3 in Exercises 34–36. E X A M P L E 3 Finding Measures Using Tangents and Secants Find the value of x. A B EHG and ⁀ EG joined ⁀ together make a whole circle. So m ⁀ EHG = 360° - 132° = 228° (m ⁀RS - m ⁀QS ) (174° - 98°) x = 1_ 2 = 1_ 2 = 38° 3. Find the value of x. E X A M P L E 4 Biology Application (m ⁀EHG - m ⁀EG ) (

228° - 132°) x = 1_ 2 = 1_ 2 = 48° When a person is farsighted, light rays enter the eye and are focused behind the retina. In the eye shown, light rays converge at R. If m ⁀PS = 60° and m ⁀QT = 14°, what is m∠PRS? (m ⁀PS - m ⁀QT ) m∠PRS = 1_ 2 = 1_ 2 = 1_ 2 (60° - 14° ) (46°) = 23° 4. Two of the six muscles that control eye movement are attached to the eyeball and intersect behind the eye. If m ⁀AEB = 225°, what is m∠ACB? 784 784 Chapter 11 Circles �� Angle Relationships in Circles VERTEX OF THE ANGLE MEASURE OF ANGLE On a circle Half the measure of its intercepted arc DIAGRAMS m∠1 = 60° m∠2 = 100° (44° + 86° ) m∠1 = 1 _ 2 = 65° Inside a circle Half the sum of the measures of its intercepted arcs Outside a circle Half the difference of the measures of its intercepted arcs (202° - 78°) m∠1 = 1_ 2 = 62° (125° - 45°) m∠2 = 1_ 2 = 40° E X A M P L E 5 Finding Arc Measures Find m ⁀ AF. Step 1 Find m ⁀ ADB. m∠ABC = 1 _ m ⁀ ADB 2 110° = 1 _ m ⁀ ADB 2 m ⁀ ADB = 220° Step 2 Find m ⁀ AD. If a tangent and secant intersect on a ⊙ at the pt. of tangency, then the measure of the ∠ formed is half the measure of its intercepted arc. Substitute 110 for m∠ABC. Mult. both sides by 2. ADB = m ⁀ AD + m ⁀ DB Arc Add. Post. m ⁀ 220° = m ⁀ AD + 160° m ⁀ AD = 60° Substitute. Subtract 160 from both sides. Step 3 Find m ⁀ AF

. m ⁀ AF = 360° - (m ⁀ AD + m ⁀ DB + m ⁀ BF ) = 360° - (60° + 160° + 48° ) = 92° Def. of a ⊙ Substitute. Simplify. 5. Find m ⁀ LP. 11-5 Angle Relationships in Circles 785 785 �� THINK AND DISCUSS 1. Explain how the measure of an angle formed by two chords of a circle is related to the measure of the angle formed by two secants. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box write a theorem and draw a diagram according to where the angle’s vertex is in relationship to the circle. 11-5 Exercises Exercises GUIDED PRACTICE Find each measure. p. 782 1. m∠DAB 2. m ⁀ AC 3. m ⁀ PN 4. m∠MNP KEYWORD: MG7 11-5 KEYWORD: MG7 Parent. m∠STU 6. m∠HFG 7. m∠NPK p. 783 Find the value of x. p. 784 8. 9. 10. 784 11. Science A satellite orbits Mars. When it reaches S it is about 12,000 km above the planet. How many arc degrees of the planet are visible to a camera in the satellite? 786 786 Chapter 11 Circles ��ABCSx°38°ge07se_c11l05004aAB��. 785 Multi-Step Find each measure. 12. m ⁀ DF 13. m ⁀ CD 14. m ⁀ PN 15. m ⁀ KN PRACTICE AND PROBLEM SOLVING Find each measure. 16. m∠BCD 17. m∠ABC 18. m∠XZW 19. m ⁀ XZV Independent Practice For See Exercises Example 16–19 20–22 23–25 26 27–30 1 2 3 4 5 TEKS TEKS TAKS TAKS 20. m∠QPR 21. m∠ABC 22

. m∠MKJ Skills Practice p. S25 Application Practice p. S38 Find the value of x. 23. 24. 25. Archaeology Outside of Hunt, Texas, is a replica of Stonehenge. It is 60 percent as tall as the original and 90 percent as large in circumference. 26. Archaeology Stonehenge is a circular arrangement of massive stones near Salisbury, England. A viewer at V observes the monument from a point where two of the stones A and B are aligned with stones at the endpoints of a diameter of the circular shape. Given that m ⁀ AB = 48°, what is m∠AVB? Multi-Step Find each measure. 27. m ⁀ EG 28. m ⁀ DE 29. m ⁀ PR 30. m ⁀ LP 11-5 Angle Relationships in Circles 787 787 �� In the diagram, m∠ABC = x °. Write an expression in terms of x for each of the following. 31. m ⁀ AB 33. m ⁀ AEB 32. m∠ABD 34. Given: Tangent CD and secant CA (m ⁀ AD - m ⁀ BD ) Prove: m∠ACD = 1 _ 2 Plan: Draw auxiliary line segment ̶ BD. Use the Exterior Angle Theorem to show that m∠ACD = m∠ABD - m∠BDC. Then use the Inscribed Angle Theorem and Theorem 11-5-1. 35. Given: Tangents  FG (m ⁀ Prove: m∠EFG = 1 _ 2  FE and EHG - m ⁀ EG ) 36. Given: Secants ̶ LN (m ⁀ JN - m ⁀ KM ) Prove: m∠JLN = 1 _ 2 ̶ LJ and 37. Critical Thinking Suppose two secants intersect in the exterior of a circle as shown. What is greater, m∠1 or m∠2? Justify your answer

. 38. Write About It The diagrams show the intersection of perpendicular lines on a circle, inside a circle, and outside a circle. Explain how you can use these to help you remember how to calculate the measures of the angles formed. Algebra Find the measures of the three angles of △ABC. 39. 40. 41. This problem will prepare you for the Multi-Step TAKS Prep on page 806. The design was made by placing six equally-spaced points on a circle and connecting them. a. Find m∠BHC. b. Find m∠EGD. c. Classify △EGD by its angle measures and by its side lengths. 788 788 Chapter 11 Circles ��ABEDCFge07sec11l05007aABHG 42. What is m∠DCE? 19° 21° 79° 101° 43. Which expression can be used to calculate m∠ABC? 1 _ (m ⁀ DE - m ⁀ AF ) 2 1 _ (m ⁀ AD - m ⁀ AF ) 2 1 _ (m ⁀ AD + m ⁀ AF ) 2 1 _ (m ⁀ DE + m ⁀ AF ) 2 44. Gridded Response In ⊙Q, m ⁀ MN = 146° and m∠JLK = 45°. Find the degree measure of ⁀ JK. CHALLENGE AND EXTEND 45. Prove Theorem 11-5-1.  BC and secant Given: Tangent Prove: m∠ABC = 1 _ m ⁀ AB 2 (Hint: Consider two cases, one where  BA ̶ AB is a diameter and one where ̶ AB is not a diameter.) ̶ WZ are tangent to ⊙X. m ⁀ WY = 90° 46. Given: ̶ YZ and Prove: WXYZ is a square. 47. Find x. 48. Find m ⁀ GH. SPIRAL REVIEW Determine whether the ordered pair (7, -8) is a solution of the following functions. (Previous course) 49. g (x) = 2 x 2 - 15x - 1

50. f (x) = 29 - 3x 51. y = - 7 _ x 8 Find the volume of each pyramid or cone. Round to the nearest tenth. (Lesson 10-7) 52. regular hexagonal pyramid with a base edge of 4 m and a height of 7 m 53. right cone with a diameter of 12 cm and lateral area of 60π cm 2 54. regular square pyramid with a base edge of 24 in. and a surface area of 1200 in 2 In ⊙P, find each angle measure. (Lesson 11-4) 55. m∠BCA 56. m∠DBC 57. m∠ADC 11-5 Angle Relationships in Circles 789 789 �� 11-6 Use with Lesson 11-6 Activity 1 Explore Segment Relationships in Circles When secants, chords, or tangents of circles intersect, they create several segments. You will measure these segments and investigate their relationships. TEKS G.9.C Congruence and the geometry of size: formulate … conjectures about … circles and the lines that intersect them …. Also G.2.A, G.2.B, G.3.B, G.5.A KEYWORD: MG7 Lab11 1 Construct a circle with center A. Label the point on the circle as B. Construct two secants CD and   EF that intersect outside the circle at G. Hide B if desired. (It controls the circle’s size.) 2 Measure ̶̶ GC, ̶̶̶ GD, ̶̶ GE, and around the circle and examine the changes in the measurements. ̶̶ GF. Drag points 3 Fill in the segment lengths in a chart like the one below. Find the products of the lengths of segments on the same secant. Can you make a conjecture about the relationship of the segments formed by intersecting secants of a circle? GC GD GC ⋅ GD GE GF GE ⋅ GF Try This 1. Make a sketch of the diagram from Activity 1, ̶̶ DE to create △CFG and ̶̶ CF and and create △EDG as shown. 2. Name pairs of congruent angles in the diagram. How are △CFG and △

EDG related? Explain your reasoning. 3. Write a proportion involving sides of the triangles. Cross-multiply and state the result. What do you notice? Activity 2 1 Construct a new circle with center A. Label the point on the circle as B. Create a radius segment from A to a new point C on the circle. 2 Construct a line through C perpendicular to radius point D on this line, which is tangent to circle A at C. Hide radius ̶̶ AC. Create a new ̶̶ AC. 790 790 Chapter 11 Circles 3 Create a secant line through D that intersects the circle at two new points E and F, as shown. 4 Measure ̶̶ DC, ̶̶ DE, and ̶̶ DF. Drag points around the circle and examine the changes in the measurements. Fill in the measurements in a chart like the one below. Can you make a conjecture about the relationship between the segments of a tangent and a secant of a circle? DE DF DE ⋅ DF DC? Try This 4. How are the products for a tangent and a secant similar to the products for secant segments? 5. Try dragging E and F so they overlap (to make the secant segment look like a tangent segment). What do you notice about the segment lengths you measured in Activity 2? Can you state a relationship about two tangent segments from the same exterior point? 6. Challenge Write a formal proof of the relationship you found in Problem 2. Activity 3 1 Construct a new circle with two chords that intersect inside the circle at G. ̶̶ CD and ̶̶ EF 2 Measure ̶̶ GC, ̶̶̶ GD, ̶̶ GE, and the circle and examine the changes in the measurements. ̶̶ GF. Drag points around 3 Fill in the segment lengths in a chart like the ones used in Activities 1 and 2. Find the products of the lengths of segments on the same chord. Can you make a conjecture about the relationship of the segments formed by intersecting chords of a circle? Try This 7. Connect the endpoints of the chords to form two triangles. Name pairs of congruent angles. How are the two triangles that are formed related? Explain your reasoning. 8. Examine the conclusions you made in all three activities about segments formed by secants, chords, and tangents in a circle. Summarize your findings. 11- 6 Technology Lab 791 791 11-6

Segment Relationships in Circles TEKS G.5.A Geometric patterns: use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Also G.1.A, G.2.B Objectives Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems. Vocabulary secant segment external secant segment tangent segment Who uses this? Archaeologists use facts about segments in circles to help them understand ancient objects. (See Example 2.) In 1901, divers near the Greek island of Antikythera discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. The following theorem describes the relationship among the four segments that are formed when two chords intersect in the interior of a circle. Theorem 11-6-1 Chord-Chord Product Theorem THEOREM HYPOTHESIS CONCLUSION If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. AE ⋅ EB = CE ⋅ ED ̶̶ AB and ̶̶ CD Chords intersect at E. You will prove Theorem 11-6-1 in Exercise 28. E X A M P L E 1 Applying the Chord-Chord Product Theorem Find the value of x and the length of each chord. PQ ⋅ QR = SQ ⋅ QT 6 (4) = x ( 8) 24 = 8x 3 = x PR = 6 + 4 = 10 ST = 3 + 8 = 11 1. Find the value of x and the length of each chord. 792 792 Chapter 11 Circles �� E X A M P L E 2 Archaeology Application Archaeologists discovered a fragment of an ancient disk. To calculate its original diameter, they drew a chord ̶̶ PQ. Find the disk’s diameter. bisector ̶̶̶ PQ is the perpendicular bisector of ̶̶ PR is a diameter of the disk. ̶̶ AB and its perpendicular Since a chord, AQ ⋅ QB = PQ ⋅ QR 5 (5) = 3 (QR) 25 = 3QR 8 1 _ 3 in. = QR = 11 1 _ PR = 3 + 8 1 _ 3 3 in. 2. What if…? Suppose the length of chord ̶̶ AB that the archaeologists

drew was 12 in. In this case how much longer is the disk’s diameter compared to the disk in Example 2? A secant segment is a segment of a secant with at least one endpoint on the circle. An external secant segment is a secant segment that lies in the exterior of the circle with one endpoint on the circle. ̶̶̶ NM, ̶̶̶ KM, and ̶̶ ̶̶̶ PM, JM are secant segments of ⊙Q. ̶̶ ̶̶̶ NM and JM are external secant segments. Theorem 11-6-2 Secant-Secant Product Theorem THEOREM HYPOTHESIS CONCLUSION If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole ⋅ outside = whole ⋅ outside) ̶̶ AE and ̶̶ CE Secants intersect at E. AE ⋅ BE = CE ⋅ DE PROOF PROOF Secant-Secant Product Theorem Given: Secant segments Prove: AE ⋅ BE = CE ⋅ DE ̶̶ AE and ̶̶ CE Proof: Draw auxiliary line segments ̶̶ AD and ̶̶ CB. ∠EAD and ∠ECB both intercept ⁀ BD, so ∠EAD ≅ ∠ECB. ∠E ≅ ∠E by the Reflexive Property of ≅. Thus △EAD ∼ △ECB by AA Similarity. Therefore corresponding sides are proportional, and AE ___ = DE ___ BE CE AE ⋅ BE = CE ⋅ DE.. By the Cross Products Property, 11-6 Segment Relationships in Circles 793 793 3 in.5 in.PBARQge07sec11l06002a�� E X A M P L E 3 Applying the Secant-Secant Product Theorem Find the value of x and the length of each secant segment. RT ⋅ RS = RQ ⋅ RP 10 (4) = (x + 5) 5 40 = 5x + 25 15 = 5x 3 = x RT = 4 + 6 = 10 RQ = 5 + 3 = 8 3. Find the value of z and the length of

each secant segment. A tangent segment is a segment of a tangent with one endpoint on the circle. ̶̶ ̶̶ AC are tangent segments. AB and Theorem 11-6-3 Secant-Tangent Product Theorem THEOREM HYPOTHESIS CONCLUSION If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole ⋅ outside = tangent 2 ) AC ⋅ BC = DC 2 ̶̶ AC and tangent ̶̶ DC Secant intersect at C. You will prove Theorem 11-6-3 in Exercise 29. E X A M P L E 4 Applying the Secant-Tangent Product Theorem Find the value of x. SQ ⋅ RQ = PQ 2 9 (4) = x 2 36 = x 2 ±6 = x The value of x must be 6 since it represents a length. 4. Find the value of y. 794 794 Chapter 11 Circles �� THINK AND DISCUSS 1. Does the Chord-Chord Product Theorem apply when both chords are diameters? If so, what does the theorem tell you in this case? 2. Given A in the exterior of a circle, how many different tangent segments can you draw with A as an endpoint? 3. GET ORGANIZED Copy and complete the graphic organizer. 11-6 Exercises Exercises KEYWORD: MG7 11-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary   AB intersects ⊙P at exactly one point. Point A is in the exterior ̶̶ AB is a(n)?. (tangent segment or external ̶̶̶̶ of ⊙P, and point B lies on ⊙P. secant segment Find the value of the variable and the length of each chord. p. 792 2. 3. 4. Engineering A section of an aqueduct p. 793 is based on an arc of a circle as shown. ̶̶̶ ̶̶ EF is the perpendicular bisector of GH. GH = 50 ft, and EF = 20 ft. What is the diameter of the circle Find the value of

the variable and the length of each secant segment. p. 794 6. 7. 8. 11-6 Segment Relationships in Circles 795 795 �� Find the value of the variable. p. 794 9. 10. 11. Independent Practice For See Exercises Example 12–14 15 16–18 19–21 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 PRACTICE AND PROBLEM SOLVING Find the value of the variable and the length of each chord. 12. 13. 14. 15. Geology Molokini is a small, crescent- shaped island 2 1 __ 2 miles from the Maui coast. It is all that remains of an extinct volcano. To approximate the diameter of the mouth of the volcano, a geologist can use a diagram like the one shown. What is the approximate diameter of the volcano’s mouth to the nearest foot? Find the value of the variable and the length of each secant segment. 16. 17. 18. Find the value of the variable. 19. 20. 21. Use the diagram for Exercises 22 and 23. 22. M is the midpoint of ̶̶ PQ. RM = 10 cm, and PQ = 24 cm. a. Find MS. b. Find the diameter of ⊙O. ̶̶ PQ.The diameter of ⊙O is 13 in., 23. M is the midpoint of and RM = 4 in. a. Find PM. b. Find PQ. 796 796 Chapter 11 Circles �� Multi-Step Find the value of both variables in each figure. 24. 25. 26. Meteorology A weather satellite S orbits Earth at a distance SE of 6000 mi. Given that the diameter of the earth is approximately 8000 mi, what is the distance from the satellite to P? Round to the nearest mile. 27. /////ERROR ANALYSIS///// The two solutions show how to find the value of x. Which solution is incorrect? Explain the error. Meteorology Satellites are launched to an area above the atmosphere where

there is no friction. The idea is to position them so that when they fall back toward Earth, they fall at the same rate as Earth’s surface falls away from them. 28. Prove Theorem 11-6-1. ̶̶ AB and Given: Chords Prove: AE ⋅ EB = CE ⋅ ED ̶̶ CD intersect at point E. Plan: Draw auxiliary line segments ̶̶ AC and ̶̶ BD. Show that △ECA ∼ △EBD. Then write a proportion comparing the lengths of corresponding sides. 29. Prove Theorem 11-6-3. Given: Secant segment Prove: AC ⋅ BC = DC 2 ̶̶ AC, tangent segment ̶̶ DC 30. Critical Thinking A student drew a circle and two secant segments. By measuring with a ruler, he found ̶̶ PQ ≅ the student’s conclusion? Why or why not? ̶̶ PS. He concluded that ̶̶ ST. Do you agree with ̶̶ QR ≅ 31. Write About It The radius of ⊙A is 4. CD = 4, ̶̶ CB is a tangent segment. Describe two different and methods you can use to find BC. 32. This problem will prepare you for the Multi-Step TAKS Prep on page 806. Some Native American designs are based on eight points that are placed around the circumference of a circle. In ⊙O, BE = 3 cm. AE = 5.2 cm, and EC = 4 cm. a. Find DE to the nearest tenth. b. What is the diameter of the circle to the nearest tenth? c. What is the length of ̶̶ OE to the nearest hundredth? 11-6 Segment Relationships in Circles 797 797 ��ge07se_c11l06005aABSEP�� 33. Which of these is closest to the length of tangent ̶̶ PQ? 6.9 9.2 9.9 10.6 34. What is the length of ̶̶ UT? 5 7 12 14 35. Short Response In ⊙A, ̶̶ AB is the

perpendicular ̶̶ CD. CD = 12, and EB = 3. Find the radius bisector of of ⊙A. Explain your steps. CHALLENGE AND EXTEND 36. Algebra ̶̶ KL is a tangent segment of ⊙N. 37. a. Find the value of x. b. Classify △KLM by its angle measures. Explain. ̶̶ PQ is a tangent segment of a circle with radius 4 in. Q lies on the circle, and PQ = 6 in. Find the distance from P to the circle. Round to the nearest tenth of an inch. 38. The circle in the diagram has radius c. Use this diagram and the Chord-Chord Product Theorem to prove the Pythagorean Theorem. 39. Find the value of y to the nearest hundredth. SPIRAL REVIEW 40. An experiment was conducted to find the probability of rolling two threes in a row on a number cube. The probability was 3.5%. How many trials were performed in this experiment if 14 favorable outcomes occurred? (Previous course) 41. Two coins were flipped together 50 times. In 36 of the flips, at least one coin landed heads up. Based on this experiment, what is the experimental probability that at least one coin will land heads up when two coins are flipped? (Previous course) Name each of the following. (Lesson 1-1) 42. two rays that do not intersect 43. the intersection of  AC and  CD 44. the intersection of  CA and  BD Find each measure. Give your answer in terms of π and rounded to the nearest hundredth. (Lesson 11-3) 45. area of the sector XZW 46. arc length of ⁀ XW 47. m∠YZX if the area of the sector YZW is 40π ft 2 798 798 Chapter 11 Circles �� 11-7 Circles in the Coordinate Plane TEKS G.2.B Geometric structure: make conjectures about … circles … choosing from a variety of approaches such as coordinate …. Also G.1.A, G.4.A, G.5.A Objectives Write equations and graph circles in

the coordinate plane. Use the equation and graph of a circle to solve problems. Who uses this? Meteorologists use circles and coordinates to plan the location of weather stations. (See Example 3.) The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center = √  ( = √  (x - h ) 2 + (y - k ) 2 r 2 = (x - h) 2 + (y - k) 2 Distance Formula Substitute the given values. Square both sides. Theorem 11-7-1 Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h) 2 + (y - k Writing the Equation of a Circle Write the equation of each circle. A ⊙A with center A (4, -2) and radius 3 (x - 4) 2 + (y - (-2) ) (x - h) 2 + (y - kx - 4) 2 + (y + 2) 2 = 9 Equation of a circle Substitute 4 for h, -2 for k, and 3 for r. Simplify. B ⊙B that passes through (-2, 6) and has center B (-6, 3) r =  2 √ ( -2 - (-6) ) + (6 - 3) 2 Distance Formula = √  25 = 5 2 (x - (-6) ) Simplify. + (y - 3) 2 = 5 2 (x + 6) 2 + (y - 3) 2 = 25 Simplify. Substitute -6 for h, 3 for k, and 5 for r. Write the equation of each circle. 1a. ⊙P with center P (0, -3) and radius 8 1b. ⊙Q that passes through (2, 3) and has center Q (2, -1) 11-7 Circles in the Coordinate Plane 799 799 �� If you are given the equation of

a circle, you can graph the circle by making a table or by identifying its center and radius. E X A M P L E 2 Graphing a Circle Graph each equation. x 2 + y 2 = 25 Step 1 Make a table of values. A Since the radius is √  25, or 5, use ±5 and the values between for x-values. x y -5 -4 -3 0 3 4 0 ±3 ±4 ±5 ±4 ±3 5 0 Step 2 Plot the points and connect them to form a circle. Always compare the equation to the form (x - h) 2 + (y - k) 2 = r 2. B (x + 1) 2 + (y - 2) 2 = 9 The equation of the given circle can be written as (x - (-1) ) + (y - 2) 2 = 3 2. So h = -1, k = 2, and r = 3. The center is (-1, 2), and the radius is 3. Plot the point (-1, 2). Then graph a circle having this center and radius 3. 2 Graph each equation. 2a. x 2 + y 2 = 9 2b. (x - 3) 2 + (y + 2) 2 = 4 Graphing Circles I found a way to use my calculator to graph circles. You first need to write the circle’s equation in y = form. For example, to graph x 2 + y 2 = 16, first solve for y. y 2 = 16 - x 2 y = ± √  16 - x 2 Christina Avila Crockett High School Now enter and graph the two equations y 1 = √  16 - x 2 and y 2 = - √  16 - x 2. 800 800 Chapter 11 Circles �� E X A M P L E 3 Meteorology Application Meteorologists are planning the location of a new weather station to cover Osceola, Waco, and Ireland, Texas. To optimize radar coverage, the station must be equidistant from the three cities which are located on a coordinate plane at A (2, 5), B (3, -2), and C (-5, -2). a. What are the coordinates where the station should be built? b.

If each unit of the coordinate plane represents 8.5 miles, what is the diameter of the region covered by the radar? The perpendicular bisectors of a triangle are concurrent at a point equidistant from each vertex. Step 1 Plot the three given points. Step 2 Connect A, B, and C to form a triangle. Step 3 Find a point that is equidistant from the three points by constructing the perpendicular bisectors of two of the sides of △ABC. The perpendicular bisectors of the sides of △ABC intersect at a point that is equidistant from A, B, and C. The intersection of the perpendicular bisectors is P (-1, 1). P is the center of the circle that passes through A, B, and C. The weather station should be built at P (-1, 1), Clifton, Texas. There are approximately 10 units across the circle. So the diameter of the region covered by the radar is approximately 85 miles. 3. What if…? Suppose the coordinates of the three cities in Example 3 are D (6, 2), E (5, -5), and F (-2, -4). What would be the location of the weather station? THINK AND DISCUSS 1. What is the equation of a circle with radius r whose center is at the origin? 2. A circle has a diameter with endpoints (1, 4) and (-3, 4). Explain how you can find the equation of the circle. 3. Can a circle have a radius of -6? Justify your answer. 4. GET ORGANIZED Copy and complete the graphic organizer. First select values for a center and radius. Then use the center and radius you wrote to fill in the other circles. Write the corresponding equation and draw the corresponding graph. 11-7 Circles in the Coordinate Plane 801 801 1800200020001800xyBCAWacoIrelandOsceolaxyBCAWacoIrelandOsceolaHolt, Rinehart & WinstonGeometry © 2007ge07sec11107003a Texas Grid map3rd proof1800200020001800xyPBCAWacoIrelandOsceolaCliftonxyPBCAWacoIrelandOsceolaCliftonHolt, Rinehart & WinstonGeometry © 2007ge07sec11107004a Texas Grid map3rd proof�� 11-7 Exercises Exercises

GUIDED PRACTICE Write the equation of each circle. p. 799 1. ⊙A with center A (3, -5) and radius 12 2. ⊙B with center B (-4, 0) and radius 7 KEYWORD: MG7 11-7 KEYWORD: MG7 Parent 3. ⊙M that passes through (2, 0) and that has center M (4, 0) 4. ⊙N that passes through (2, -2) and that has center N (-1, 2 Multi-Step Graph each equation. p. 800 5. (x - 3) 2 + (y - 3) 2 = 4 7. (x + 3) 2 + (y + 4) 2 = 1 6. (x - 1) 2 + (y + 2) 2 = 9 8. (x - 3) 2 + (y + 4) 2 = 16. 801 9. Communications A radio antenna tower is kept perpendicular to the ground by three wires of equal length. The wires touch the ground at three points on a circle whose center is at the base of the tower. The wires touch the ground at A (2, 6), B (-2, -2), and C (-5, 7). a. What are the coordinates of the base of the tower? b. Each unit of the coordinate plane represents 1 ft. What is the diameter of the circle? PRACTICE AND PROBLEM SOLVING Independent Practice Write the equation of each circle. For See Exercises Example 10. ⊙R with center R (-12, -10) and radius 8 10–13 14–17 18 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S25 Application Practice p. S38 11. ⊙S with center S (1.5, -2.5) and radius √  3 12. ⊙C that passes through (2, 2) and that has center C (1, 1) 13. ⊙D that passes through (-5, 1) and that has center D (1, -2) 13. 15. (x + 1) 2 - y 2 = 16 17. x 2 + (y + 2) 2 = 4 Multi-Step Graph each equation. 14. x 2 + (y - 2) 2 = 9 16. x 2 + y 2 = 100 18.

Anthropology Hundreds of stone circles can be found along the Gambia River in western Africa. The stones are believed to be over 1000 years old. In one of the circles at Ker Batch, three stones have approximate coordinates of A (3, 1), B (4, -2), and C (-6, -2). a. What are the coordinates of the center of the stone circle? b. Each unit of the coordinate plane represents 1 ft. What is the diameter of the stone circle? 802 802 Chapter 11 Circles Entertainment The wooden carousel at Fair Park in Dallas, Texas, was manufactured by Gustav Dentzel and his family around 1914, It has 50 elaborately carved jumping horses, 16 standing horses, and 2 chariots. Algebra Write the equation of each circle. 19. 20. 21. Entertainment In 2004, the world’s largest carousel was located at the House on the Rock, in Spring Green, Wisconsin. Suppose that the center of the carousel is at the origin and that one of the animals on the circumference of the carousel has coordinates (24, 32). a. If one unit of the coordinate plane equals 1 ft, what is the diameter of the carousel? b. As the carousel turns, the animals follow a circular path. Write the equation of this circle. Determine whether each statement is true or false. If false, explain why. 22. The circle x 2 + y 2 = 7 has radius 7. 23. The circle (x - 2) 2 + (y + 3) 2 = 9 passes through the point (-1, -3). 24. The center of the circle (x - 6)2 + (y + 4)2 = 1 lies in the second quadrant. 25. The circle (x + 1) 2 + (y - 4) 2 = 4 intersects the y-axis. 26. The equation of the circle centered at the origin with diameter 6 is x 2 + y 2 = 36. 27. Estimation You can use the graph of a circle to estimate its area. a. Estimate the area of the circle by counting the number of squares of the coordinate plane contained in its interior. Be sure to count partial squares. b. Find the radius of the circle. Then use the area formula to calculate the circle’s area to the nearest tenth. c. Was your estimate in part a an overestimate or an underestimate? 28. Consider the circle whose equation is (x -

4) 2 + (y + 6) 2 = 25. Write, in point-slope form, the equation of the line tangent to the circle at (1, -10). 29. This problem will prepare you for the Multi-Step TAKS Prep on page 806. A hogan is a traditional Navajo home. An artist is using a coordinate plane to draw the symbol for a hogan. The symbol is based on eight equally spaced points placed around the circumference of a circle. a. She positions the symbol at A (-3, 5) and C (0, 2). What are the coordinates of E and G? b. What is the length of a diameter of the symbol? c. Use your answer from part b to write an equation of the circle. 11-7 Circles in the Coordinate Plane 803 803 �� Geology Geology The New Madrid earthquake of 1811 was the largest earthquake known in American history. Large areas sank into the earth, new lakes were formed, forests were destroyed, and the course of the Mississippi River was changed. Find the center and radius of each circle. 30. (x - 2)2 + (y + 3)2 = 81 31. x 2 + (y + 15)2 = 25 32. (x + 1)2 + y 2 = 7 Find the area and circumference of each circle. Express your answer in terms of π. Find the area and circumference of each circle. Express your answer in terms of 33. circle with equation (x + 2) 2 + (y - 7) 2 = 9 33. 34. circle with equation (x - 8)2 + (y + 5)2 = 7 34. 34. 35. circle with center (-1, 3) that passes through (2, -1) 36. Critical Thinking Describe the graph of the equation x 2 + y 2 = r 2 when r = 0. 37. Geology A seismograph measures ground motion during an earthquake. To find the epicenter of an earthquake, scientists take readings in three different locations. Then they draw a circle centered at each location. The radius of each circle is the distance the earthquake is from the seismograph. The intersection of the circles is the epicenter. Use the data below to find the epicenter of the New Madrid earthquake. Seismograph Location Distance to Earthquake A B C (-200, 200) (400, -100)

(100, -500) 300 mi 600 mi 500 mi 38. For what value(s) of the constant k is the circle x 2 + (y - k) 2 = 25 tangent to the x-axis? 39. ⊙A has a diameter with endpoints (-3, -2) and (5, -2). Write the equation of ⊙A. 40. Recall that a locus is the set of points that satisfy a given condition. Draw and describe the locus of points that are 3 units from (2, 2). 41. Write About It The equation of ⊙P is (x - 2) 2 + (y - 1) 2 = 9. Without graphing, explain how you can determine whether the point (3, -1) lies on ⊙P, in the interior of ⊙P, or in the exterior of ⊙P. 42. Which of these circles intersects the x-axis? (x - 3) 2 + (y + 3) 2 = 4 (x + 1) 2 + (y - 4) 2 = 9 (x + 2) 2 + (y + 1) 2 = 1 (x + 1) 2 + (y + 4) 2 = 9 43. What is the equation of a circle with center (-3, 5) that passes through the point (1, 5)? (x + 3) 2 + (y - 5) 2 = 4 (x - 3) 2 + (y + 5) 2 = 4 (x + 3) 2 + (y - 5) 2 = 16 (x - 3) 2 + (y + 5) 2 = 16 44. On a map of a park, statues are located at (4, -2), (-1, 3), and (-5, -5). A circular path connects the three statues, and the circle has a fountain at its center. Find the coordinates of the fountain. (-1, -2) (2, 1) (-2, 1) (1, -2) 804 804 Chapter 11 Circles CharlestonDetroitMinneapolisEpicenterHolt, Rinehart & WinstonGeometry © 2007ge07sec11107009a Epicenter map2nd proof CHALLENGE AND EXTEND 45. In three dimensions, the equation of a sphere is similar to that of a circle. The equation of a sphere with center (h, j,

k) and radius r is (x - h) 2 + (y - j) 2 + (z - k) 2 = r 2. a. Write the equation of a sphere with center (2, -4, 3) that contains the point (1, -2, -5). b.   AC and   BC are tangents from the same exterior point. If AC = 15 m, what is BC? Explain. 46. Algebra Find the point(s) of intersection of the line x + y = 5 and the circle x 2 + y 2 = 25 by solving the system of equations. Check your result by graphing the line and the circle. 47. Find the equation of the circle with center (3, 4) that is tangent to the line whose equation is y = 2x + 3. (Hint: First find the point of tangency.) SPIRAL REVIEW Simplify each expression. (Previous course) 18a + 4 (9a + 3) __ 6 2 x 2 - 2 (4 x 2 + 1) __ 2 49. 48. 50. 3 (x + 3y) - 4 (3x + 2y) - (x - 2y) In isosceles △DEF, EF = 4y - 1. Find the value of each variable. (Lesson 4-8) ̶̶ DE ≅ ̶̶ EF. m∠E = 60°, and m∠D = (7x + 4) °. DE = 2y + 10, and 51. x 52. y Find each measure. (Lesson 11-5) 53. m ⁀ LNQ 54. m∠NMP KEYWORD: MG7 Career Q: What math classes did you take in high school? A: I took Algebra 1 and Geometry. I also took Drafting and Woodworking. Those classes aren’t considered math classes, but for me they were since math was used in them. Q: What type of furniture do you make? A: I mainly design and make household furniture, such as end tables, bedroom furniture, and entertainment centers. Q: How do you use math? A: Taking appropriate and precise measurements is very important. If wood is not measured correctly, the end result doesn’t turn out as expected. Understanding angle measures is also important. Some

of the furniture I build has 30° or 40° angles at the edges. Q: What are your future plans? A: Someday I would love to design all the furniture in my own home. It would be incredibly satisfying to know that all my furniture was made with quality and attention to detail. 11-7 Circles in the Coordinate Plane 805 805 Bryan Moreno Furniture Maker �� SECTION 11B Angles and Segments in Circles Native American Design The members of a Native American cultural center are painting a circle of colors on their gallery floor. They start by laying out the circle and chords shown. Before they apply their paint to the design, they measure angles and lengths to check for accuracy. 1. The circle design is based on twelve equally spaced points placed around the circumference of the circle. As the group lays out the design, what should be m∠AGB? 2. What should be m∠KAE? Why? 3. What should be m∠KMJ? Why? 4. The diameter of the circle is 22 ft. KM ≈ 4.8 ft, and JM ≈ 6.4 ft. What should be the length of ̶̶̶ MB? 5. The group members use a coordinate plane to help them position the design. Each square of a grid represents one square foot, and the center of the circle is at (20, 14). What is the equation of the circle? 6. What are the coordinates of points L, C, F, and I? 806 806 Chapter 11 Circles FEGDHCIBJAKMLge07sec11ac2003a SECTION 11B Quiz for Lessons 11-4 Through 11-7 11-4 Inscribed Angles Find each measure. 1. m∠BAC 2. m ⁀ CD 3. m∠FGH 4. m ⁀ JFG 11-5 Angle Relationships in Circles Find each measure. 5. m∠ RST 6. m∠AEC 7. A manufacturing company is creating a plastic stand for DVDs. They want to make the stand with m ⁀ MN = 102°. What should be the measure of ∠MPN? 11-6 Segment Relationships in Circles Find the value of the variable and the length of each chord or secant segment. 8. 9. 10. An archaeologist discovers a portion of a circular stone wall, shown by �

�� ST in the figure. ST = 12.2 m, and UR = 3.9 m. What was the diameter of the original circular wall? Round to the nearest hundredth. 11-7 Circles in the Coordinate Plane Write the equation of each circle. 11. ⊙A with center A (-2, -3) and radius 3 12. ⊙B that passes through (1, 1) and that has center B (4, 5) 13. A television station serves residents of three cities located at J (5, 2), K (-7, 2), and L (-5, -8). The station wants to build a new broadcast facility that is equidistant from the three cities. What are the coordinates of the location where the facility should be built? Ready to Go On? 807 807 �� EXTENSION EXTENSION Polar Coordinates TEKS G.1.A Geometric structure: develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. Objectives Convert between polar and rectangular coordinates. Plot points using polar coordinates. Vocabulary polar coordinate system pole polar axis In a Cartesian coordinate system, a point is represented by the two coordinates x and y. In a polar coordinate system, a point A is represented by its distance from the origin r, and an angle θ. θ is measured counterclockwise from  OA. The ordered pair (r, θ ) the horizontal axis to represents the polar coordinates of point A. In a polar coordinate system, the origin is called the pole. The horizontal axis is called the polar axis. y You can use the equation of a circle r 2 = x 2 + y 2 and the tangent ratio θ = convert rectangular coordinates to polar coordinates. __ x to E X A M P L E 1 Converting Rectangular Coordinates to Polar Coordinates Convert (3, 4) to polar coordinates = 25 r = 5 tan θ = 4 _ 3 θ = tan -1(4_ 3) ≈ 53° The polar coordinates are (5, 53°). 1. Convert (4, 1) to polar coordinates. You can use the relationships x = r cos θ and y = r sin θ to convert polar coordinates to rectangular coordinates. E X A M P L E

2 Converting Polar Coordinates to Rectangular Coordinates Convert (2, 130°) to rectangular coordinates. x = r cos θ x = 2 cos 130° ≈ -1.29 y = r sinθ y = 2 sin 130° ≈ 1.53 The rectangular coordinates are (-1.29, 1.53). 2. Convert (4, 60°) to rectangular coordinates. 808 808 Chapter 11 Circles �� E X A M P L E 3 Plotting Polar Coordinates Plot the point (4, 225°). Step 1 Measure 225° counterclockwise from the polar axis. Step 2 Locate the point on the ray that is 4 units from the pole. 3. Plot the point (4, 300°). E X A M P L E 4 Graphing Polar Equations Graph r = 4. Make a table of values and plot the points. θ r 0° 4 45° 135° 270° 300° 4 4 4 4 4. Graph r = 2. EXTENSION Exercises Exercises Convert to polar coordinates. 1. (2, 2) 2. (1, 0) Convert to rectangular coordinates. 6. (5, 214°) 5. (3, 150°) 3. (3, 7) 4. (0, 15) 7. (4, 303°) 8. (4.5, 90°) Plot each point. 9. (4, 45°) 10. (3, 165°) 11. (1, 240°) 12. (3.5, 315°) 13. Critical Thinking Graph the equation r = 5. What can you say about the graph of an equation of the form r = a, where a is a positive real number? Technology Graph each equation. 14. r = -5 sin θ 17. r = 5 cos 3θ 15. r = 3 sin 4θ 18. r = 3 cos 2θ 16. r = -4 cos θ 19. r = 2 + 4 sin θ Chapter 11 Extension 809 809 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary adjacent arcs............... 757 exterior of a circle.....