jacobcd52/hs_math_cleaned · Datasets at Hugging Face

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the pattern below. 1, 3, 7, 13, 21, … 15. In △XYZ, ∠X and ∠Z are remote interior angles of exterior ∠XYT. If m∠X = (x + 15) °, m∠Z = (50 - 3x) °, and m∠XYT = (4x - 25) °, what is the value of x? 16. In △ABC and △DEF, ∠A ≅ ∠F. If EF = 4.5, DF = 3, ̶̶ AB would let you and AC = 1.5, what length for conclude that △ABC ∼ △FED? STANDARDIZED TEST PREP Short Response 17. A building casts a shadow that is 85 ft long when the angle of elevation to the sun is 34°. a. What is the height of the building? Round to the nearest inch and show your work. b. What is the angle of elevation to the sun when the shadow is 42 ft 6 in. long? Round to the nearest tenth of a degree and show your work. 18. Use the figure to find each of the following. Round to the nearest tenth of a centimeter and show your work. a. the length of b. the length of ̶̶ DC ̶̶ AB Extended Response 19. Tony and Paul are taking a vacation with their cousin, Greg. Tony and Paul live in the same house. Paul will go directly to the vacation spot, but Tony has to pick up Greg. Tony travels 90 miles at a bearing of N 25° E to get to his cousin’s house. He then travels due east for 50 miles to get to the vacation spot. Paul travels on one highway to get from his house to the vacation spot. For each of the following, explain in words how you found your answer and round to the nearest tenth. a. Write the vectors in component form for the route from Tony and Paul’s house to their cousin’s house and the route from their cousin’s house to the vacation spot. b. What are the direction and magnitude of Paul’s direct route from his house to the vacation spot? c. Tony and Paul leave the house at the same time and arrive at the vacation spot at the same time. If Tony traveled at an average speed of 50 mi/h, what was Paul’s average speed? Cumulative Assessment, Chapters 1
–8 581 581 �� T E X A S TAKS Grades 9–11 Obj. 10 �� Reunion Tower The 55-story Reunion Tower is one of the most recognized buildings in the Dallas skyline. Built as a part of the Hyatt Regency Hotel, the tower is topped by a geodesic dome that houses a revolving restaurant. �� The tower itself consists of four concrete cylinders arranged in a �� triangular pattern. The center cylinder contains an elevator, which �� takes visitors on a 68-second ride to the top of the tower. �� Choose one or more strategies to solve each problem. 1. The building’s observation deck, the Lookout, is on the fifty-third floor, approximately 540 feet above street level. The deck is equipped with telescopes that offer close-up views of the Dallas area. Using one of the telescopes, a visitor spots a sculpture in a nearby park. The angle of depression to the sculpture is 10°. To the nearest foot, how far is the sculpture from the base of Reunion Tower? For 2–4, use the table. 2. At noon on May 15, the shadow of Reunion Tower is 150 ft long. Find the height of the tower to the nearest foot. 3. How long is the shadow of the tower at noon on October 15? Round to the nearest foot. 4. On which of the dates shown is the tower’s shadow the longest? What is the length of the shadow to the nearest foot? 582 582 Chapter 8 Right Triangles and Trigonometry Elevation of the Sun in Dallas, Texas Date January 15 February 15 March 15 April 15 May 15 June 15 July 15 August 15 September 15 October 15 November 15 December 15 Angle of Elevation at Noon (°) 36 44 54 66 75 79 77 70 60 48 39 34 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 Lyndon B. Johnson’s Birthplace Lyndon Baines Johnson (1908–1973), the thirtysixth president of the United States, was born in Stonewall, Texas, and moved to what is now Johnson City at the age of 5. Visitors to the Lyndon B. Johnson National Historical Park can tour the home where President Johnson spent much of his childhood.
Choose one or more strategies to solve each problem. 1. In 1973, the National Park Service finished restoring President Johnson’s house to its appearance during Johnson’s childhood. The blueprint below shows the layout of the house after the restoration. Suppose there is a border of wallpaper along the edge of the ceiling around the dining room. About how long is the border? 2. During the restoration of the house, the wooden floor of the parlor, dining room, and kitchen was replaced with new wood. About how many square feet of wood was used? 3. Estimate the percentage of the square footage of the home that is occupied by porches. 4. In 1934, the Johnson family spent $666.09 for improvements to their home. The total cost consisted of materials and labor. The materials cost $416.09 more than the labor. How much did they spend on labor for the improvements? �� Problem Solving on Location 583 583 Extending Perimeter, Circumference, and Area 9A Developing Geometric Formulas 9-1 Developing Formulas for Triangles and Quadrilaterals Lab Develop π 9-2 Developing Formulas for Circles and Regular Polygons 9-3 Composite Figures Lab Develop Pick’s Theorem for Area of Lattice Polygons 9B Applying Geometric Formulas 9-4 Perimeter and Area in the Coordinate Plane 9-5 Effects of Changing Dimensions Proportionally 9-6 Geometric Probability Lab Use Geometric Probability to Estimate π KEYWORD: MG7 ChProj This map of Texas counties shows the elevations of different regions. 584 584 Chapter 9 Vocabulary Match each term on the left with a definition on the right. 1. area A. a polygon that is both equilateral and equiangular 2. kite 3. perimeter 4. regular polygon B. a quadrilateral with exactly one pair of parallel sides C. the number of nonoverlapping unit squares of a given size that exactly cover the interior of a figure D. a quadrilateral with exactly two pairs of adjacent congruent sides E. the distance around a closed plane figure Convert Units Use multiplication or division to change from one unit of measure to another. 5. 12 mi = yd Length 6. 7.3 km =
m 7. 6 in. = ft 8. 15 m = mm Metric 1 kilometer = 1000 meters 1 meter = 100 centimeters 1 centimeter = 10 millimeters Customary 1 mile = 1760 yards 1 mile = 5280 feet 1 yard = 3 feet 1 foot = 12 inches Pythagorean Theorem Find x in each right triangle. Round to the nearest tenth, if necessary. 9. 11. 10. Measure with Customary and Metric Units Measure each segment to the nearest eighth of an inch and to the nearest half of a centimeter. 12. 14. 13. Solve for a Variable Solve each equation for the indicated variable. 15. A = 1_ 2 17. A = 1_ 2(b 1 + b 2)h for b 1 bh for b 16. P = 2b + 2h for h 18. A = 1_ 2 d 1d 2 for d 1 Extending Perimeter, Circumference, and Area 585 585 �� Key Vocabulary/Vocabulario apothem apotema center of a circle centro de un circulo center of a regular polygon centro de un poligono regular central angle of a regular polygon ángulo central de un poligono circle circulo composite figure figuras compuestas geometric probability probabilidad geométrica Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. How can you use the everyday meaning of the word center to understand the term center of a circle? 2. The word composite means “of separate parts.” What do you think the term composite figure means? 3. What does the word probability mean? How do you think geometric probability differs from theoretical probability? 4. The word apothem begins with the root apo-, which means “away from.” The apothem of a regular polygon is measured “away from” the center to the midpoint of a side. What do you think is true about the apothem and the side of the polygon? Geometry TEKS G.3.C Geometric structure* use logical reasoning to prove statements are true... G.3.E Geometric structure* use deductive reasoning to prove a statement Les. 9-1 ★ ★ 9-
2 Geo. Lab Les. 9-2 Les. 9-3 9-3 Geo. Lab Les. 9-4 Les. 9-5 Les. 9-6 9-6 Geo. Lab G.5.A Geometric patterns* use... geometric patterns to develop ★ ★ ★ algebraic expressions representing geometric properties G.5.B Geometric patterns* use... geometric patterns to make generalizations about geometric properties, including... ratios in similar figures... G.7.A Dimensionality and the geometry of location* use... two- dimensional coordinate systems to represent... figures G.7.B Dimensionality and the geometry of location* use slopes... to investigate geometric relationships... ★ ★ ★ ★ G.8.A Congruence and the geometry of size* find areas of... circles, ★ ★ ★ ★ ★ ★ and composite figures G.8.C Congruence and the geometry of size... use the ★ ★ Pythagorean Theorem G.11.D Similarity and the geometry of shape describe the effect on perimeter, area,... when one or more dimensions of a figure are changed... ★ * Knowledge and skills are written out completely on pages TX28–TX35. 586 586 Chapter 9 Study Strategy: Memorize Formulas Throughout a geometry course, you will learn many formulas, theorems, postulates, and corollaries. You may be required to memorize some of these. In order not to become overwhelmed by the amount of information, it helps to use flash cards. In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c. Theorem 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2 To create a flash card, write the name of the formula or theorem on the front of the card. Then clearly write the appropriate information on the back of the card. Be sure to include a labeled diagram. Front Back �� Try This 1. Choose a lesson from this book that you
have already studied, and make flash cards of the formulas or theorems from the lesson. 2. Review your flash cards by looking at the front of each card and trying to recall the information on the back of the card. Extending Perimeter, Circumference, and Area 587 587 �� Literal Equations Algebra A literal equation contains two or more variables. Formulas you have used to find perimeter, circumference, area, and side relationships of right triangles are examples of literal equations. See Skills Bank page S59 If you want to evaluate a formula for several different values of a given variable, it is helpful to solve for the variable first. Example Danielle plans to use 50 feet of fencing to build a dog run. Use the formula P = 2ℓ + 2w to find the length ℓ when the width w is 4, 5, 6, and 10 feet. Solve the equation for ℓ. First solve the formula for the variable. P = 2ℓ + 2w Write the original equation. P - 2w = 2ℓ P - 2w _ 2 = ℓ Subtract 2w from both sides. Divide both sides by 2. Use your result to find ℓ for each value of w. ℓ = P - 2w _ = 2 ℓ = P - 2w _ = 2 50 - 2 (4) _ 2 50 - 2 (5) _ 2 = 21 ft = 20 ft ℓ = P - 2w _ = 2 50 - 2 (6) _ 2 = 19 ft ℓ = P - 2w _ = 2 50 - 2 (10) _ 2 = 15 ft Substitute 50 for P and 4 for w. Substitute 50 for P and 5 for w. Substitute 50 for P and 6 for w. Substitute 50 for P and 10 for w. Try This TAKS Grades 9–11 Obj. 1 1. A rectangle has a perimeter of 24 cm. Use the formula P = 2ℓ + 2w to find the width when the length is 2, 3, 4, 6, and 8 cm. 2. A right triangle has a hypotenuse of length c = 65 ft. Use the Pythagorean Theorem to find the length of leg a when the length of leg b is 16, 25, 33, and 39 feet. 3. The perimeter of △ABC is 112 in. Write an expression for
a in terms of b and c, and use it to complete the following table. a b 48 36 14 c 35 36 50 588 588 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-1 Developing Formulas for Triangles and Quadrilaterals TEKS G.5.A Geometric patterns: use... geometric patterns to develop algebraic expressions representing geometric properties. Also G.1.B, G.3.C, G.3.E, G.8.C Objectives Develop and apply the formulas for the areas of triangles and special quadrilaterals. Solve problems involving perimeters and areas of triangles and special quadrilaterals. Why learn this? You can use formulas for area to help solve puzzles such as the tangram. A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. The area of a figure made with all the pieces is the sum of the areas of the pieces. Postulate 9-1-1 Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate to see that a parallelogram has the same area as a rectangle with the same base and height. � � � � A triangle is cut off one side and translated to the other side. Area Parallelogram The area of a parallelogram with base b and height h is A = bh. Remember that rectangles and squares are also parallelograms. The area of a square with side s is A = s 2, and the perimeter is P = 4s. E X A M P L E 1 Finding Measurements of Parallelograms The height of a parallelogram is measured along a segment perpendicular to a line containing the base. Find each measurement. A the area of the parallelogram Step 1 Use the Pythagorean Theorem to find the height h Step 2 Use h to find the area of the parallelogram. A = bh A = 6 (4) A = 24 in 2 Substitute 6 for b and 4 for h. Area of a parallelogram Simplify. 9- 1 Developing Formulas for Triangles and Quadrilaterals 589 589 �� Find each measurement. B the height of a rectangle in
which b = 5 cm and A = (5 x 2 - 5x) cm 2 A = bh 5 x 2 - 5x = 5h 5 ( x 2 - x) = 5h x 2 - x = h Area of a rectangle Substitute 5 x 2 - 5x for A and 5 for b. Factor 5 out of the expression for A. Divide both sides by 5. h = ( x 2 - x) cm Sym. Prop. of = C the perimeter of the rectangle, in which A = 12 x ft 2 Step 1 Use the area and the height to find the base. A = bh 12x = b (6) 2x = b Area of a rectangle Substitute 12x for A and 6 for h. Divide both sides by 6. The perimeter of a rectangle with base b and height h is P = 2b + 2h, or P = 2 (b + h). Step 2 Use the base and the height to find the perimeter. P = 2b + 2h P = 2 (2x) + 2 (6) P = (4x + 12) ft. Substitute 2x for b and 6 for h. Perimeter of a rectangle Simplify. 1. Find the base of a parallelogram in which h = 56 yd and A = 28 yd 2. To understand the formula for the area of a triangle or trapezoid, notice that two congruent triangles or two congruent trapezoids fit together to form a parallelogram. Thus the area of a triangle or trapezoid is half the area of the related parallelogram. � � �� Area Triangles and Trapezoids The area of a triangle with base b and height h is A = 1 __ 2 bh. The area of a trapezoid with bases b 1 and b 2 and height h is A = 1 __ 2 ( b 1 + b 2 ) h, or __ Finding Measurements of Triangles and Trapezoids Find each measurement. A the area of a trapezoid in which b 1 = 9 cm, b 2 = 12 cm, and h = 3 cm 9 + 12) 3 2 A = 31.5 cm 2 Area of a trapezoid Substitute 9 for b 1, 12 for b 2, and 3 for h. Simplify. 590 590 Chapter 9 Extending Perimeter, Circumference, and Area �� Find each
measurement. B the base of the triangle, in which A = x 2 in 2 bx bh 2x = b b = 2x in. Area of a triangle Substitute x 2 for A and x for h. Divide both sides by x. Multiply both sides by 2. Sym. Prop. of = C Area of a trapezoid b 2 of the trapezoid, in which A = 8 ft 3 + b 2 ) (2 ft Sym. Prop. of = Substitute 8 for A, 3 for b 1, and 2 for h. Multiply 1 __ 2 Subtract 3 from both sides. by 2. 2. Find the area of the triangle. A kite or a rhombus with diagonals d 1 and d 2 can be divided into two congruent triangles with a base of d 1 and a height of 1 __ 2 d 2. area of each triangle total area �� Area Rhombuses and Kites The area of a rhombus or kite with diagonals d 1 and d 2 is A = 1 __ Finding Measurements of Rhombuses and Kites Find each measurement. A Area of a kite d 2 of a kite in which d 1 = 16 cm and A = 48 cm 48 = 1 _ (16 cm Solve for d 2. Sym. Prop. of = Substitute 48 for A and 16 for d 1. 9- 1 Developing Formulas for Triangles and Quadrilaterals 591 591 �� Find each measurement. B the area of the rhombus 6x + 4) (10x + 10) 2 A = 1 _ (60 x 2 + 100x + 40) 2 Substitute (6x + 4) for d 1 and (10x + 10) for d 2. Multiply the binomials (FOIL). The diagonals of a rhombus or kite are perpendicular, and the diagonals of a rhombus bisect each other. A = (30 x 2 + 50x + 20) in 2 Distrib. Prop. C the area of the kite Step 1 The diagonals d 1 and d 2 form four right triangles. Use the Pythagorean Theorem to find x and y. 9 2 + x 2 = 41 2 x 2 = 1600
x = 40 9 2 + y 2 = 15 2 y 2 = 144 y = 12 Step 2 Use d 1 and d 2 to find the area. d 1 is equal to x + y, which is 52. Half of d 2 is equal to 9, so d 2 is equal to 1852) (18) 2 A = 468 ft 2 Substitute 52 for d 1 and 18 for d 2. Area of a kite Simplify. 3. Find d 2 of a rhombus in which d 1 = 3x m and A = 12xy Games Application The pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the red square. Perimeter: Each side of the red square is the diagonal of a square of the grid. Each grid square has a side length of 1 cm, so the diagonal is √  2 cm. The perimeter of the red square is P = 4s = 4 √  2 cm. Area: Method 1 The red square is also a rhombus. The diagonals d 1 and d 2 each measure 2 cm. So its area is 2) (2) = 2 cm 2. 2 2 Method 2 The side length of the red square is √  2 cm, so the area is cm. 2 4. In the tangram above, find the perimeter and area of the large green triangle. 592 592 Chapter 9 Extending Perimeter, Circumference, and Area �� THINK AND DISCUSS 1. Explain why the area of a triangle is half the area of a parallelogram with the same base and height. 2. Compare the formula for the area of a trapezoid with the formula for the area of a rectangle. 3. GET ORGANIZED Copy and complete the graphic organizer. Name all the shapes whose area is given by each area formula and sketch an example of each shape. 9-1 Exercises Exercises GUIDED PRACTICE Find each measurement. KEYWORD: MG7 9-1 KEYWORD: MG7 Parent. 589 1. the area of the parallelogram 2. the height of the rectangle, in which A = 10 x 2 ft 2 3. the perimeter of a square in which A = 169 cm. the area of the trapezoid 5. the base of the
triangle, in which p. 590 A = 58.5 in 2 6. b 1 of a trapezoid in which A = (48x + 68) in 2, h = 8 in., and b 2 = (9x + 12) in. the area of the rhombus p. 591 8. d 2 of the kite, in which A = 187.5 m 2 9. d 2 of a kite in which A = 12 x 2 y 3 cm 2, d 1 = 3xy cm 10. Art The stained-glass window shown is a rectangle p. 592 with a base of 4 ft and a height of 3 ft. Use the grid to find the area of each piece. 9- 1 Developing Formulas for Triangles and Quadrilaterals 593 593 �� Independent Practice For See Exercises Example 11–13 14–16 17–19 20–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING Find each measurement. 11. the height of the parallelogram, 12. the perimeter of the rectangle in which A = 7.5 m 2 13. the area of a parallelogram in which b = (3x + 5) ft and h = (7x - 1) ft 13. 14. the area of the triangle 15. the height of the trapezoid, in which A = 280 cm 2 16. the area of a triangle in which b = (x + 1) ft and h = 8x ft 17. the area of the kite 18. d 2 of the rhombus, in which A = (3 x 2 + 6x) m 2 19. the area of a kite in which d 1 = (6x + 5) ft and d 2 = (4x + 8) ft Crafts In origami, a square base is the starting point for the creation of many figures, such as a crane. In the pattern for the square base, ABCD is a square, and E, F, G, and H are the midpoints of the sides. If AB = 6 in., find the area of each shape. 20. rectangle ABFH 21. �
�AEJ 22. trapezoid ABFJ � � � � � � � � � Multi-Step Find the area of each figure. Round to the nearest tenth, if necessary. 23. 24. 25. Write each area in terms of x. 26. equilateral triangle 27. 30°-60°-90° triangle 28. 45°-45°-90° triangle 594 594 Chapter 9 Extending Perimeter, Circumference, and Area �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A sign manufacturer makes yield signs by cutting an equilateral triangle from a square piece of aluminum with the dimensions shown. a. Find the height of the yield sign to the nearest tenth. b. Find the area of the sign to the nearest tenth. c. How much material is left after a sign is made? Find the missing measurements for each rectangle. Base b Height h Area A Perimeter P 12 17 30. 31. 32. 33. 16 11 136 216 50 66 34. The perimeter of a rectangle is 72 in. The base is 3 times the height. Find the area of the rectangle. 35. The area of a triangle is 50 cm 2. The base of the triangle is 4 times the height. Find the height of the triangle. 36. The perimeter of an isosceles trapezoid is 40 ft. The bases of the trapezoid are 11 ft and 19 ft. Find the area of the trapezoid. Use the conversion table for Exercises 37–42. 37. 1 yd 2 = 38. 1 m 2 = 39. 1 cm 2 = 40. 1 mi 2 =? ft 2 ̶̶̶̶? mm 2 ̶̶̶̶? cm 2 ̶̶̶̶? in 2 ̶̶̶̶ History 41. A triangle has a base of 3 yd and a height of 8 yd. Find the area in square feet. 42. A rhombus has diagonals 500 yd and 800 yd in length. Find the area in square miles. Conversion Factors Metric 1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm Customary 1 mi = 1760 yd 1 mi = 5280 ft 1 yd = 3 ft 1 ft =
12 in. President James Garfield was a classics professor and a major general in the Union Army. He was assassinated in 1881. Source: www.whitehouse.gov 43. The following proof of the Pythagorean Theorem was discovered by President James Garfield in 1876 while he was a member of the House of Representatives. a. Write the area of the trapezoid in terms of a and b. b. Write the areas of the three triangles in terms of a, b, and c. c. Use the Area Addition Postulate to write an equation relating your results from parts a and b. Simplify the equation to prove the Pythagorean Theorem. 44. Use the diagram to prove the formula for the area of a rectangle, given the formula for the area of a square. Given: Rectangle with base b and height h Prove: The area of the rectangle is A = bh. Plan: Use the formula for the area of a square to find the areas of the outer square and the two squares inside the figure. Write and solve an equation for the area of the rectangle. 9- 1 Developing Formulas for Triangles and Quadrilaterals 595 595 �� Prove each area formula. 45. Given: Parallelogram with area A = bh Prove: The area of the triangle is A = 1 __ 2 bh. 46. Given: Triangle with area A = 1__ 2bh Prove: The area of the trapezoid is. 2 47. Measurement Choose an appropriate unit of measurement and measure the base and height of each parallelogram. a. Find the area of each parallelogram. Give your answer with the correct precision. b. Which has the greatest area? 48. Hobbies Tina is making a kite according to the plans at right. The fabric weighs about 40 grams per square meter. The diagonal braces, or spars, weigh about 20 grams per meter. Estimate the weight of the kite. 49. Home Improvement Tom is buying tile for a 12 ft by 18 ft rectangular kitchen floor. He needs to buy 15% extra in case some of the tiles break. The tiles are squares with 4 in. sides that come in cases of 100. How many cases should he buy? �� 50. Critical Thinking If the maximum error in the given measurements of the rectangle is 0.1 cm, what is the greatest possible error in the area? Explain. ��
51. Write About It A square is also a parallelogram, a rectangle, and a rhombus. Prove that the area formula for each shape gives the same result as the formula for the area of a square. 52. Which expression best represents the area of the rectangle? 2x + 2 (x - c) x (x - c) x 2 + (x - c) 2 2x (x - c) 53. The length of a rectangle is 3 times the width. The perimeter is 48 inches. Which system of equations can be used to find the dimensions of the rectangle? ℓ = w + 3 2 (ℓ + w) = 48 ℓ = 3w 2ℓ + 6w = 48 ℓ = 3w 2 (ℓ + w) = 48 ℓ = w + 3 2ℓ + 6w = 48 596 596 Chapter 9 Extending Perimeter, Circumference, and Area �� 54. A 16- by 18-foot rectangular section of a wall will be covered by square tiles that measure 2 feet on each side. If the tiles are not cut, how many of them will be needed to cover the section of the wall? 288 144 72 17 55. The area of trapezoid HJKM is 90 square centimeters. Which is closest to the length of 10 centimeters ̶̶ JK? 11.7 centimeters 10.5 centimeters 16 centimeters 56. Gridded Response A driveway is shaped like a parallelogram with a base of 28 feet and a height of 17 feet. Covering the driveway with crushed stone will cost $2.75 per square foot. How much will it cost to cover the driveway with crushed stone? �� CHALLENGE AND EXTEND Multi-Step Find h in each parallelogram. 57. �� 58. �� 59. Algebra A rectangle has a perimeter of (26x + 16) cm and an area of (42 x 2 + 51x + 15) cm 2. Find the dimensions of the rectangle in terms of x. 60. Prove that the area of any quadrilateral with perpendicular diagonals is 1 __ 2 d 1 d 2. 61. Gardening A gardener has 24 feet of fencing to enclose a rectangular garden. a. Let x and y represent the side
lengths of the rectangle. Solve the perimeter formula 2x + 2y = 24 for y, and substitute the expression into the area formula A = xy. b. Graph the resulting function on a coordinate plane. What are the domain and range of the function? c. What are the dimensions of the rectangle that will enclose the greatest area? d. Write About It How would you find the dimensions of the rectangle with the least perimeter that would enclose a rectangular area of 100 square feet? SPIRAL REVIEW Determine the range of each function for the given domain. (Previous course) 62. f (x) = x - 3, domain: -4 ≤ x ≤ 6 63. f (x) = - x 2 + 2, domain: -2 ≤ x ≤ 2 Find the perimeter and area of each figure. Express your answers in terms of x. (Lesson 1-5) 64. �� 65. � �� Write each vector in component form. (Lesson 8-6) 66.  LM with L (4, 3) and M (5, 10) 67.  ST with S (-2, -2) and T (4, 6) 9- 1 Developing Formulas for Triangles and Quadrilaterals 597 597 9-2 Develop π The ratio of the circumference of a circle to its diameter is defined as π. All circles are similar, so this ratio is the same for all circles: π = circumference __. diameter Use with Lesson 9-2 TEKS G.5.A Geometric patterns: use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Activity 1 1 Use your compass to draw a large circle on a piece of cardboard and then cut it out. 2 Use a measuring tape to measure the circle’s diameter and circumference as accurately as possible. 3 Use the results from your circle to estimate π. Compare your answers with the results of the rest of the class. Try This 1. Do you think it is possible to draw a circle whose ratio of circumference to diameter is not π? Why or why not? 2. How does knowing the relationship between circumference, diameter, and π help you determine the formula for circumference? 3. Use a ribbon to make a π measuring tape. Mark off increments of π inches or π cm on your ribbon as accurately as
possible. How could you use this π measuring tape to find the diameter of a circular object? Use your π measuring tape to measure 5 circular objects. Give the circumference and diameter of each object. 598 598 Chapter 9 Extending Perimeter, Circumference, and Area Archimedes used inscribed and circumscribed polygons to estimate the value of π. His “method of exhaustion” is considered to be an early version of calculus. In the figures below, the circumference of the circle is less than the perimeter of the larger polygon and greater than the perimeter of the smaller polygon. This fact is used to estimate π. Activity 2 1 Construct a large square. Construct the perpendicular bisectors of two adjacent sides. 2 Use your compass to draw an inscribed circle as shown. 3 Connect the midpoints of the sides to form a square that is inscribed in the circle. 4 Let P 1 represent the perimeter of the smaller square, P 2 represent the perimeter of the larger square, and C represent the circumference of the circle. Measure the squares to find P 1 and P 2 and substitute the values into the inequality below. P 1 < C < P 2 5 Divide each expression in the inequality by the diameter of the circle. Why does this give you an inequality in terms of π? Complete the inequality below.? < π < ̶̶̶̶̶? ̶̶̶̶̶ Try This 4. Use the perimeters of the inscribed and circumscribed regular hexagons to write an inequality for π. Assume the diameter of each circle is 2 units. 5. Compare the inequalities you found for π. What do you think would be true about your inequality if you used regular polygons with more sides? How could you use inscribed and circumscribed regular polygons to estimate π? 6. An alternate definition of π is the area of a circle with radius 1. How could you use this definition and the figures above to estimate the value of π? 9- 2 Geometry Lab 599 599 9-2 Developing Formulas for Circles and Regular Polygons TEKS G.8.A Congruence and the geometry of size: find areas of regular polygons, circles,... Also G.5.A, G.8.C Objectives Develop and apply the formulas for the area and circumference of a circle. Develop and apply the formula for the area of a regular polygon. Vocabulary circle center of a circle center
of a regular polygon apothem central angle of a regular polygon Who uses this? Drummers use drums of different sizes to produce different notes. The pitch is related to the area of the top of the drum. (See Example 2.) A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol ⊙ and its center. ⊙A has radius r = AB and diameter d = CD. The irrational number π is defined as the ratio of the circumference C to the diameter d, or π = C__ d Solving for C gives the formula C = πd. Also d = 2r, so C = 2πr.. You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram. The base of the parallelogram is about half the circumference, or πr, and the height is close to the radius r. So A ≅ πr · r = πr 2. The more pieces you divide the circle into, the more accurate the estimate will be. Circumference and Area Circle A circle with diameter d and radius r has circumference C = πd or C = 2πr and area Finding Measurements of Circles Find each measurement. A the area of ⊙P in terms of π A = πr 2 A = π (8) 2 A = 64π cm 2 Area of a circle Divide the diameter by 2 to find the radius, 8. Simplify. 600 600 Chapter 9 Extending Perimeter, Circumference, and Area �� Find each measurement. B the radius of ⊙X in which C = 24π in. C = 2πr 24π = 2πr r = 12 in. Circumference of a circle Substitute 24π for C. Divide both sides by 2π. C the circumference of ⊙S in which A = 9x 2 π cm 2 Step 1 Use the given area to solve for r. A = π r 2 9x 2 π = π r 2 9x 2 = r 2 3x = r Area of a circle Substitute 9x 2 π for A. Divide both sides by π. Take the square root of both sides. Step 2 Use the value of r to find
the circumference. C = 2πr C = 2π (3x) Substitute 3x for r. C = 6xπ cm Simplify. 1. Find the area of ⊙A in terms of π in which C = (4x - 6) π m. E X A M P L E 2 Music Application A drum kit contains three drums with diameters of 10 in., 12 in., and 14 in. Find the area of the top of each drum. Round to the nearest tenth. The π key gives the best possible approximation for π on your calculator. Always wait until the last step to round. 10 in. diameter A = π ( 5 2 ) r = 10 _ 2 ≅ 78.5 in 2 12 in. diameter 14 in. diameter = 5 A = π ( 6 2 ) r = 12 _ 2 ≅ 113.1 in 2 = 6 A = π (7) 2 r = 14 _ 2 = 7 ≅ 153.9 in 2 2. Use the information above to find the circumference of each drum. The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. ____ n. Each central angle measure of a regular n-gon is 360° To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles. area of each triangle: 1 _ as 2 total area of the polygon: A = n ( 1 _ as), or A = 1 _ 2 2 aP The perimeter is P = ns. Area Regular Polygon The area of a regular polygon with apothem a and perimeter P is A = 1 _ 2 aP. 9- 2 Developing Formulas for Circles and Regular Polygons 601 601 �� E X A M P L E 3 Finding the Area of a Regular Polygon Find the area of each regular polygon. Round to the nearest tenth. A a regular hexagon with side length 6 m The perimeter is 6 (6) = 36 m. The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°-90
° Triangle Theorem, the apothem is 3 √  3 m3 √  3 ) (36) 2 aP Area of a regular polygon Substitute 3 √  3 for a and 36 for P. The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525. A = 54 √  3 ≅ 93.5 m 2 Simplify. B a regular pentagon with side length 8 in. Step 1 Draw the pentagon. Draw an isosceles triangle with its vertex at the center of the ____ 5 = 72°. pentagon. The central angle is 360° Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Step 2 Use the tangent ratio to find the apothem. tan 36° = 4 _ a 4 _ a = tan 36° The tangent of an angle is opp. leg _______. adj. leg Solve for a. Step 3 Use the apothem and the given side length to find the area. aP tan 36° 2 A ≅ 110.1 in 2 ) (40) Area of a regular polygon The perimeter is 8 (5) = 40 in. Simplify. Round to the nearest tenth. 3. Find the area of a regular octagon with a side length of 4 cm. THINK AND DISCUSS 1. Describe the relationship between the circumference of a circle and π. 2. Explain how you would find the central angle of a regular polygon with n sides. 3. GET ORGANIZED Copy and complete the graphic organizer. 602 602 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-2 Exercises Exercises KEYWORD: MG7 9-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe how to find the apothem of a square with side length s Find each measurement. p. 600 2. the circumference of ⊙C 3. the area of ⊙A in terms of π 4. the circumference of ⊙P in which A = 36π in. Food A pizza parlor offers
pizzas with diameters of 8 in., 10 in., and 12 in. p. 601 Find the area of each size pizza. Round to the nearest tenth Find the area of each regular polygon. Round to the nearest tenth. p. 602 6. 7. Independent Practice For See Exercises Example 10–12 13 14–17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 8. an equilateral triangle with an apothem of 2 ft 9. a regular dodecagon with a side length of 5 m PRACTICE AND PROBLEM SOLVING Find each measurement. Give your answers in terms of π. 10. the area of ⊙M 11. the circumference of ⊙Z 12. the diameter of ⊙G in which C = 10 ft. 13. Sports A horse trainer uses circular pens that are 35 ft, 50 ft, and 66 ft in diameter. Find the area of each pen. Round to the nearest tenth. Find the area of each regular polygon. Round to the nearest tenth, if necessary. 14. 15. 16. a regular nonagon with a perimeter of 144 in. 17. a regular pentagon with an apothem of 2 ft. 9- 2 Developing Formulas for Circles and Regular Polygons 603 603 �� Biology Dendroclimatologists study tree rings for evidence of changes in weather patterns over time. Find the central angle measure of each regular polygon. (Hint: To review polygon names, see page 382.) 18. equilateral triangle 19. square 20. pentagon 21. hexagon 22. heptagon 23. octagon 24. nonagon 25. decagon Find the area of each regular polygon. Round to the nearest tenth. 26. 29. 27. 30. 28. 31. 32. Biology You can estimate a tree’s age in years by using the formula a = r __ w, where r is the tree’s radius without bark and w is the average thickness of the tree’s rings. The circumference of a white oak tree is 100 in. The bark is 0.5 in. thick, and the average width of a ring is 0.2 in. Estimate the tree’s age. 33. /////ERROR ANALYSIS///// A circle has a
circumference of 2π in. Which calculation of the area is incorrect? Explain. Find the missing measurements for each circle. Give your answers in terms of π. Diameter d Radius r Area A Circumference C 6 34. 35. 36. 37. 100 17 36 π 38. Multi-Step Janet is designing a garden around a gazebo that is a regular hexagon with side length 6 ft. The garden will be a circle that extends 10 feet from the vertices of the hexagon. What is the area of the garden? Round to the nearest square foot. 39. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A stop sign is a regular octagon. The signs are available in two sizes: 30 in. or 36 in. a. Find the area of a 30 in. sign. Round to the nearest tenth. b. Find the area of a 36 in. sign. Round to the nearest tenth. c. Find the percent increase in metal needed to make a 36 in. sign instead of a 30 in. sign. 604 604 Chapter 9 Extending Perimeter, Circumference, and Area �� 40. Measurement A trundle wheel is used to measure distances by rolling it on the ground and counting its number of turns. If the circumference of a trundle wheel is 1 meter, what is its diameter? 41. Critical Thinking Which do you think would seat more people, a 4 ft by 6 ft rectangular table or a circular table with a diameter of 6 ft? How many people would you sit at each table? Explain your reasoning. 42. Write About It The center of each circle in the figure lies on the number line. Describe the relationship between the circumference of the largest circle and the circumferences of the four smaller circles. 43. Find the perimeter of the regular octagon to the nearest centimeter. 5 40 20 68 44. Which of the following ratios comparing a circle’s circumference C to its diameter d gives the value of π? C _ d 4C _ d 2 d _ C d _ 2C 45. Alisa has a circular tabletop with a 2-foot diameter. She wants to paint a pattern on
the table top that includes a 2-foot-by-1-foot rectangle and 4 squares with sides 0.5 foot long. Which information makes this scenario impossible? There will be no room left on the tabletop after the rectangle has been painted. A 2-foot-long rectangle will not fit on the circular tabletop. Squares cannot be painted on the circle. There will not be enough room on the table to fit all the 0.5-foot squares. CHALLENGE AND EXTEND 46. Two circles have the same center. The radius of the larger circle is 5 units longer than the radius of the smaller circle. Find the difference in the circumferences of the two circles. 47. Algebra Write the formula for the area of a circle in terms of its circumference. 48. Critical Thinking Show that the formula for the area of a regular n-gon approaches the formula for the area of a circle as n gets very large. SPIRAL REVIEW Write an equation for the linear function represented by the table. (Previous course) 49. x y -2 0 -19 -13 5 2 10 17 50. x y -3 2 0 4 9 -1 -5 -10 Find each value. (Lesson 4-8) 51. m∠B 52. AB Find each measurement. (Lesson 9-1) 53. d 2 of a kite if A = 14 cm 2 and d 1 = 20 cm 54. the area of a trapezoid in which b 1 = 3 yd, b 2 = 6 yd, and h = 4 yd 9- 2 Developing Formulas for Circles and Regular Polygons 605 605 �� 9-3 Composite Figures TEKS G.8.A Congruence and the geometry of size: find areas of... composite figures. Objectives Use the Area Addition Postulate to find the areas of composite figures. Use composite figures to estimate the areas of irregular shapes. Vocabulary composite figure Who uses this? Landscape architects must compute areas of composite figures when designing gardens. (See Example 3.) A composite figure is made up of simple shapes, such as triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, find the areas of the simple shapes and then use the Area Addition Postulate. E X A M P L E 1 Finding the Areas of Composite Figures by Adding Find the shaded area. Round to the nearest tenth, if
necessary. A B Divide the figure into rectangles. Divide the figure into parts. The base of the triangle is √  10. 2 2 - 4. 8 2 = 9 ft. area of top rectangle: A = bh = 12 (15) = 180 c m 2 area of bottom rectangle: A = bh = 9 (27) = 243 c m 2 shaded area: 180 + 243 = 423 c m 2 area of triangle: A = 1 __ 2 bh = 1 __ 2 (9) (4.8) = 21.6 ft 2 area of rectangle: A = bh = 9 (3) = 27 ft 2 area of half circle: A = 1 __ 2 π r 2 = 1 __ 2 π (4. 5 2 ) = 10.125π f t 2 shaded area: 21.6 + 27 + 10.125π ≈ 80.4 ft 2 1. Find the shaded area. Round to the nearest tenth, if necessary. 606 606 Chapter 9 Extending Perimeter, Circumference, and Area �� Sometimes you need to subtract to find the area of a composite figure. E X A M P L E 2 Finding the Areas of Composite Figures by Subtracting Find the shaded area. Round to the nearest tenth, if necessary. A B Subtract the area of the triangle from the area of the rectangle. area of rectangle: A = bh = 18 (36) = 648 m 2 area of triangle: A = 1 __ 2 bh = 1 __ 2 (36) (9) = 162 m 2 area of figure: A = 648 - 162 = 486 m 2 The two half circles have the same area as one circle. Subtract the area of the circle from the area of the rectangle. area of the rectangle: A = bh = 33 (16) = 528 f t 2 area of circle ) = 64π f t 2 area of figure: A = 528 - 64π ≈ 326.9 ft 2 2. Find the shaded area. Round to the nearest tenth, if necessary. E X A M P L E 3 Landscaping Application Katie is using the given plan to convert part of her lawn to a xeriscape garden.
A newly planted xeriscape uses 17 gallons of water per square foot per year. How much water will the garden require in one year? To find the area of the garden in square feet, divide the garden into parts. The area of the top rectangle is 28.5 (7.5) = 213.75 f t 2. The area of the center trapezoid is 1 __ 2 (12 + 18) (6) = 90 f t 2. The area of the bottom rectangle is 12 (6) = 72 f t 2. The total area of the garden is 213.75 + 90 + 72 = 375.75 f t 2. The garden will use 375.75 (17) = 6387.75 gallons of water per year. Landscaping The rainwater harvesting system at the Lady Bird Johnson Wildflower Center in Austin, Texas, collects approximately 10,200 gallons of water per inch of rain. Approximately 300,000 gallons are collected annually. 3. The lawn that Katie is replacing requires 79 gallons of water per square foot per year. How much water will Katie save by planting the xeriscape garden? 9- 3 Composite Figures 607 607 ��ge07sec09l03002aaABeckmann28.5 ft12 ft10.5 ft7.5 ft6 ft19.5 ftge07se_c09l03003aABeckmann28.5 ft12 ft6 ft7.5 ft6 ft18 ft To estimate the area of an irregular shape, you can sometimes use a composite figure. First, draw a composite figure that resembles the irregular shape. Then divide the composite figure into simple shapes. E X A M P L E 4 Estimating Areas of Irregular Shapes Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 cm. Draw a composite figure that approximates the irregular shape. Find the area of each part of the composite figure. area of triangle a: bh = 1 _ A = 1 _ (3) (1) = 1.5 c m 2 2 2 area of parallelogram b: A = bh = 3 (1) = 3 c m 2 area of trapezoid c: A = 1 _ (3 + 2) (1) = 2.5 c m 2 2 area of triangle d: A = 1 _ (2) (1) = 1 c m 2 2 area of composite figure: 1.5
+ 3 + 2.5 + 1 = 8 cm 2 The shaded area is about 8 c m 2. 4. Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 ft. THINK AND DISCUSS 1. Describe a composite figure whose area you could find by using subtraction. 2. Explain how to find the area of an irregular shape by using a composite figure. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the given composite figure. 608 608 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-3 Exercises Exercises KEYWORD: MG7 9-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Draw a composite figure that is made up of two rectangles Multi-Step Find the shaded area. Round to the nearest tenth, if necessary. p. 606 2. p. 607 3. 5. Interior Decorating Barbara is getting carpet p. 607 installed in her living room and hallway. The cost of installation is $6 per square yard. What is the total cost of installing the carpet. 608 Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 in. 7. 8. PRACTICE AND PROBLEM SOLVING Independent Practice Multi-Step Find the shaded area. Round to the nearest tenth, if necessary. For See Exercises Example 9. 9–10 11–12 13 14–15 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 11. 10. 12. 9- 3 Composite Figures 609 609 �� 13. Drama Pat is painting a stage backdrop for a play. The paint he is using covers 90 square feet per quart. How many quarts of paint should Pat buy? Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 m. 14. 15. Find the area of each figure first by adding and then by subtracting. Compare your answers. 16. 17. Find the area of each figure.
Give your answers in terms of π. 18. 19. 20. 21. Geography Use the grid on the map of Lake Superior to estimate the area of the surface of the lake. Each square on the grid has a side length of 100 miles. 22. Critical Thinking A trapezoid can be divided into a rectangle and two triangles. Show that the area formula for a trapezoid gives the same result as the sum of the areas of the rectangle and triangles. 23. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A school crossing sign has the dimensions shown. a. Find the area of the sign. b. A manufacturer has a rectangular sheet of metal measuring 45 in. by 105 in. Draw a figure that shows how 6 school crossing signs can be cut from this sheet of metal. c. How much metal will be left after the six signs are made? 610 610 Chapter 9 Extending Perimeter, Circumference, and Area 30 ft15 ft22 ftge07sec09l03005a��Lake SuperiorWISCONSINMINN.CANADAMICHIGAN ge07se_c09l03006a�� Math History � � � � � � Hippocrates attempted to use lunes to solve a �� problem that has since been proven impossible: constructing a square with the same area as a given circle. Multi-Step Use a ruler and compass to draw each figure and then find the area. 24. A rectangle with a base length of b = 3 cm and a height of h = 4 cm has a circle with a radius of r = 1 cm removed from the interior. 25. A square with a side length of s = 4 in. shares a side with a triangle with a height of h = 5 in. and a base length of b = 4 in. and shares another side with a half circle with d = 4 in. 26. A circle with a radius of r = 5 cm has a right triangle with a base of b = 8 cm and a height of h = 6 cm removed from its interior. � 27. Multi-Step A lune is a crescent-shaped figure bounded by two intersecting circles. Find the shaded area in each of the first three diagrams, and then use your results to � find the area of the lune. � � � � �� Estimation Trace
each irregular shape and draw a composite figure that approximates it. Measure the composite figure and use it to estimate the area of the irregular shape. 28. 29. 30. Write About It Explain when you would use addition to find the area of a composite figure and when you would use subtraction. 31. Which equation can be used to find the area of the composite figure? A = bh + 1 _ (h) 2 2 A = bh + h 2 A = h + 2b + h 2 A = h + 2b + 1 _ h 2 2 32. Use a ruler to measure the dimensions of the composite figure to the nearest tenth of a centimeter. Which of the following best represents the area of the composite figure? 4 c m 2 19 cm 2 22 cm 2 42 c m 2 9- 3 Composite Figures 611 611 �� 33. Find the area of the unshaded part of the rectangle. 1800 m 2 2250 m 2 2925 m 2 4725 m 2 CHALLENGE AND EXTEND 34. An annulus is the region between two circles that have the same center. Write the formula for the area of the annulus in terms of the outer radius R and the inner radius r. 35. Draw two composite figures with the same area: one made up of two rectangles and the other made up of a rectangle and a triangle. 36. Draw a composite figure that has a total area of 10π c m 2 and is made up of a rectangle and a half circle. Label the dimensions of your figure. SPIRAL REVIEW Find each sale price. (Previous course) 37. 20% off a regular price of $19.95 38. 15% off a regular price of $34.60 Find the length of each segment. (Lesson 7-4) 39. ̶̶ BC 40. ̶̶ CD Find the area of each regular polygon. Round to the nearest tenth. (Lesson 9-2) 41. an equilateral triangle with a side length of 3 cm 42. a regular hexagon with an apothem of 4 √  3 m Q: What math classes did you take in high school? A: In high school I took Algebra 1, Geometry, Algebra 2, and Trigonometry. KEYWORD: MG7 Career Q: What math classes did you take in college? A: In college I took Precalculus, Calculus, and Statistics. Q:
What technical materials do you write? A: I write training manuals for computer software packages. Q: How do you use math? A: Some manuals I write are for math programs, so I use a lot of formulas to describe patterns and measurements. Q: What are your future plans? A: After I get a few more years experience writing manuals, I would like to train others who use these programs. Anessa Liu Technical writer 612 612 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-3 Use with Lesson 9-3 Develop Pick’s Theorem for Area of Lattice Polygons A lattice polygon is a polygon drawn on graph paper so that all its vertices are on intersections of grid lines, called lattice points. The lattice points of the grid at right are shown in red. In this lab, you will discover a formula called Pick’s Theorem, which is used to find the area of lattice polygons. Activity TEKS G.8.A Congruence and the geometry of size: find areas of... composite figures. 1 Find the area of each figure. Create a table like the one below with a row for each shape to record your answers. The first one is done for you. 2 Count the number of lattice points on the boundary of each figure. Record your answers in the table. 3 Count the number of lattice points in the interior of each figure. Record your answers in the table. Figure Area Number of Lattice Points On Boundary In Interior 2.5 5 1 A B C D E F Try This 1. Make a Conjecture What do you think is true about the relationship between the area of a figure and the number of lattice points on the boundary and in the interior of the figure? Write your conjecture as a formula in terms of the number of lattice points on the boundary B and the number of lattice points in the interior I. 2. Test your conjecture by drawing at least three different figures on graph paper and by finding their areas. 3. Estimate the area of the curved figure by using a lattice polygon. 4. Find the shaded area in the figure by subtracting. Test your formula on this figure. Does your formula work for figures with holes in them? 9- 3 Geometry Lab 613 613 �� SECTION 9A Developing Geometric Formulas Traffic Signs Traffic
signs are usually made of reflective aluminum. A manufacturer of traffic signs begins with a rectangular sheet of aluminum that measures 60 in. by 90 in. 1. A railroad crossing sign is a circle with a diameter of 30 in. The manufacturer can make 6 of these signs from the sheet of aluminum by arranging the signs as shown. How much aluminum is left over once the signs have been made? 2. A stop sign is a regular octagon. The manufacturer can use the sheet of aluminum to make 6 stop signs as shown. How much aluminum is left over in this case? 3. A yield sign is an equilateral triangle with sides 30 in. long. By arranging the triangles as shown, the manufacturer can use the sheet of aluminum to make 10 yield signs. How much aluminum is left over when yield signs are made? 4. The making of which type of sign results in the least amount of waste? 614 614 Chapter 9 Extending Perimeter, Circumference, and Area �� SECTION 9A Quiz for Lessons 9-1 Through 9-3 9-1 Developing Formulas for Triangles and Quadrilaterals Find each measurement. 1. the area of the parallelogram 2. the base of the rectangle, in which A = (24 x 2 + 8x) m 2 3. d 1 of the kite, in which A = 126 ft 2 4. the area of the rhombus 5. The tile mosaic shown is made up of 1 cm squares. Use the grid to find the perimeter and area of the green triangle, the blue trapezoid, and the yellow parallelogram. 9-2 Developing Formulas for Circles and Regular Polygons Find each measurement. 6. the circumference of ⊙R in terms of π 7. the area of ⊙E in terms of π Find the area of each regular polygon. Round to the nearest tenth. 8. a regular hexagon with apothem 6 ft 9. a regular pentagon with side length 12 m 9-3 Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. 10. 11. 12. Shelby is planting grass in an irregularly shaped garden as shown. The grid has squares with side lengths of 1 yd. Estimate the area of the garden. Given that grass cost $6.50 per square yard, find the cost of the grass. Ready to Go On? 615 6
15 �� 9-4 Perimeter and Area in the Coordinate Plane TEKS G.7.A Dimensionality and the geometry of location: use... two-dimensional coordinate systems to represent... figures. Also G.7.B, G.8.A Objective Find the perimeters and areas of figures in a coordinate plane. Why learn this? You can use figures in a coordinate plane to solve puzzles like the one at right. (See Example 4.) In Lesson 9-3, you estimated the area of irregular shapes by drawing composite figures that approximated the irregular shapes and by using area formulas. Another method of estimating area is to use a grid and count the squares on the grid. E X A M P L E 1 Estimating Areas of Irregular Shapes in the Coordinate Plane Estimate the area of the irregular shape. Method 1: Draw a composite Method 2: Count the number of figure that approximates the irregular shape and find the area of the composite figure. squares inside the figure, estimating half squares. Use a ■ for a whole for a half square. square and a The area is approximately 4 + 6.5 + 5 + 4 + 5 + 3.5 + 3 + 3 + 2 = 36 unit s 2. There are approximately 31 whole squares and 13 half squares, so the area is about 31 + 1 __ 2 (13) = 37.5 unit s 2. 1. Estimate the area of the irregular shape. 616 616 Chapter 9 Extending Perimeter, Circumference, and Area �� E X A M P L E 2 Finding Perimeter and Area in the Coordinate Plane Draw and classify the polygon with vertices A (-4, 1), B (2, 4), C (4, 0), and D (-2, -3). Find the perimeter and area of the polygon. ( x 2 - x 1 ) The distance from ( x 1, y 1 ) to ( x 2, y 2 ) in a coordinate plane is d = √  2 2 + ( y 2 - y 1 ), and the slope of the line containing the y 2 - y 1 ______ x 2 - x 1. points is
m = See pages 44 and 182. Step 1 Draw the polygon. slope of Step 2 ABCD appears to be a rectangle. To verify this, use slopes to show that the sides are perpendicular. ̶̶ = 3 _ AB : - (-4) ̶̶ BC : 0 - 4 _ = -4 _ 4 - 2 2 ̶̶ = -3 _ CD : -3 - 0 _ = 1 _ -6 -2 - 4 2 1- (-3) = 4 _ _ -2 -4 - (-2) slope of slope of slope of ̶̶ DA : = -2 = -2 ̶̶ CD be the base and The consecutive sides are perpendicular, so ABCD is a rectangle. ̶̶ BC be the height of the rectangle. Step 3 Let Use the Distance Formula to find each side length. b = CD = √  h = BC = √  perimeter of ABCD: P = 2b + 2h = 2 (3 √  5 ) + 2 (2 √  5 ) = 10 √  5 units area of ABCD: A = bh = (3 √  5 ) (2 √  5 ) = 30 units 2. (-2 - 4) 2 + (-3 - 0) 2 = √  45 = 3 √  5 (4 - 2) 2 + (0 - 4) 2 = √  20 = 2 √  5 2. Draw and classify the polygon with vertices H (-3, 4), J (2, 6), K (2, 1), and L (-3, -1). Find the perimeter and area of the polygon. For a figure in a coordinate plane that does not have an area formula, it may be easier to enclose the figure in a rectangle and subtract the areas of the parts of the rectangle that are not included in the figure. E X A M P L E 3 Finding Areas in the Coordinate Plane by Subtracting Find the area of the polygon with vertices W (1, 4), X (4, 2), Y (2
, -3), and Z (-4, 0). Draw the polygon and enclose it in a rectangle. area of the rectangle: A = bh = 8 (7) = 56 units 2 area of the triangles: bh = 1 _ a: A = 1 _ (5) (4) = 10 units 2 2 2 bh = 1 _ b: A = 1 _ (3) (2) = 3 units 2 2 2 bh = 1 _ c: A = 1 _ (2) (5) = 5 units 2 2 2 bh = 1 _ d: A = 1 _ (6) (3) = 9 units 2 2 2 The area of the polygon is 56 - 10 - 3 - 5 - 9 = 29 units 2. 3. Find the area of the polygon with vertices K (-2, 4), L (6, -2), M (4, -4), and N (-6, -2). 9- 4 Perimeter and Area in the Coordinate Plane 617 617 �� E X A M P L E 4 Problem-Solving Application In the puzzle, the two figures are made up of the same pieces, but one figure appears to have a larger area. Use coordinates to show that the area does not change when the pieces are rearranged. Understand the Problem The parts of the puzzle appear to form two triangles with the same base and height that contain the same shapes, but one appears to have an area that is larger by one square unit. Make a Plan Find the areas of the shapes that make up each figure. If the corresponding areas are the same, then both figures have the same area by the Area Addition Postulate. To explain why the area appears to increase, consider the assumptions being made about the figure. Each figure is assumed to be a triangle with a base of 8 units and a height of 3 units. Both figures are divided into several smaller shapes. Solve Find the area of each shape. Top figure Bottom figure red triangle: bh = 1 _ A = 1 _ (5) (2) = 5 unit s 2 2 2 blue triangle: bh = 1 _ A = 1 _ (3) (1) = 1.5 units 2 2 2 green rectangle: A = bh = (3) (1) = 3 units 2 red triangle: bh = 1 _ A = 1 _ (5) (2) = 5 u n
its 2 2 2 blue triangle: bh = 1 _ A = 1 _ (3) (1) = 1.5 u nits 2 2 2 green rectangle: A = bh = (3) (1) = 3 units 2 yellow rectangle: A = bh = (2) (1) = 2 units 2 yellow rectangle: A = bh = (2) (1) = 2 units 2 The areas are the same. Both figures have an area of 5 + 1.5 + 3 + 2 = 11.5 units 2. If the figures were triangles, their areas would be A = 1 __ 2 (8) (3) = 12 units 2. By the Area Addition Postulate, the area is only 11.5 units 2, so the figures must not be triangles. Each figure is a quadrilateral whose shape is very close to a triangle. Look Back The slope of the hypotenuse of the red triangle is 2 __ 5. The slope of the hypotenuse of the blue triangle is 1 __ 3. Since the slopes are unequal, the hypotenuses do not form a straight line. This means the overall shapes are not triangles. 4. Create a figure and divide it into pieces so that the area of the figure appears to increase when the pieces are rearranged. 618 618 Chapter 9 Extending Perimeter, Circumference, and Area ��1234 THINK AND DISCUSS 1. Describe two ways to estimate the area of an irregular shape in a coordinate plane. 2. Explain how you could use the Distance Formula to find the area of a special quadrilateral in a coordinate plane. 3. GET ORGANIZED Copy the graph and the graphic organizer. Complete the graphic organizer by writing the steps used to find the area of the parallelogram. 9-4 Exercises Exercises GUIDED PRACTICE Estimate the area of each irregular shape. p. 616 1. 2. KEYWORD: MG7 9-4 KEYWORD: MG7 Parent. 617 Multi-Step Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 3. V (-3, 0), W (3, 0), X (0, 3) 4. F (2, 8), G (4, 4), H (2, 0) 5. P (-2, 5), Q (8, 5), R
(8, 1), S (-2, 1) 6. A (-4, 2), B (-2, 6), C (6, 6), D (8, 2 Find the area of each polygon with the given vertices. p. 617 7. S (3, 8), T (8, 3), U (2, 1) 8. L (3, 5), M (6, 8), N (9, 6), P (5, 0. 618 9. Find the area and perimeter of each polygon shown. Use your results to draw a polygon with a perimeter of 12 units and an area of 4 un its 2 and a polygon with a perimeter of 12 units and an area of 3 un its 2. 9- 4 Perimeter and Area in the Coordinate Plane 619 619 �� Independent Practice For See Exercises Example 10–11 12–15 16–17 18 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S21 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING Estimate the area of each irregular shape. 10. 11. Multi-Step Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 12. H (-3, -3), J (-3, 3), K (5, 3) 13. L (7, 5), M (5, 0), N (3, 5), P (5, 10) 14. X (2, 1), Y (5, 3), Z (7, 1) 15. A (-3, 5), B (2, 7), C (2, 1), D (-3, 3) Find the area of each polygon with the given vertices. 16. A (9, 9), B (4, -4), C (-4, 1) 17. T (-4, 4), U (5, 3), V (4, -5), W (-5, 1) 18. In which two figures do the rectangles cover the same area? Explain your reasoning. Algebra Graph each set of lines to form a triangle. Find the area and perimeter. 19. y = 2, x =
5, and y = x 20. y = -5, x = 2, and y = -2x + 7 21. Transportation The graph shows the speed of a boat versus time. a. If the base of each square on the graph represents 1 hour and the height represents 20 miles per hour, what is the area of one square on the graph? Include units in your answer. b. Estimate the shaded area in the graph. c. Critical Thinking Use your results from part a to interpret the meaning of the area you found in part b. (Hint: Look at the units.) 22. Write About It Explain how to find the perimeter of the polygon with vertices A (2, 3), B (4, 0), C (3, -2), D (-1, -1), and E (-2, 0). 23. This problem will prepare you for the Multi-Step TAKS Prep on page 638. A carnival game uses a 10-by-10 board with three targets. Each player throws a dart at the board and wins a prize if it hits a target. a. One target is a parallelogram as shown. Find its area. b. What should the coordinates be for points C and H so that the triangular target △ABC and the kite-shaped target EFGH have the same area as the parallelogram? 620 620 Chapter 9 Extending Perimeter, Circumference, and Area �� 24. A circle with center (0, 0) passes through the point (3, 4). What is the area of the circle to the nearest tenth of a square unit? 15.7 25.0 31.4 78.5 25. △ABC with vertices A (1, 1) and B (3, 5) has an area of 10 unit s 2. Which is NOT a possible location of the third vertex? C (-4, 1) C (7, 3) C (6, 1) C (3, -3) 26. Extended Response Mike estimated the area of the irregular figure to be 64 units 2. a. Explain why his answer is not very accurate. b. Explain how to use a composite figure to estimate the area. c. Explain how to estimate the area by averaging the areas of two squares. CHALLENGE AND
EXTEND Algebra Estimate the shaded area under each curve. 28. y = x 2 for 0 ≤ x ≤ 3 27. y = 2 x for 0 ≤ x ≤ 3 29. y = √  x for 0 ≤ x ≤ 9 30. Estimation Use a composite figure and the Distance Formula to estimate the perimeter of the irregular shape. 31. Graph a regular octagon on the coordinate plane with vertices on the x-and y-axes and on the lines y = x and y = -x so that the distance between opposite vertices is 2 units. Find the area and perimeter of the octagon. SPIRAL REVIEW Solve and graph each compound inequality. (Previous course) 32. -4 < x + 3 < 7 ̶̶ BC, ∠DCA ≅ ∠ACB Prove: ∠DAC ≅ ∠BAC (Lesson 4-6) 35. Given: ̶̶ DC ≅ 33. 0 < 2a + 4 < 10 34. 12 ≤ -2m + 10 ≤ 20 Find each measurement. (Lesson 9-2) 36. the area of ⊙C if the circumference is 16π cm 37. the diameter of ⊙H if the area is 121π f t 2 9- 4 Perimeter and Area in the Coordinate Plane 621 621 �� 9-5 Effects of Changing Dimensions Proportionally TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area,... when one or more dimensions of a figure are changed.... Also G.5.A, G.5.B Objectives Describe the effect on perimeter and area when one or more dimensions of a figure are changed. Apply the relationship between perimeter and area in problem solving. Why learn this? You can analyze a graph to determine whether it is misleading or to explain why it is misleading. (See Example 4.) In the graph, the height of each DVD is used to represent the number of DVDs shipped per year. However as the height of each DVD increases, the width also increases, which can create a misleading effect. E X A M P L E 1 Effects of Changing One Dimension Describe the effect of each change on the area of the given figure. A The height of the parallelogram is doubled. double the height: A = bh = 12 (18) =
216 cm 2 original dimensions: A = bh = 12 (9) = 108 cm 2 Notice that 216 = 2 (108). If the height is doubled, the area is also doubled. B The base length of the triangle with vertices A(1, 1), B(6, 1), and C (3, 5) is multiplied by 1 __. 2 Draw the triangle in a coordinate plane and find the base and height. original dimensions: bh = 1 _ A = 1 _ (5) (4) = 10 units 2 2 2 base multiplied by 1 __ 2 : bh = 1 _ A = 1 _ (2.5) (4) = 5 units 2 2 2 Notice that 5 = 1 __ 2 (10). If the base length is multiplied by 1 __ 2, the area is multiplied by 1 __ 2. 1. The height of the rectangle is tripled. Describe the effect on the area. 622 622 Chapter 9 Extending Perimeter, Circumference, and Area ge07se_c09l05001a AB�� E X A M P L E 2 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter or circumference and the area of the given figure. A The base and height of a rectangle with base 8 m and height 3 m are If the radius of a circle or the side length of a square is changed, the size of the entire figure changes proportionally. both multiplied by 5. original dimensions: P = 2 (8) + 2 (3) = 22 m A = 83 = 24 m 2 dimensions multiplied by 5: P = 2 (40) + 2 (15) = 110 m A = 40 (15) = 600 m 2 The perimeter is multiplied by 5. The area is multiplied by 5 2, or 25. B The radius of ⊙A is multiplied by 1 __. 3 original dimensions: C = 2π (9) = 18π in. A = π (9) 2 = 81π in 2 dimensions multiplied by 1 __ 3 : C = 2π (3) = 6π in. A = π (3) 2 = 9π in 2 P = 2b + 2h A = bh 5 (8) = 40; 5 (3) = 15 5 (22) = 110 25 (24) = 600 C = 2πr A = π r 2 1 __ (
9) = 3 3 The perimeter is multiplied by 1 __ 3. The area is multiplied by ( 1 __ 3 ), or 1 __ 9. 2 1 __ (18π) = 6π 3 1 __ (81π) = 9π 9 2. The base and height of the triangle with vertices P (2, 5), Q (2, 1) and R (7, 1) are tripled. Describe the effect on its area and perimeter. When all the dimensions of a figure are changed proportionally, the figure will be similar to the original figure. Effects of Changing Dimensions Proportionally Change in Dimensions Perimeter or Circumference Area All dimensions multiplied by a Changes by a factor of a Changes by a factor of Effects of Changing Area A A square has side length 5 cm. If the area is tripled, what happens to the side length? The area of the original square is A = s 2 = 5 2 = 25 cm 2. If the area is tripled, the new area is 75 cm 2. s 2 = 75 s = √  75 = 5 √  3 Set the new area equal to s 2. Take the square root of both sides and simplify. Notice that 5 √  3 = √  3 (5). The side length is multiplied by √  3. 9- 5 Effects of Changing Dimensions Proportionally 623 623 �� B A circle has a radius of 6 in. If the area is doubled, what happens to the circumference? The original area is A = π r 2 = 36π in 2, and the circumference is C = 2πr = 12π in. If the area is doubled, the new area is 72π in 2. π r 2 = 72π r 2 = 72 r 2 = √  72 = 6 √  2 C = 2πr = 2π ( 6 √  2 ) = 12 √  2 π Set the new area equal to π r 2. Divide both sides by π. Take the square root of both sides and simplify. Substitute 6 √  2 for r and simplify. Notice that 12 √  2 π = √  2 (12π). The circumference is multiplied by √  2. 3. A square has a perimeter of 36 mm. If the area
is multiplied by 1 __ 2, what happens to the side length? E X A M P L E 4 Entertainment Application The graph shows that DVD shipments totaled about 182 million in 2000, 364 million in 2001, and 685 million in 2002. The height of each DVD is used to represent the number of DVDs shipped. Explain why the graph is misleading. The height of the DVD representing shipments in 2002 is about 3.8 times the height of the DVD representing shipments in 2002. This means that the area of the DVD is multiplied by about 3.8 2, or 14.4, so the area of the larger DVD is about 14.4 times the area of the smaller DVD. The graph gives the misleading impression that the number of shipments in 2002 was more than 14 times the number in 2000, but it was actually closer to 4 times the number shipped in 2000. 4. Use the information above to create a version of the graph that is not misleading. THINK AND DISCUSS 1. Discuss how changing both dimensions of a rectangle affects the area and perimeter. 2. GET ORGANIZED Copy and complete the graphic organizer. 624 624 Chapter 9 Extending Perimeter, Circumference, and Area ��20002001YearDVDs shipped (millions)2002100200300400500600ge06se_c09105002aDVD Shipments 9-5 Exercises Exercises KEYWORD: MG7 9-5 KEYWORD: MG7 Parent GUIDED PRACTICE Describe the effect of each change on the area of the given figure. p. 622 1. The height of the triangle is doubled. 2. The height of a trapezoid with base lengths 12 cm and 18 cm and height 5 cm is multiplied by 1 __. 623 Describe the effect of each change on the perimeter or circumference and the area of the given figure. 3. The base and height of a triangle with base 12 in. and height 6 in. are both tripled. 4. The base and height of the rectangle are both multiplied by 1 __. A square has an area of 36 m 2. If the area is doubled, what happens to the p. 623 side length? 6. A circle has a diameter of 14 ft. If the area is tripled, what happens to the circumference. Business A restaurant has a weekly ad in a
local newspaper that is 2 inches wide p. 624 and 4 inches high and costs $36.75 per week. The cost of each ad is based on its area. If the owner of the restaurant decides to double the width and height of the ad, how much will the new ad cost? Independent Practice Describe the effect of each change on the area of the given figure. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–9 10–11 12–13 14 1 2 3 4 8. The height of the triangle with vertices (1, 5), (2, 3), and (-1, -6) is multiplied by 4. 9. The base of the parallelogram is multiplied by 2 __ 3. Describe the effect of each change on the perimeter or circumference and the area of the given figure. TEKS TEKS TAKS TAKS 10. The base and height of the triangle are both doubled. Skills Practice p. S21 Application Practice p. S36 11. The radius of the circle with center (0, 0) that passes through (5, 0) is multiplied by 3 __ 5. 12. A circle has a circumference of 16π mm. If you multiply the area by 1 __ 3, what happens to the radius? 13. A square has vertices (3, 2), (8, 2,) (8, 7), and (3, 7). If you triple the area, what happens to the side length? 14. Entertainment Two televisions have rectangular screens with the same ratio of base to height. One has a 32 in. diagonal, and the other has a 36 in. diagonal. a. What is the ratio of the height of the larger screen to that of the smaller screen? b. What is the ratio of the area of the larger screen to that of the smaller screen? 9- 5 Effects of Changing Dimensions Proportionally 625 625 �� Describe the effect of each change on the area of the given figure. 15. The diagonals of a rhombus are both multiplied by 8. 16. The circumference of a circle is multiplied by 2.4. 17. The base of a rectangle is multiplied by 4, and the height is multiplied by 7. 18. The apothem of a regular octagon is tripled. 19. The diagonal of a square is divided by 4. 20. One diagonal of a kite is multiplied
by 1 _. 7 21. 21. The perimeter of an equilateral triangle is doubled. 22. 22. Find the area of the trapezoid. Describe the effect of each change on the area. a. a. The length of the top base is doubled. b. The length of both bases is doubled. c. The height is doubled. d. Both bases and the height are doubled. 23. Geography A map has the scale 1 inch = 10 miles. On the map, the area of Big Bend National Park in Texas is about 12.5 square inches. Estimate the actual area of the park in acres. (Hint: 1 square mile = 640 acres) 24. Critical Thinking If you want to multiply the dimensions of a figure so that the area is 50% of the original area, what is your scale factor? Multi-Step For each figure in the coordinate plane, describe the effect on the area that results from each change. a. Only the x-coordinates of the vertices are multiplied by 3. b. Only the y-coordinates of the vertices are multiplied by 3. c. Both the x- and y-coordinates of the vertices are multiplied by 3. 25. 26. 27. Geography Geography The altitude in Big Bend National Park ranges from approximately 1800 feet along the Rio Grande to 7800 feet in the Chisos Mountains. 28. Write About It How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram? if the parallelogram does have to be similar to the original parallelogram? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 638. To win a prize at a carnival, a player must toss a beanbag onto a circular disk with a diameter of 8 in. a. The organizer of the game wants players to win twice as often, so he changes the disk so that it has twice the area. What is the diameter of the new disk? b. Suppose the organizer wants players to win half as often. What should be the disk’s diameter in this case? 626 626 Chapter 9 Extending Perimeter, Circumference, and Area �� 30. Which of the following describes the effect on the area of a square when the side length is doubled? The area remains constant
. The area is reduced by a factor of 1 __. 2 The area is doubled. The area is increased by a factor of 4. 31. If the area of a circle is increased by a factor of 4, what is the change in the diameter of the circle? The diameter is 1 __ of the original diameter. 2 The diameter is 2 times the original diameter. The diameter is 4 times the original diameter. The diameter is 16 times the original diameter. 32. Tina and Kieu built rectangular play areas for their dogs. The play area for Tina’s dog is 1.5 times as long and 1.5 times as wide as the play area for Kieu’s dog. If the play area for Kieu’s dog is 60 square feet, how big is the play area for Tina’s dog? 40 ft 2 90 ft 2 135 ft 2 240 ft 2 33. Gridded Response Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled. Find the perimeter of the new triangle in inches. CHALLENGE AND EXTEND 34. Algebra A square has a side length of (2x + 5) cm. If the side length is multiplied by 5, what is the area of the new square? 35. Algebra A circle has a diameter of 6 in. If the circumference is multiplied by (x + 3), what is the area of the new circle? 36. Write About It How could you change the dimensions of the composite figure to double the area if the resulting figure does not have to be similar to the original figure? if the resulting figure does have to be similar to the original figure? SPIRAL REVIEW Write an equation that can be used to determine the value of the variable in each situation. (Previous course) 37. Steve can make 2 tortillas per minute. He makes t tortillas in 36 minutes. 38. A car gets 25 mi/gal. At the beginning of a trip of m miles, the car’s gas tank contains 13 gal of gas. At the end of the trip, the car has 8 gal of gasoline left. Find the measure of each interior and each exterior angle of each regular polygon. Round to the nearest tenth, if necessary. (Lesson 6-1) 39. heptagon 40. decagon 41. 14-gon Find the area of each polygon with the given vertices. (Lesson 9-4) 42. L (-1, 1),
M (5, 2), and N (1, -5) 43. A (-4, 2), M (-2, 4), C (4, 2) and D (2, -4) 9- 5 Effects of Changing Dimensions Proportionally 627 627 �� Probability Probability An experiment is an activity in which results are observed. Each result of an experiment is called an outcome. The sample space is the set of all outcomes of an experiment. An event is any set of outcomes. See Skills Bank page S77 The probability of an event is a number from 0 to 1 that tells you how likely the event is to happen. The closer the probability is to 0, the less likely the event is to happen. The closer it is to 1, the more likely the event is to happen. An experiment is fair if all outcomes are equally likely. The theoretical probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space. P (E) = number of outcomes in event E ___ number of possible outcomes Example 1 A fair number cube has six faces, numbered 1 through 6. An experiment consists of rolling the number cube. A What is the sample space of the experiment? The sample space has 6 possible outcomes. The outcomes are 1, 2, 3, 4, 5, and 6. B What is the probability of the event “rolling a 4”? The event “rolling a 4” contains only 1 outcome. The probability is P (E) = number of outcomes in event E___ number of possible outcomes = 1_. 6 C What are the outcomes in the event “rolling an odd number”? What is the probability of rolling an odd number? The event “rolling an odd number” contains 3 outcomes. The outcomes are 1, 3, and 5. The probability is P (E) = number of outcomes in event E = 3 _ = 1 _ ___. 2 6 number of possible outcomes If two events A and B have no outcomes in common, then the probability that A or B will happen is P (A) + P (B). The complement of an event is the set of outcomes that are not in the event. If the probability of an event is p, then the probability of the complement of the event is 1 - p. 628 628 Chapter 9 Extending Perimeter, Circumference, and Area Example 2 The tiles shown
below are placed in a bag. An experiment consists of drawing a tile at random from the bag. A What is the sample space of the experiment? The sample space has 9 possible outcomes. The outcomes are 1, 2, 3, 4, A, B, C, D, E, and F. B What is the probability of choosing a 3 or a vowel? The event “choosing a 3” contains only 1 outcome. The probability is P (A) = number of outcomes in event A ___ number of possible outcomes = 1 _. 9 The event “choosing a vowel” has 2 outcomes, A and E. The probability is P (B) = number of outcomes in event B ___ number of possible outcomes = The probability of choosing a 3 or a vowel is 1 _. 3 9 9 9 C What is the probability of not choosing a letter? The event “choosing a letter” contains 5 outcomes, A, B, C, D, and E. The probability is P (E) = number of outcomes in event E ___ number of possible outcomes = 5 _. 9 The event of not choosing a letter is the complement of the event of choosing a letter. The probability of not choosing a letter is 1 - 5 __ 9 = 4 __ 9. Try This TAKS Grades 9–11 Obj. 8, 9 An experiment consists of randomly choosing one of the given shapes. 1. What is the probability of choosing a circle? 2. What is the probability of choosing a shape whose area is 36 cm 2? 3. What is the probability of choosing a quadrilateral or a triangle? 4. What is the probability of not choosing a triangle? On Track for TAKS 629 629 �� 9-6 Geometric Probability TEKS G.8.A Congruence and the geometry of size: find areas of... regular polygons, circles, and composite figures. Objectives Calculate geometric probabilities. Use geometric probability to predict results in realworld situations. Vocabulary geometric probability Why learn this? You can use geometric probability to estimate how long you may have to wait to cross a street. (See Example 2.) Remember that in probability, the set of all possible outcomes of an experiment is called the sample space. Any set of outcomes is called an event. If every outcome in the sample space is equally likely, the theoretical probability of an event is P = _____________________________ number of outcomes
in the event number of outcomes in the sample space. Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure. Three models for geometric probability are shown below. Model Length Angle Measure Area Geometric Probability Example Sample space All points on ̶̶̶ AD All points in the circle All points in the rectangle ̶̶ BC All points in the All points in the triangle All points on shaded region P = BC _ AD P = measure of angle __ 360° P = area of triangle __ area of rectangle Event Probability E X A M P L E 1 Using Length to Find Geometric Probability A point is chosen randomly on ̶̶ AD. Find the probability of each event. If an event has a probability p of occurring, the probability of the event not occurring is 1 - p. A The point is on ̶̶ AC. P = AC _ AD = 7 _ 12 B The point is not on ̶̶ AB. First find the probability that the point is on P ( ̶̶ AB ) = AB _ AD = 4 _ 12 = 1 _ 3 ̶̶ AB. Subtract from 1 to find the probability that the point is not on P (not on ̶̶ = 2 _ AB ) = 1 - 1 _ 3 3 ̶̶ AB. 630 630 Chapter 9 Extending Perimeter, Circumference, and Area �� A point is chosen randomly on Find the probability of each event. ̶̶ AD. C The point is on P ( ̶̶ AB or ̶̶ CD ) = P ( ̶̶ ̶̶ CD. AB or ̶̶ AB ) + P ( ̶̶ CD ) = 4 _ 12 + 5 _ 12 = 9 _ 12 = 3 _ 4 1. Use the figure above to find the probability that the point is ̶̶ BD. on E X A M P L E 2 Transportation Application A stoplight has the following cycle: green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. A What is the probability that the light will be yellow when you arrive? To find the probability, draw a segment to represent the number of seconds that each color light is on. P = 5 _ 60 = 1 _ 12 ≈ 0.08 The light is yellow for 5
out of every 60 seconds. B If you arrive at the light 50 times, predict about how many times you will have to stop and wait more than 10 seconds. In the model, the event of stopping and waiting more than 10 seconds is represented by a segment that starts at C and ends 10 units from D. The probability of stopping and waiting more than 10 seconds is P = 20 __ 60 = 1 __ 3. If you arrive at the light 50 times, you will probably stop and wait more than 10 seconds about 1 __ 3 (50) ≈ 17 times. 2. Use the information above. What is the probability that the light will not be red when you arrive? E X A M P L E 3 Using Angle Measures to Find Geometric Probability Use the spinner to find the probability of each event. A the pointer landing on red = 2 _ 9 P = 80 _ 360 The angle measure in the red region is 80°. B the pointer landing on purple or blue = 135 _ 360 75 +60 _ 360 = 3 _ 8 P = The angle measure in the purple region is 75°. The angle measure in the blue region is 60°. In Example 3C, you can also find the probability of the pointer landing on yellow, and subtract from 1. C the pointer not landing on yellow P = 360 - 100 _ 360 = 13 _ 18 = 260 _ 360 The angle measure in the yellow region is 100°. Substract this angle measure from 360°. 3. Use the spinner above to find the probability of the pointer landing on red or yellow. 9- 6 Geometric Probability 631 631 �� Geometric Probability I like to write a probability as a percent to see if my answer is reasonable. The probability of the pointer landing on red is 80° ____ = 2 __ ≈ 22%. 9 360° The angle measure is close to 90°, which is 25% of the circle, so the answer is reasonable. Jeremy Denton Memorial High School E X A M P L E 4 Using Area to Find Geometric Probability Find the probability that a point chosen randomly inside the rectangle is in each given shape. Round to the nearest hundredth. A the equilateral triangle The area of the triangle is A = 1 _ aP 2 = 1 _ (6) (36 √  3 ) ≈ 187 m 2. 2 The area of the rectangle is A = bh = 45 (20
) = 900 m 2. The probability is P = 187 _ 900 ≈ 0.21. B the trapezoid The area of the trapezoid is 3 + 12) (10) = 75 m 2. 2 The area of the rectangle is A = bh = 45 (20) = 900 m 2. The probability is P = 75 _ 900 ≈ 0.08. C the circle The area of the circle is ) = 36π ≈ 113.1 m 2. The area of the rectangle is A = bh = 45 (20) = 900 m 2. The probability is P = 113.1 _ ≈ 0.13. 900 4. Use the diagram above. Find the probability that a point chosen randomly inside the rectangle is not inside the triangle, circle, or trapezoid. Round to the nearest hundredth. 632 632 Chapter 9 Extending Perimeter, Circumference, and Area �� THINK AND DISCUSS 1. Explain why the ratio used in theoretical probability cannot be used to find geometric probability. 2. A spinner is one-half red and one-third blue, and the rest is yellow. How would you find the probability of the pointer landing on yellow? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, give an example of the geometric probability model. 9-6 Exercises Exercises KEYWORD: MG7 9-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of a model used to find geometric probability. 630 A point is chosen randomly on each event. ̶̶ WZ. Find the probability of 2. The point is on 4. The point is on ̶̶ XZ. ̶̶̶ WX or ̶̶ YZ. 3. The point is not on ̶̶̶ WY. 5. The point is on ̶̶ XY.. 631 Transportation A bus comes to a station once every 10 minutes and waits at the station for 1.5 minutes. 6. Find the probability that the bus will be at the station when you arrive. 7. If you go to the station 20 times, predict about how many times you will have to wait less than 3 minutes Use the spinner to find the probability of each event. p. 631 8. the pointer landing on green 9. the pointer landing on orange or blue 10
. the pointer not landing on red 11. the pointer landing on yellow or blue. 632 Multi-Step Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth. 12. the triangle 13. the trapezoid 14. the square 15. the part of the rectangle that does not include the square, triangle, or trapezoid 9- 6 Geometric Probability 633 633 �� Independent Practice For See Exercises Example 16–19 20–22 23–26 27–30 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S21 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING A point is chosen randomly on Find the probability of each event. Round to the nearest hundredth. ̶̶̶ HM. 16. The point is on 18. The point is on ̶̶ JK. ̶̶ HJ or ̶̶ KL. 17. The point is not on 19. The point is not on ̶̶̶ LM. ̶̶ JK or ̶̶̶ LM. Communications A radio station gives a weather report every 15 minutes. Each report lasts 45 seconds. Suppose you turn on the radio at a random time. 20. Find the probability that the weather report will be on when you turn on the radio. 21. Find the probability that you will have to wait more than 5 minutes to hear the weather report. 22. If you turn on the radio at 50 random times, predict about how many times you will have to wait less than 1 minute before the start of the next weather report. Use the spinner to find the probability of each event. 23. the pointer landing on red 24. the pointer landing on yellow or blue 25. the pointer not landing on green 26. the pointer landing on red or green Multi-Step Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth, if necessary. 27. the equilateral triangle 28. the square 29. the part of the circle that does not include the square 30. the part of the rectangle that does not include the square, circle, or triangle 31. /////ERROR ANALYSIS///// In the spinner at right, the angle measure of the red region is
90°. The angle measure of the yellow region is 135°, and the angle measure of the blue region is 135°. Which value of the probability of the spinner landing on yellow is incorrect? Explain. Algebra A point is chosen randomly inside rectangle ABCD with vertices A (2, 8), B (15, 8), C (15, 1), and D (2, 1). Find the probability of each event. Round to the nearest hundredth. 32. The point lies in △KLM with vertices K (4, 3), L (5, 7), and M (9, 5). 33. The point does not lie in ⊙P with center P (2, 5) and radius 3. (Hint: draw the rectangle and circle.) 634 634 Chapter 9 Extending Perimeter, Circumference, and Area �� Algebra A point is chosen at random in the coordinate plane such that -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5. Find the probability of each event. Round to the nearest hundredth. Sports 34. The point is inside the parallelogram. 35. The point is inside the circle. 36. The point is inside the triangle or the circle. 37. The point is not inside the triangle, the parallelogram, or the circle. 38. Sports The point value of each region of an Olympic archery target is shown in the diagram. The outer diameter of each ring is 12.2 cm greater than the inner diameter. a. What is the probability of hitting the center? b. What is the probability of hitting a blue or black ring? c. What is the probability of scoring higher than five points? d. Write About It In an actual event, why might the probabilities be different from those you calculated in parts a, b, and c? Olympic archers stand 70 m from their targets. From that distance, the target appears about the size of the head of a thumbtack held at arm’s length. Source: www.olympic.org A point is chosen randomly in
each figure. Describe an event with a probability of 1 __. 2 39. 41. 40. 42. If a fly lands randomly on the tangram, what is the probability that it will land on each of the following pieces? a. the blue parallelogram b. the medium purple triangle c. the large yellow triangle d. Write About It Do the probabilities change if you arrange the tangram pieces differently? Explain. 43. Critical Thinking If a rectangle is divided into 8 congruent regions and 4 of them are shaded, what is the probability that you will randomly pick a point in the shaded area? Does it matter which four regions are shaded? Explain. 44. This problem will prepare you for the Multi-Step TAKS Prep on page 638. A carnival game board consists of balloons that are 3 inches in diameter and are attached to a rectangular board. A player who throws a dart at the board wins a prize if the dart pops a balloon. a. Find the probability of winning if there are 40 balloons on the board. b. How many balloons must be on the board for the probability of winning to be at least 0.25? 9- 6 Geometric Probability 635 635 �� 45. What is the probability that a ball thrown randomly at the backboard of the basketball goal will hit the inside rectangle? 0.14 0.21 0.26 0.27 46. Point B is between A and C. If AB = 18 inches and BC = 24 inches, what is the probability that a point chosen at random is on ̶̶ AB? 0.18 0.43 0.57 0.75 47. A skydiver jumps from an airplane and parachutes down to the 70-by-100-meter rectangular field shown. What is the probability that he will miss all three targets? 0.014 0.180 0.089 0.717 �� 48. Short Response A spinner is divided into 12 congruent regions, colored red, blue, and green. Landing on red is twice as likely as landing on blue. Landing on blue and landing on green are equally likely. a. What is the probability of landing on green? Show your work or explain in words how you got your answer. b. How many regions of the spinner are colored green? Explain your reasoning. CHALLENGE AND EXTEND 49. If you randomly choose a point
on the grid, what is the probability that it will be in a red region? 50. You are designing a target that is a square inside an 18 ft by 24 ft rectangle. What size should the square be in order for the target to have a probability of 1 __ 3? to have a probability of 3 __ 4? 51. Recreation How would you design a spinner so that 1 point is earned for landing on yellow, 3 points for landing on blue and 6 points for landing on red? Explain. SPIRAL REVIEW Simplify each expression. (Previous course) 52. (3 x 2 y) (4 x 3 y 2 ) 55. Given: A (0, 4), B (4, 6), C (4, 2), D (8, 8), and E (8, 0) 2 53. (2 m 5 ) Prove: △ABC ∼ △ADE (Lesson 7-6) 54. -8 a 4 b 6 _ 2a b 3 Find the shaded area. Round to the nearest tenth, if necessary. (Lesson 9-3) 56. 57. 636 636 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-6 Use with Lesson 9-6 Activity Use Geometric Probability to Estimate π In this lab, you will use geometric probability to estimate π. The squares in the grid below are the same width as the diameter of a penny: 0.75 in., or 19.05 mm. TEKS G.8.A Congruence and the geometry of size: find areas of... regular polygons, circles, and composite figures. 1 Toss a penny onto the grid 20 times. Let x represent the number of times the penny lands touching or covering an intersection of two grid lines. 2 Estimate π using the formula π ≈ 4 · x_. 20 Try This 1. How close is your result to π? Average the results of the entire class to get a more accurate estimate. 2. In order for a penny to touch or cover an intersection, the center of the penny can land anywhere in the shaded area. a. Find the area of the shaded region. (Hint: Each corner part is one fourth of the circle. Put the four corner parts together to form a circle with radius r.) b. Find the area of the square. c. Write
the expressions as a ratio and simplify to determine the probability of the center of the penny landing in the shaded area. 3. Explain why the formula in the activity can be used to estimate π. 9-6 Geometry Lab 637 637 � SECTION 9B Applying Geometric Formulas Step Right Up! A booster club organizes a carnival to raise money for sports uniforms. The carnival features several games that give visitors chances to win prizes. �� 1. The balloon game consists of 15 balloons attached to a vertical rectangular board with the dimensions shown. Each balloon has a diameter of 4 in. Each player throws a dart at the board and wins a prize if the dart pops a balloon. Assuming that all darts hit the board at random, what is the probability of winning a prize? �� 2. The organizers decide to make the game easier, so they double the diameter of the balloons. How does this affect the probability of winning? 3. The bean toss consists of a horizontal rectangular board that is divided into a grid. The board has coordinates (0, 0), (100, 0), (100, 60), and (0, 60). A quadrilateral on the board has coordinates A (60,0), B (100, 30), C (40, 60), and D (0, 40). Each player tosses a bean onto the board and wins a prize if the bean lands inside quadrilateral ABCD. Find the probability of winning a prize. 4. Of the three games described in Problems 1, 2, and 3, which one gives players the best chance of winning a prize? 638 638 Chapter 9 Extending Perimeter, Circumference, and Area SECTION 9B Quiz for Lessons 9-4 Through 9-6 9-4 Perimeter and Area in the Coordinate Plane Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 1. A (-2, 2), B (2, 4), C (2, -4), D (-2, -2) 2. E (-1, 5), F (3, 5), G (3, -3), H (-1, -3) Find the area of each polygon with the given vertices. 3. J (-3, 3), K (2, 2), L (-1, -3), M (-4, -1) 4
. N (-3, 1), P (3, 3), Q (5, 1), R (2, -4) 9-5 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter and area of the given figure. 5. The side length of the square is tripled. 6. The diagonals of a rhombus in which d 1 = 3 ft and d 2 = 9 ft are both multiplied by 1 __ 3. 7. The base and height of the rectangle are both doubled. 8. The base and height of a right triangle with base 15 in. and height 8 in. are multiplied by 1 __ 3. 9. A square has vertices (-1, 2), (3, 2), (3, -2), and (-1, -2). If you quadruple the area, what happens to the side length? 10. A restaurant sells pancakes in two sizes, silver dollar and regular. The silver-dollar pancakes have a 4-inch diameter and require 1 __ 8 cup of batter per pancake. The diameter of a regular pancake is 2.5 times the diameter of a silver-dollar pancake. About how much batter is required to make a regular pancake? 9-6 Geometric Probability Use the spinner to find the probability of each event. 11. the pointer landing on red 12. the pointer landing on red or yellow 13. the pointer not landing on green 14. the pointer landing on yellow or blue 15. A radio station plays 12 commercials per hour. Each commercial is 1 minute long. If you turn on the radio at a random time, find the probability that a commercial will be playing. Ready to Go On? 639 639 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary apothem............................. 601 circle................................ 600 center of a circle...................... 600 composite figure...................... 606 center
of a regular polygon............ 601 geometric probability................. 630 central angle of a regular polygon...... 601 Complete the sentences below with vocabulary words from the list above. 1. A(n)? is the length of a segment perpendicular to a side of a regular polygon. ̶̶̶̶ 2. The point that is equidistant from every point on a circle is the?. ̶̶̶̶ 3.? is based on a ratio of geometric measures. ̶̶̶̶ 9-1 Developing Formulas for Triangles and Quadrilaterals (pp. 589–597) E X A M P L E S EXERCISES TEKS G.1.B, G.3.C, G.3.E, G.5.A, G.8.C Find each measurement. ■ the perimeter of a square in which A = 36 in 2 Find each measurement. 4. the area of a square in which P = 36 in. A = s 2 = 36 in 2 S = √  36 = 6 in. P = 4s = 4 ⋅ 6 = 24 in. Use the Area Formula to find the side length. ■ the area of the triangle By the Pythagorean Theorem, 8 2 + b 2 = 17 2 64 + b 2 = 289 b 2 = 225, so b = 15 ft. bh = 1 _ A = 1 _ (15) (8) = 60 ft 2 2 2 ■ the diagonal d 2 of a rhombus in which A = 6 x 3 y 3 m and 4 x 2 y) d 2 2 d 2 = 3x y 2 Substitute the given values. Solve for d 2. 5. the perimeter of a rectangle in which b = 4 cm and A = 28 cm 2 6. the height of a triangle in which A = 6 x 3 y in 2 and b = 4xy in. 7. the height of the trapezoid, in which A = 48xy ft 2 8. the area of a rhombus in which d 1 = 21 yd and d 2 = 24 yd 9. the diagonal d 2 of the rhombus, in which A = 630 x 3 y 7 in 2 10. the area of a kite in which d 1 = 32 m and d
2 = 18 m 640 640 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-2 Developing Formulas for Circles and Regular Polygons (pp. 600–605) E X A M P L E S Find each measurement. ■ the circumference and area of ⊙B in terms of π C = 2πr = 2π (5xy) = 10xyπ m A = π r 2 = π (5xy) 2 = 25 x 2 y 2 π m 2 EXERCISES TEKS G.5.A, G.8.A, G.8.C Find each measurement. Round to the nearest tenth, if necessary. 11. the circumference of ⊙G 12. the area of ⊙J in which C = 14π yd ■ the area, to the nearest tenth, of a regular 13. the diameter of ⊙K in which A = 64 x 2 π m 2 hexagon with apothem 9 yd By the 30°-60°-90° Triangle Theorem, x = 9 √  3 ____ 3 = 3 √3. So s = 2x = 6 √  3, and P = 6 (6 √  3 ) = 36 √  3. aP = 1 _ A = 1 _ (9) (36 √  3 ) = 162 √  3 ≈ 280.6 yd 2 2 2 14. the area of a regular pentagon with side length 10 ft 15. the area of an equilateral triangle with side length 4 in. 16. the area of a regular octagon with side length 8 cm 17. the area of the square 9-3 Composite Figures (pp. 606–612) TEKS G.8.A E X A M P L E EXERCISES ■ Find the shaded area. Round to the nearest tenth, if necessary. Find the shaded area. Round to the nearest tenth, if necessary. 18. 19. The area of the triangle is A = 1 _ (18) (20) = 180 cm 2. 2 The area of the parallelogram is A = bh = 20 (10) = 200 cm 2. The area of the figure is the sum of the two areas. 180 + 200 = 380
cm 2 20. Study Guide: Review 641 641 �� 9-4 Perimeter and Area in the Coordinate Plane (pp. 616–621) TEKS G.7.A, G.7.B EXERCISES G.8.A Estimate the area of each irregular shape. 21. 22. Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 23. H (0, 3), J (3, 0), K (0, -3), L (-3, 0) 24. M (-2, 5), N (3, -2), P (-2, -2) 25. A (-2, 3), B (2, 3), C (4, -1), D (-4, -1) 26. E (-1, 3), F (3, 3), G (1, 0), H (-3, 0) Find the area of the polygon with the given vertices. 27. Q (1, 4), R (4, 3), S (2, -4), T (-3, -2) 28. V (-2, 2), W (4, 0), X (2, -3), Y (-3, 0) 29. A (1, 4), B (2, 3), C (0, -3), D (-2, -1) 30. E (-1, 2), F (2, 0), G (1, -3), H (-4, -1) E X A M P L E S ■ Estimate the area of the irregular shape. The shape has 28 approximately whole squares and 17 approximately half squares. The total area is approximately 28 + 1 _ (17) = 36.5 units 2. 2 ■ Draw and classify the polygons with vertices R (2, 4), S (3, 1), T (2, -2), and U (1, 1). Find the perimeter and area of the polygon. RSTU appears to be a rhombus. Verify this by showing that the four sides are congruent. By the Distance Formula, UR = RS = ST = TU = √  10 units.
The perimeter is 4 √  10 units. US ⋅ RT = 1 _ d 1 d 2 = 1 _ The area is A = 1 _ (2 ⋅ 6) 2 2 2 = 6 units 2. ■ Find the area of the polygon with vertices A (-3, 4), B (2, 3), C (0, -2), and D (-5, -1). area of rectangle: 7 (6) = 42 units 2 area of triangles: a: A = 1 _ (2) (5) 2 = 5 units 2 b: A = 1 _ (5) (1) 2 = 2.5 units 2 c: A = 1 _ (2) (5) = 5 units 2 2 d: A = 1 _ (5) (1) = 2.5 units 2 2 area of polygon: A = 42 - 5 - 2.5 - 5 - 2.5 = 27 units 2 642 642 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-5 Effects of Changing Dimensions Proportionally (pp. 622–627) E X A M P L E EXERCISES TEKS G.5.A, G.5.B, G.11.D ■ The base and height of a rectangle with base 10 cm and height 15 cm are both doubled. Describe the effect on the area and perimeter of the figure. original: P = 2b + 2h = 2 (10) + 2 (15) = 50 cm A = bh = 10 (15) = 150 cm 2 doubled: P = 2b + 2h = 2 (20) + 2 (30) = 100 cm A = bh = 20 (30) = 600 cm 2 The perimeter increases by a factor of 2. The area increases by a factor of 4. Describe the effect of each change on the perimeter or circumference and area of the given figure. 31. The base and height of the triangle with vertices X (-1, 3), Y (-3, -2), and Z (2, -2) are tripled. 32. The side length of the square with vertices P (-1, 1), Q (3, 1), R (3, -3), and S (-1, -3) is doubled. 33. The radius of ⊙
A with radius 11 m is multiplied by 1 _. 2 34. The base and height of a triangle with base 8 ft and height 20 ft are both multiplied by 4. 9-6 Geometric Probability (pp. 630–636) TEKS G.8.A E X A M P L E S EXERCISES A point is chosen randomly on probability of each event. ̶̶ WZ. Find the A point is chosen randomly on probability of each event. ̶̶ AD. Find the ■ The point is on P (XZ) = XZ _ WZ ■ The point is on = 5 _ 6 ̶̶ XZ. = 15 _ 18 ̶̶̶ ̶̶ WX or YZ. ̶̶̶ WX ) + P ( ̶̶ YZ ) = P ( P ( = 10 _ 18 ̶̶̶ WX or = 5 _ 9 ̶̶ YZ ) = 3 _ 18 + 7 _ 18 ■ Find the probability that a point chosen randomly inside the rectangle is inside the equilateral triangle. area of rectangle A = bh = 20 (10) = 200 ft 2 area of triangle ( 5 √  3 aP = = 43.3 _ 200 ≈ 0.22 ) (30) = 25 √  3 ≈ 43.3 ft 2 35. The point is on ̶̶ AB. 36. The point is not on 37. The point is on 38. The point is on ̶̶ CD. ̶̶ CD. ̶̶ CD. ̶̶ AB or ̶̶ BC or Find the probability that a point chosen randomly inside the 40 m by 24 m rectangle is in each shape. Round to the nearest hundredth. 39. the regular hexagon 40. the triangle 41. the circle or the triangle 42. inside the rectangle but not inside the hexagon, triangle, or circle Study Guide: Review 643 643 �� Find each measurement. 1. the height h of a triangle in which A = 12 x 2 y ft 2 and b = 3x ft 2. the base b 1 of a trapezoid in which A = 161.5 cm 2, h = 17 cm, and b 2 = 13 cm 3. the area A of a kite in which d 1 = 25 in. and d 2 = 12 in. 4. Find the circumference and area of ⊙A
with diameter 12 in. Give your answers in terms of π. 5. Find the area of a regular hexagon with a side length of 14 m. Round to the nearest tenth. Find the shaded area. Round to the nearest tenth, if necessary. 6. 7. 8. The diagram shows a plan for a pond. Use a composite figure to estimate the pond’s area. The grid has squares with side lengths of 1 yd. 9. Draw and classify the polygon with vertices A (1, 5), B (2, 3), C (-2, 1), and D (-3, 3). Find the perimeter and area of the polygon. Find the area of each polygon with the given vertices. 10. E (-3, 4), F (1, 1), G (0, -4), H (-4, 1) 11. J (3, 4), K (4, -1), L (-2, -4), M (-3, 3) Describe the effect of each change on the perimeter or circumference and area of the given figure. 12. The base and height of a triangle with base 10 cm and height 12 cm are multiplied by 3. 13. The radius of a circle with radius 12 m is multiplied by 1 _. 2 14. A circular garden plot has a diameter of 21 ft. Janelle is planning a new circular plot with an area 1 __ 9 as large. How will the circumference of the new plot compare to the circumference of the old plot? A point is chosen randomly on probability of each event. ̶̶ NS. Find the 15. The point is on ̶̶̶ NQ. 16. The point is not on ̶̶̶ NQ or 17. The point is on ̶̶ RS. ̶̶ QR. 18. A shuttle bus for a festival stops at the parking lot every 18 minutes and stays at the lot for 2 minutes. If you go to the festival at a random time, what is the probability that the shuttle bus will be at the parking lot when you arrive? 644 644 Chapter 9 Extending Perimeter, Circumference, and Area �� FOCUS ON SAT STUDENT-PRODUCED RESPONSES There are two types of questions in the mathematics sections of the SAT: multiple-choice questions, where you select the correct answer from five choices, and
student-produced response questions, for which you enter the correct answer in a special grid. On the SAT, the student-produced response items do not have a penalty for incorrect answers. If you are uncertain of your answer and do not have time to rework the problem, you should still grid in the answer you have. You may want to time yourself as you take this practice test. It should take you about 9 minutes to complete. 1. A triangle has two angles with a measure of 60° and one side with a length of 12. What is the perimeter of the triangle? 4. Three overlapping squares and the coordinates of a corner of each square are shown above. What is the y-intercept of line ℓ? 2. The figure above is composed of four congruent trapezoids arranged around a shaded square. What is the area of the shaded square? 3. If △PQR ∼ △STU, m∠P = 22°, m∠Q = 57°, and m∠U = x°, what is the value of x? 5. In the figure above, what is the value of y? 6. The three angles of a triangle have measures 12x°, 3x°, and 7y°, where 7y > 60. If x and y are integers, what is the value of x? College Entrance Exam Practice 645 645 �� Any Question Type: Use a Formula Sheet When you take a standardized mathematics test, you may be given a formula sheet or a mathematics chart that accompanies the test. Although many common formulas are given on these sheets, you still need to know when the formulas are applicable, and what the variables in the formulas represent. Mathematics Chart Perimeter rectangle P = 2ℓ + 2w or P = 2 (ℓ + w) Circumference circle C = 2πr or C = πd Area rectangle A = ℓw or A = bh triangle bh or A = bh _ A = 1 _ 2 2 trapezoid or circle A = π r 2 Multiple Choice In the figure, a rectangle is inscribed in a circle. Which best represents the shaded area to the nearest tenth of a square meter? 3.4 m 2 7.6 m 2 12.6 m 2 17.2 m 2 Which formula(s) do I need? area
of a circle, area of a rectangle What do I substitute for each variable in the formulas? To use the formula for the area of a circle, I need to know the radius. The diameter of the circle is 5 m, so the radius is 2.5 m. I should substitute 2.5 for r and 3.14 for π. To use the formula for the area of a rectangle, I need to know its base and height. The base b is 4 m. To find the height, I can use the Pythagorean Theorem. 4 2 + h 2 = 5 2 16 + h 2 = 25 h 2 = 9 h = 3 What are the areas of the shapes? circle: A = π r 2 = π (2.5) 2 = 6.25π m 2 rectangle: A = bh = 4 (3) = 12 m 2 What do I do with the areas to find the answer? shaded area = area of circle - area of rectangle = 6.25π - 12 ≈ 7.6 m 2 Choice B is the correct answer. 646 646 Chapter 9 Extending Perimeter, Circumference, and Area �� Read each test problem and answer the questions that follow. Use the formula sheet below, if applicable. �� Before you begin a test, quickly review the formulas included on your formula sheet. Perimeter rectangle P = 2ℓ + 2w or P = 2 (ℓ + w) Circumference circle Area rectangle triangle trapezoid C = 2πr or C = πd A = ℓw or A = bh bh or A = bh or circle A = π r 2 Pi π π ≈ 3.14 or π ≈ 22 _ 7 Item A The circumference of a circle is 48π meters. What is the radius in meters? 6.9 meters 24 meters 12 meters 36 meters 1. Which formula would you use to solve this problem? 2. After substituting the variables in the formula, what would you need to do to find the correct answer? Item B Gridded Response The area of a trapezoid is 171 square meters. The height is 9 meters, and one base length is 23 meters. What is the other base length of the trapezoid in meters? Item C Gridded Response The area of the rectangle is 48 square miles. What is the perimeter in
miles? 5. What formula(s) would you use to solve this problem? 6. What would you substitute for each variable in the formula? 7. After substituting the variables in the formula, what would you need to do to find the correct answer? Item D Gridded Response A point is chosen randomly inside the rectangle. What is the probability that the point does not lie inside the triangle or the trapezoid? Round to the nearest hundredth. 8. Which formulas would you use to solve this problem? 9. What would you substitute for each variable 3. What formula(s) would you use to solve this in the formula(s)? problem? 4. What would you substitute for each variable in the formula? 10. After substituting the variables in the formula, what would you need to do to find the correct answer? TAKS Tackler 647 647 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–9 Multiple Choice 1. The floor of a tent is a regular hexagon. If the side length of the tent floor is 5 feet, what is the area of the floor? Round to the nearest tenth. 5. If ABCD is a rhombus in which m∠1 = (x + 15) ° and m∠2 = (2x + 12) °, what is the value of x? 32.5 square feet 65.0 square feet 75.0 square feet 129.9 square feet 2. If J is on the perpendicular bisector of ̶̶ KL, what is 3 9 18 21 the length of KL? 6. What is the area of the shaded portion of the rectangle? 34 square centimeters 36 square centimeters 38 square centimeters 50 square centimeters 7. If △XYZ is isosceles and m∠Y > 100°, which of the following must be true? m∠X < 40° m∠X > 40° ̶̶ XZ ≅ ̶̶ XY ≅ ̶̶ YZ ̶̶ XZ 8. The Eiffel Tower in Paris, France, is 300 meters tall. The first level of the tower has a height of 57 meters. A scale model of the Eiffel Tower in Shenzhen, China, is 108 meters tall. What is the height of the first level of the model? Round to the nearest tenth. 15.8 meters 20
.5 meters 56.8 meters 61.6 meters 12 18 24 36 3. What is the length of ̶̶ VY? 1.6 2 2.5 4 4. A sailor on a ship sights the light of a lighthouse at an angle of elevation of 15°. If the light in the lighthouse is 189 feet higher than the sailor’s line of sight, what is the horizontal distance between the ship and the lighthouse? Round to the nearest foot. 49 feet 51 feet 705 feet 730 feet 648 648 Chapter 9 Extending Perimeter, Circumference, and Area �� There is often more than one way to find a missing side length or angle measure in a figure. For example, you might be able to find a side length of a right triangle by using either the Pythagorean Theorem or a trigonometric ratio. Check your answer by using a different method than the one you originally used. 9. The lengths of both bases of a trapezoid are tripled. What is the effect of the change on the area of the trapezoid? The area remains the same. The area is tripled. The area increases by a factor of 6. The area increases by a factor of 9. 10. If ∠1 and ∠2 form a linear pair, which of the following must also be true about these angles? They are adjacent. They are complementary. They are congruent. They are vertical. 11. In △ABC, AB = 8, BC = 17, and AC = 2x + 1. Which of the following is a possible value of x? 3 4 9 12 12. Which line is parallel to the line with the equation y = -3x + 4? y - 3x = 8 4y - 12x = 1 3y - x = 3 2y + 6x = 5 Gridded Response 13. What is the radius of a circle in inches if the ratio of its area to its circumference is 2.5 square inches : 1 inch? 14. △JLM ∼ △RST. If JL = 5, LM = 4, RS = 3x - 1, and ST = x + 2, what is the value of x? 15. If the two diagonals of a kite measure 16 centimeters and 10 centimeters, what is the area of the kite in square centimeters? STANDARDIZED TEST PREP Short
Response 16. Two gas stations on a straight highway are 8 miles apart. If a car runs out of gas at a random point between the two gas stations, what is the probability that the car will be at least 2 miles from either gas station? Draw a diagram or write and explanation to show how you determined your answer. 17. Use the figure below to find each measure. Show your work or explain in words how you found your answers. Round the angle measure to the nearest degree. a. m∠A b. AC 18. Given that ̶̶ DE, ̶̶ DF, and ̶̶ EF are midsegments of △ABC, determine m∠C to the nearest degree. Show your work or explain in words how you determined your answer. Extended Response 19. Quadrilateral LMNP has vertices at L (1, 4), M (4, 4), N (1, 0) and P (-2, 0). a. Write a coordinate proof showing that LMNP is a parallelogram. b. Draw a rectangle with the same area as figure LMNP. Explain how you know that the figures have the same area. c. Does the rectangle you drew have the same perimeter as figure LMNP? Explain. Cumulative Assessment, Chapters 1–9 649 649 �� Spatial Reasoning 10A Three-Dimensional Figures 10-1 Solid Geometry 10-2 Representations of Three-Dimensional Figures Lab Use Nets to Create Polyhedrons 10-3 Formulas in Three Dimensions 10B Surface Area and Volume 10-4 Surface Area of Prisms and Cylinders Lab Model Right and Oblique Cylinders 10-5 Surface Area of Pyramids and Cones 10-6 Volume of Prisms and Cylinders 10-7 Volume of Pyramids and Cones 10-8 Spheres Lab Compare Surface Areas and Volumes Ext Spherical Geometry KEYWORD: MG7 ChProj Four Chromatic Gates by Herbert Bayer stands at Ervay and Federal Streets in downtown Dallas. 650 650 Chapter 10 Vocabulary Match each term on the left with a definition on the right. 1. equilateral A. the distance from the center of a regular polygon to a side of 2. parallelogram 3. apothem 4. composite figure the polygon B. a quadrilateral with four right angles C. a quadrilateral with two pairs of parallel sides D. having all sides con
gruent E. a figure made up of simple shapes, such as triangles, rectangles, trapezoids, and circles Find Area in the Coordinate Plane Find the area of each figure with the given vertices. 5. △ABC with A (0, 3), B (5, 3), and C (2, -1) 6. rectangle KLMN with K (-2, 3), L (-2, 7), M (6, 7), and N (6, 3) 7. ⊙P with center P (2, 3) that passes through the point Q (-6, 3) Circumference and Area of Circles Find the circumference and area of each circle. Give your answers in terms of π. 8. 10. 9. Distance and Midpoint Formulas Find the length and midpoint of the segment with the given endpoints. 11. A (-3, 2) and B (5, 6) 13. E (0, 1) and F (-3, 4) 12. C (-4, -4) and D (2, -3) 14. G (2, -5) and H (-2, -2) Evaluate Expressions Evaluate each expression for the given values of the variables. 15. √A_ 16. 2A_ P π for A = 121π cm 2 for A = 128 ft 2 and P = 32 ft 17. √ c 2 - a 2 for a = 8 m and c = 17 m 18. 2A_ h - b 1 for A = 60 in 2, b 1 = 8 in., and h = 6 in. Spatial Reasoning 651 651 �� Key Vocabulary/Vocabulario cone cylinder net polyhedron prism pyramid sphere cono cilindro plantilla poliedro prisma pirámide esfera surface area área total volume volumen Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word polyhedron begins with the root poly-. List some other words that begin with poly-. What do all of these words have in common? 2. The word cone comes from the root ko-, which means “to sharpen.” Think of sharp
ening a pencil. How do you think this relates to a cone? 3. What does the word surface mean? What do you think the surface area of a three-dimensional figure is? Geometry TEKS Les. 10-1 Les. 10-2 10-3 Geo. Lab Les. 10-3 Les. 10-4 10-4 Geo. Lab Les. 10-5 Les. 10-6 Les. 10-7 Les. 10-8 G.1.C Geometric structure* compare and contrast … Euclidean and non-Euclidean geometries G.6.A Dimensionality and the geometry of location* describe and draw the intersection of a given plane with various three-dimensional … figures ★ ★ ★ G.6.B Dimensionality and the geometry of location* ★ ★ ★ ★ ★ 10-8 Tech. Lab Ext. ★ use nets to represent and construct threedimensional geometric figures G.6.C Dimensionality and the geometry of location* use orthographic and isometric views of threedimensional geometric figures … ★ G.7.C Dimensionality and the geometry of location* ★ … use formulas involving length, slope, and midpoint G.8.D Congruence and the geometry of size* find ★ ★ ★ ★ ★ surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites … G.9.D Congruence and the geometry of size* ★ ★ ★ ★ ★ analyze the characteristics of polyhedra … 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 … * Knowledge and skills are written out completely on pages TX28–TX35. 652 652 Chapter 10 Writing Strategy: Draw Three-Dimensional Figures When you encounter a three-dimensional figure such as a cylinder, cone, sphere, prism, or pyramid, it may help you to make a quick sketch so that you can visualize its shape. Use these tips to help you draw quick sketches of three-dimensional figures. ��
�� Try This 1. Explain and show how to draw a cube, a prism with equal length, width, and height. �� 2. Draw a prism, starting with two hexagons. (Hint: Draw the �� hexagons as if you were viewing them at an angle.) 3. Draw a pyramid, starting with a triangle and a point above the triangle. Spatial Reasoning 653 653 10-1 Solid Geometry TEKS G.6.A Dimensionality and the geometry of location: describe and draw the intersection of a given plane with various three-dimensional geometric figures. Objectives Classify three-dimensional figures according to their properties. Use nets and cross sections to analyze threedimensional figures. Vocabulary face edge vertex prism cylinder pyramid cone cube net cross section Why learn this? Some farmers in Japan grow cube-shaped watermelons to save space in small refrigerators. Each fruit costs about the equivalent of U.S. $80. (See Example 4.) Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces. Three-Dimensional Figures TERM EXAMPLE A prism is formed by two parallel congruent polygonal faces called bases connected by faces that are parallelograms. A cylinder is formed by two parallel congruent circular bases and a curved surface that connects the bases. Also G.2.B, G.6.B, G.9.D A pyramid is formed by a polygonal base and triangular faces that meet at a common vertex. A cone is formed by a circular base and a curved surface that connects the base to a vertex. A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of
their bases. Triangular prism Rectangular prism Pentagonal prism Hexagonal prism Triangular pyramid Rectangular pyramid Pentagonal pyramid Hexagonal pyramid 654 654 Chapter 10 Spatial Reasoning �� E X A M P L E 1 Classifying Three-Dimensional Figures Classify each figure. Name the vertices, edges, and bases. A � B � � � � rectangular pyramid vertices: A, B, C, D, E edges: ̶̶ AB, ̶̶ BE, ̶̶ BC, ̶̶ CE, ̶̶ CD, ̶̶ DE ̶̶ AD, ̶̶ AE, � � cylinder vertices: none edges: none base: rectangle ABCD bases: ⊙P and ⊙Q Classify each figure. Name the vertices, edges, and bases. 1a. � 1b. � � � � � � A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face. E X A M P L E 2 Identifying a Three-Dimensional Figure From a Net Describe the three-dimensional figure that can be made from the given net. A B The net has two congruent triangular faces. The remaining faces are parallelograms, so the net forms a triangular prism. The net has one square face. The remaining faces are triangles, so the net forms a square pyramid. Describe the three-dimensional figure that can be made from the given net. 2a. 2b. 10- 1 Solid Geometry 655 655 A cross section is the intersection of a three-dimensional figure and a plane. E X A M P L E 3 Describing Cross Sections of Three-Dimensional Figures Describe each cross section. A B The cross section is a triangle. The cross section is a circle. Describe each cross section. 3a. 3b. E X A M P L E 4 Food Application A chef is slicing a cube-shaped watermelon for a buffet. How can the chef cut the watermelon to make a slice of each shape? A a square B a hexagon Cut parallel to the bases. Cut through the midpoints of the edges. 4. How can a chef cut a cube-shaped watermelon to make slices that are triangles? THINK AND DIS
CUSS 1. Compare prisms and cylinders. 2. GET ORGANIZED Copy and complete the graphic organizer. 656 656 Chapter 10 Spatial Reasoning �� 10-1 Exercises Exercises KEYWORD: MG7 10-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A? has two circular bases. (prism, cylinder, or cone) ̶̶̶̶ Classify each figure. Name the vertices, edges, and bases. p. 655 2. � 3. � � � 4. 4 Describe the three-dimensional figure that can be made from the given net. p. 655 5 Describe each cross section. p. 656 8. 6. 9. 7. 10. 656 Art A sculptor has a cylindrical piece of clay. How can the sculptor slice the clay to make a slice of each given shape? 11. a circle 12. a rectangle Independent Practice For See Exercises Example 13–15 16–18 19–21 22–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Classify each figure. Name the vertices, edges, and bases. 13. 14. 15. Describe the three-dimensional figure that can be made from the given net. 16. 17. 18. 10- 1 Solid Geometry 657 657 �� Describe each cross section. 19. 20. 21. Architecture An architect is drawing plans for a building that is a hexagonal prism. How could the architect draw a cutaway of the building that shows a cross section in the shape of each given figure? 22. a hexagon 23. a rectangle Name a three-dimensional figure from which a cross section in the given shape can be made. 24. square 25. rectangle 26. circle 27. hexagon Write a verbal description of each figure. 28. 29. 30. Draw and label a figure that meets each description. 31. rectangular prism with length 3 cm, width 2 cm, and height 5 cm 32. regular pentagonal prism with side length 6 in. and height 8 in. 33. cylinder with radius 4 m and height 7 m Draw a net for each three-dimensional figure. 34. 35. 36. 37. This problem will prepare you for
the Multi-Step TAKS Prep on page 678. A manufacturer of camping gear makes a wall tent in the shape shown in the diagram. a. Classify the three-dimensional figure that the wall tent forms. b. What shapes make up the faces of the tent? How many of each shape are there? c. Draw a net for the wall tent. 658 658 Chapter 10 Spatial Reasoning �� 38. /////ERROR ANALYSIS///// A regular hexagonal prism is intersected by a plane as shown. Which cross section is incorrect? Explain. 39. Critical Thinking A three-dimensional figure has 5 faces. One face is adjacent to every other face. Four of the faces are congruent. Draw a figure that meets these conditions. 40. Write About It Which of the following figures is not a net for a cube? Explain. a. b. c. d. 41. Which three-dimensional figure does the net represent? 42. Which shape CANNOT be a face of a hexagonal prism? triangle hexagon parallelogram rectangle 43. What shape is the cross section formed by a cone and a plane that is perpendicular to the base and that passes through the vertex of the cone? circle triangle trapezoid rectangle 44. Which shape best represents a hexagonal prism viewed from the top? 10- 1 Solid Geometry 659 659 �� CHALLENGE AND EXTEND A double cone is formed by two cones that share the same vertex. Sketch each cross section formed by a double cone and a plane. 45. 46. 47. Crafts Elena is designing patterns for gift boxes. Draw a pattern that she can use to create each box. Be sure to include tabs for gluing the sides together. 48. a box that is a square pyramid where each triangular face is an isosceles triangle with a height equal to three times the width 49. a box that is a cylinder with the diameter equal to the height 50. a box that is a rectangular prism with a base that is twice as long as it is wide, and with a rectangular pyramid on the top base 51. A net of a prism is shown. The bases of the prism are regular hexagons, and the rectangular faces are all congruent. a. List all pairs of parallel faces in the prism. b. Draw a net of a prism with bases that are regular pentagons. How many pairs of parallel faces does the prism
have? � � � � � � � � SPIRAL REVIEW Write the equation that fits the description. (Previous course) 52. the equation of the graph that is the reflection of the graph of y = x 2 over the x-axis 53. the equation of the graph of y = x 2 after a vertical translation of 6 units upward 54. the quadratic equation of a graph that opens upward and is wider than y = x 2 Name the largest and smallest angles of each triangle. (Lesson 5-5) 55. � 56. � � � � � � �� 57. � �� Determine whether the two polygons are similar. If so, give the similarity ratio. (Lesson 7-2) 58. � � � � � �� 59. �� 660 660 Chapter 10 Spatial Reasoning 10-2 Representations of Three-Dimensional Figures TEKS G.9.D Congruence and the geometry of size: analyze the characteristics of polyhedra and other three-dimensional figures.... Also G.6.C Objectives Draw representations of three-dimensional figures. Recognize a threedimensional figure from a given representation. Vocabulary orthographic drawing isometric drawing perspective drawing vanishing point horizon Who uses this? Architects make many different kinds of drawings to represent three-dimensional figures. (See Exercise 34.) There are many ways to represent a threedimensional object. An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side. First National Bank, San Angelo E X A M P L E 1 Drawing Orthographic Views of an Object Draw all six orthographic views of the given object. Assume there are no hidden cubes. Top: Bottom: Front: Back: Left: Right: 1. Draw all six orthographic views of the given object. Assume there are no hidden cubes. 10- 2 Representations of Three-Dimensional Figures 661 661 �� Isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern. E X A M P L E 2 Drawing an Isometric View of an Object Draw an isometric view of the given object. Assume there are no
hidden cubes. 2. Draw an isometric view of the given object. Assume there are no hidden cubes. In a perspective drawing, nonvertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing points are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing point. A two-point perspective drawing contains two vanishing points. Perspective Drawing When making a perspective drawing, it helps me to remember that all vertical lines on the object will be vertical in the drawing. Jacob Martin MacArthur High School Three-dimensional figure One-point perspective Two-point perspective 662 662 Chapter 10 Spatial Reasoning �� E X A M P L E 3 Drawing an Object in Perspective A Draw a cube in one-point perspective. Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. Then draw a square below the horizon. This is the front of the cube. From each corner of the square, lightly draw dashed segments to the vanishing point. Lightly draw a smaller square with vertices on the dashed segments. This is the back of the cube. Draw the edges of the cube, using dashed segments for hidden edges. Erase any segments that are not part of the cube. B Draw a rectangular prism in two-point perspective. In a one-point perspective drawing of a cube, you are looking at a face. In a two-point perspective drawing, you are looking at a corner. Draw a horizontal line to represent the horizon. Mark two vanishing points on the horizon. Then draw a vertical segment below the horizon and between the vanishing points. This is the front edge of the prism. From each endpoint of the segment, lightly draw dashed segments to each vanishing point. Draw two vertical segments connecting the dashed lines. These are other vertical edges of the prism. Lightly draw dashed segments from each endpoint of the two vertical segments to the vanishing points. Draw the edges of the prism, using dashed lines for hidden edges. Erase any lines that are not part of the prism. 3a. Draw the block letter L in one-point perspective. 3b. Draw the block letter L in two-point perspective. 10- 2 Representations of Three-Dimensional Figures 663 663 E X A M P L E 4 Relating Different Representations of an Object Relating Different Representations of an Object Determine whether each drawing represents the
given object. Assume there are no hidden cubes. B D A C Yes; the drawing is a one-point perspective view of the object. No; the cubes that share a face in the object do not share a face in the drawing. No; the figure in the drawing is made up of four cubes, and the object is made up of only three cubes. Yes; the drawing shows the six orthographic views of the object. 4. Determine whether the drawing represents the given object. Assume there are no hidden cubes. THINK AND DISCUSS 1. Describe the six orthographic views of a cube. 2. In a perspective drawing, are all parallel lines drawn so that they meet at a vanishing point? Why or why not? 3. GET ORGANIZED Copy and complete the graphic organizer. 664 664 Chapter 10 Spatial Reasoning �� 10-2 Exercises Exercises KEYWORD: MG7 10-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a(n)? drawing, the vanishing points are located on the ̶̶̶̶ horizon. (orthographic, isometric, or perspective Draw all six orthographic views of each object. Assume there are no hidden cubes. p. 661 2. 3. 662 Draw an isometric view of each object. Assume there are no hidden cubes. 5. 6. 4. 7. 663 Draw each object in one-point and two-point perspectives. Assume there are no hidden cubes. 8. rectangular prism 9. block letter. 664 Determine whether each drawing represents the given object. Assume there are no hidden cubes. 11. 10. 12. 13. PRACTICE AND PROBLEM SOLVING Draw all six orthographic views of each object. Assume there are no hidden cubes. 14. 15. 16. 10- 2 Representations of Three-Dimensional Figures 665 665 �� Independent Practice Draw an isometric view of each object. Assume there are no hidden cubes. For See Exercises Example 17. 18. 19. 14–16 17–19 20–21 22–25 1 2 3 4 TE
KS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 Draw each object in one-point and two-point perspective. Assume there are no hidden cubes. 20. right triangular prism 21. block letter Determine whether each drawing represents the given object. Assume there are no hidden cubes. 22. 23. 24. 25. 26. Use the top, front, side, and isometric views to build the three-dimensional figure out of unit cubes. Then draw the figure in one-point perspective. Use the top, side, and front views to draw an isometric view of each figure. 27. 28. 29. This problem will prepare you for the Multi-Step TAKS Prep on page 678. A camping gear catalog shows the three given views of a tent. a. Draw a bottom view of the tent. b. Make a sketch of the tent. c. Each edge of the three-dimensional figure from part b represents one pole of the tent. How many poles does this tent have? 666 666 Chapter 10 Spatial Reasoning �� Draw all six orthographic views of each object. 30. 31. 32. 33. Critical Thinking Describe or draw two figures that have the same left, right, front, and back orthographic views but have different top and bottom views. 34. Architecture Perspective drawings are used by architects to show what a finished room will look like. a. Is the architect’s sketch in one-point or two-point perspective? b. Write About It How would you locate the vanishing point(s) in the architect’s sketch? 35. Which three-dimensional figure has these three views? 36. Which drawing best represents the top view of the three-dimensional figure? 37. Short Response Draw a one-point perspective view and an isometric view of a triangular prism. Explain how the two drawings are different. 10- 2 Representations of Three-Dimensional Figures 667 667 �� CHALLENGE AND EXTEND Draw each figure using one-point perspective. (Hint: First lightly draw a rectangular prism. Enclose the figure in the prism.) 38. an octagonal prism 39. a cylinder 40. a cone 41. A frustum of a cone is a part of a cone with two parallel bases. Copy the diagram of the frustum of a cone. a. Draw
the entire cone. b. Draw all six orthographic views of the frustum. c. Draw a net for the frustum. 42. Art Draw a one-point or two-point perspective drawing of the inside of a room. Include at least two pieces of furniture drawn in perspective. SPIRAL REVIEW Find the two numbers. (Previous course) 43. The sum of two numbers is 30. The difference between 2 times the first number and 2 times the second number is 20. 44. The difference between the first number and the second number is 7. When the second number is added to 4 times the first number, the result is 38. 45. The second number is 5 more than the first number. Their sum is 5. For A (4, 2), B (6, 1), C (3, 0), and D (2, 0), find the slope of each line. (Lesson 3-5) 46.  AB 48.  AD 47.  AC Describe the faces of each figure. (Lesson 10-1) 49. pentagonal prism 50. cube 51. triangular pyramid Using Technology You can use geometry software to draw figures in one- and two-point perspectives. 1. a. Draw a horizontal line to represent the horizon. Create a vanishing point on the horizon. Draw a rectangle with two sides parallel to the horizon. Draw a segment from each vertex to the vanishing point. b. Draw a smaller rectangle with vertices on the segments that intersect the horizon. Hide these segments and complete the figure. c. Drag the vanishing point to different locations on the horizon. Describe what happens to the figure. 2. Describe how you would use geometry software to draw a figure in two-point perspective. 668 668 Chapter 10 Spatial Reasoning 10-3 Use with Lesson 10-3 Activity Use Nets to Create Polyhedrons A polyhedron is formed by four or more polygons that intersect only at their edges. The faces of a regular polyhedron are all congruent regular polygons, and the same number of faces intersect at each vertex. Regular polyhedrons are also called Platonic solids. There are exactly five regular polyhedrons. TEKS G.6.B Dimensionality and the geometry of location: use nets to represent and construct three-dimensional geometric figures. Also G
.9.D Use geometry software or a compass and straightedge to create a larger version of each net on heavy paper. Fold each net into a polyhedron. NAME FACES EXAMPLE NET REGULAR POLYHEDRONS Tetrahedron 4 triangles Octahedron 8 triangles Icosahedron 20 triangles Cube 6 squares Dodecahedron 12 pentagons Try This 1. Complete the table for the number of vertices V, edges E, and faces F for each of the polyhedrons you made in Activity 1. 2. Make a Conjecture What do you think is true about the relationship between the number of vertices, edges, and faces of a polyhedron? POLYHEDRON V E F V - E + F Tetrahedron Octahedron Icosahedron Cube Dodecahedron 10-3 Geometry Lab 669 669 10-3 Formulas in Three Dimensions TEKS G.7.C Dimensionality and the geometry of location: develop and use formulas involving length, slope, and midpoint. Also G.5.A, G.8.C, G.9.D Objectives Apply Euler’s formula to find the number of vertices, edges, and faces of a polyhedron. Develop and apply the distance and midpoint formulas in three dimensions. Vocabulary polyhedron space Why learn this? Divers can use a three-dimensional coordinate system to find distances between two points under water. (See Example 5.) A polyhedron is formed by four or more polygons that intersect only at their edges. Prisms and pyramids are polyhedrons, but cylinders and cones are not. Polyhedrons Not polyhedrons In the lab before this lesson, you made a conjecture about the relationship between the vertices, edges, and faces of a polyhedron. One way to state this relationship is given below. Euler’s Formula For any polyhedron with V vertices, E edges, and F faces, V - E + F = 2. E X A M P L E 1 Using Euler’s Formula Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. A B Euler is pronounced “Oiler.” V = 4, E = 6 = 10, E = 15, F = 7 Use Euler’s formula.
Simplify. 10 - 15 + 7 ≟ 2 2 = 2 Find the number of vertices, edges, and faces of the polyhedron. Use your results to verify Euler’s formula. 1a. 1b. 670 670 Chapter 10 Spatial Reasoning A diagonal of a three-dimensional figure connects two vertices of two different faces. Diagonal d of a rectangular prism is shown in the diagram. By the Pythagorean Theorem, ℓ 2 + w 2 = x 2, and x 2 + h 2 = d 2. Using substitution. Diagonal of a Right Rectangular Prism The length of a diagonal d of a right rectangular prism with length ℓ, width w, and height h is d = √  Using the Pythagorean Theorem in Three Dimensions Find the unknown dimension in each figure. A the length of the diagonal of a 3 in. by 4 in. by 5 in. rectangular prism 3 2 + 4 2 + 5 2 d = √  = √  9 + 16 + 25 = √  50 ≈ 7.1 in. Substitute 3 for ℓ, 4 for w, and 5 for h. Simplify. B the height of a rectangular prism with an 8 ft by 12 ft base and an 18 ft diagonal 18 = √  8 2 + 12 2 + h 2 18 2 = ( √  8 2 + 12 2 + h 2 ) 324 = 64 + 144 + h 2 h 2 = 116 h = √  116 ≈ 10.8 ft 2 Substitute 18 for d, 8 for ℓ, and 12 for w. Square both sides of the equation. Simplify. Solve for h 2. Take the square root of both sides. 2. Find the length of the diagonal of a cube with edge length 5 cm. Space is the set of all points in three dimensions. Three coordinates are needed to locate a point in space. A three-dimensional coordinate system has 3 perpendicular axes: the x-axis, the y-axis, and
the z-axis. An ordered triple (x, y, z) is used to locate a point. To locate the point (3, 2, 4), start at (0, 0, 0). From there move 3 units forward, 2 units right, and then 4 units up. E X A M P L E 3 Graphing Figures in Three Dimensions Graph each figure. A a cube with edge length 4 units and one vertex at (0, 0, 0) The cube has 8 vertices: (0, 0, 0), (0, 4, 0), (0, 0, 4), (4, 0, 0), (4, 4, 0), (4, 0, 4), (0, 4, 4), (4, 4, 4). 10- 3 Formulas in Three Dimensions 671 671 �� Graph each figure. B a cylinder with radius 3 units, height 5 units, and one base centered at (0, 0, 0) Graph the center of the bottom base at (0, 0, 0). Since the height is 5, graph the center of the top base at (0, 0, 5). The radius is 3, so the bottom base will cross the x-axis at (3, 0,0) and the y-axis at (0, 3, 0). Draw the top base parallel to the bottom base and connect the bases. 3. Graph a cone with radius 5 units, height 7 units, and the base centered at (0, 0, 0). You can find the distance between the two points ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) by drawing a rectangular prism with the given points as endpoints of a diagonal. Then use the formula for the length of the diagonal. You can also use a formula related to the Distance Formula. (See Lesson 1-6.) The formula for the midpoint between ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is related to the Midpoint Formula. (See Lesson 1-6.) Distance and Midpoint Formulas in Three Dimensions The distance between the points ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is d = √ �
�� (. The midpoint of the segment with endpoints ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is Finding Distances and Midpoints in Three Dimensions Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. A (0, 0, 0) and (3, 4, 12) distance: d = √  ( = √  (3 - 0) 2 + (4 - 0) 2 + (12 - 0) 2 = √  9 + 16 + 144 = √  169 = 13 units midpoint + 12 1.5, 2, 6) 672 672 Chapter 10 Spatial Reasoning �� Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. B (3, 8, 10) and (7, 12, 15) distance: d = √  midpoint ( ) 10 + 15 8 + 12 3 + 7 _ _ _ (7 - 3) 2 + (12 - 8) 2 + (15 - 10) 2,, 2 2 2 M (5, 10, 12.5) = √  = √ �
�� 16 + 16 + 25 = √  57 ≈ 7.5 units Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 4a. (0, 9, 5) and (6, 0, 12) 4b. (5, 8, 16) and (12, 16, 20) E X A M P L E 5 Recreation Application Two divers swam from a boat to the locations shown in the diagram. How far apart are the divers? Recreation Divers in the Comal Springs in New Braunfels can see a variety of plant and animal life. The springs, which are fed by the Edwards Aquifer, have a constant temperature of 68°F and are home to many endangered species. The location of the boat can be represented by the ordered triple (0, 0, 0), and the locations of the divers can be represented by the ordered triples (18, 9, -8) and (-15, -6, -12). d = √  ( = √  = √  (-15 - 18) 2 + (-6 - 9) 2 + (-12 + 8) 2 1330 ≈ 36.5 ft c10l03002a Use the Distance Formula to find the distance between the divers. 5. What if…? If both divers swam straight up to the surface, how far apart would they be? THINK AND DISCUSS 1. Explain how to find the distance between two points in a three-dimensional coordinate system. 2. GET ORGANIZED Copy and complete the graphic organizer. 10- 3 Formulas in Three Dimensions 673 673 Depth: 8 ft9 ft18 ft15 ft6 ftDepth: 12 ftge07se_c10l03002aAB�� 10-3 Exercises Exercises KEYWORD: MG7 10-3 KEYWORD:
MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why a cylinder is not a polyhedron. 670 Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 2. 3. 4 Find the unknown dimension in each figure. Round to the nearest tenth, if necessary. p. 671 5. the length of the diagonal of a 4 ft by 8 ft by 12 ft rectangular prism 6. the height of a rectangular prism with a 6 in. by 10 in. base and a 13 in. diagonal 7. the length of the diagonal of a square prism with a base edge length of 12 in. and a height of 1 in Graph each figure. p. 671 8. a cone with radius 8 units, height 4 units, and the base centered at (0, 0, 0) 9. a cylinder with radius 3 units, height 4 units, and one base centered at (0, 0, 0) 10. a cube with edge length 7 units and one vertex at (0, 0, 0. 672. 673 Independent Practice For See Exercises Example 15–17 18–20 21–23 24–26 27 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 11. (0, 0, 0) and (5, 9, 10) 12. (0, 3, 8) and (7, 0, 14) 13. (4, 6, 10) and (9, 12, 15) 14. Recreation 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? PRACTICE AND PROBLEM SOLVING Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 15. 16. 17. Find the unknown dimension in each figure. Round to the nearest tenth, if necessary. 18. the length of the diagonal of a 7 yd by 8 yd by 16 yd rectangular prism 19. the height of a rectangular prism with
a 15 m by 6 m base and a 17 m diagonal 20. the edge length of a cube with an 8 cm diagonal 674 674 Chapter 10 Spatial Reasoning Meteorology A typical cumulus cloud weighs about 1.4 billion pounds, which is more than 100,000 elephants. Source: usgs.gov Graph each figure. 21. a cylinder with radius 5 units, height 3 units, and one base centered at (0, 0, 0) 22. a cone with radius 2 units, height 4 units, and the base centered at (0, 0, 0) 23. a square prism with base edge length 5 units, height 3 units, and one vertex at (0, 0, 0) Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 24. (0, 0, 0) and (4, 4, 4) 25. (2, 3, 7) and (9, 10, 10) 26. (2, 5, 3) and (8, 8, 10) 27. Meteorology A cloud has an elevation of 6500 feet. A raindrop falling from the cloud was blown 700 feet south and 500 feet east before it hit the ground. How far did the raindrop travel from the cloud to the ground? 28. Multi-Step Find the length of a diagonal of the rectangular prism at right. If the length, width, and height are doubled, what happens to the length of the diagonal? For each three-dimensional figure, find the missing value and draw a figure with the correct number of vertices, edges, and faces. Vertices V Edges E Faces F Diagram 5 8 7 29. 30. 31. 8 12 9 5 5 7 32. Algebra Each base of a prism is a polygon with n sides. Write an expression for the number of vertices V, the number of edges E, and the number of faces F in terms of n. Use your results to show that Euler’s formula is true for all prisms. 33. Algebra The base of a pyramid is a polygon with n sides. Write an expression for the number of vertices V, the number of edges E, and the number of faces F in terms of n. Use your results to show that Euler’s formula is true for all pyramids. 34. This problem will prepare you for the Multi-Step TAK
S Prep on page 678. ̶̶̶ NM ≅ ̶̶ NP The tent at right is a triangular prism where ̶̶ KJ ≅ ̶̶ KL and has the given dimensions. and a. The tent manufacturer sets up the tent on a coordinate system so that J is at the origin and M has coordinates (7, 0, 0). Find the coordinates of the other vertices. b. The manufacturer wants to know the distance from K to P in order to make an extra support pole for the tent. Find KP to the nearest tenth. � �� 10- 3 Formulas in Three Dimensions 675 675 �� Find the missing dimension of each rectangular prism. Give your answers in simplest radical form. Length ℓ Width w Height h Diagonal d 6 in. 24 12 35. 36. 37. 38. 6 in. 18 2 6 in. 60 3 65 24 4 Graph each figure. 39. a cylinder with radius 4 units, height 5 units, and one base centered at (1, 2, 5) 40. a cone with radius 3 units, height 7 units, and the base centered at (3, 2, 6) 41. a cube with edge length 6 units and one vertex at (4, 2, 3) 42. a rectangular prism with vertices at (4, 2, 5), (4, 6, 5), (4, 6, 8), (8, 6, 5), (8, 2, 5), (8, 6, 8), (4, 2, 8), and (8, 2, 8) 43. a cone with radius 4 units, the vertex at (4, 7, 8), and the base centered at (4, 7, 1) 44. a cylinder with a radius of 5 units and bases centered at (2, 3, 7) and (2, 3, 15) Graph each segment with the given endpoints in a three-dimensional coordinate system. Find the length and midpoint of each segment. 46. (4, 3, 3) and (7, 4, 4) 45. (1, 2, 3) and (3, 2, 1) 47. (4, 7, 8) and (3, 1, 5) 48. (0, 0, 0) and (8, 3, 6) 49. (6, 1, 8) and
(2, 2, 6) 50. (2, 8, 5) and (3, 6, 3) 51. Multi-Step Find z if the distance between R (6, -1, -3) and S (3, 3, z) is 13. 52. Draw a figure with 6 vertices and 6 faces. 53. Estimation Measure the net for a rectangular prism and estimate the length of a diagonal. 54. Make a Conjecture What do you think is the longest segment joining two points on a rectangular prism? Test your conjecture using at least three segments whose endpoints are on the prism with vertices A (0, 0, 0), B (1, 0, 0), C (1, 2, 0), D (0, 2, 0), E (0, 0, 2), F (1, 0, 2), G (1, 2, 2), and H (0, 2, 2). 55. Critical Thinking The points A (3, 2, -3), B (5, 8, 6), and C (-3, -5, 3) form a triangle. Classify the triangle by sides and angles. 56. Write About It A cylinder has a radius of 4 and a height of 6. What is the length of the longest segment with both endpoints on the cylinder? Describe the location of the endpoints and explain why it is the longest possible segment. 676 676 Chapter 10 Spatial Reasoning 57. How many faces, edges, and vertices does a hexagonal pyramid have? 6 faces, 10 edges, 6 vertices 7 faces, 12 edges, 7 vertices 7 faces, 10 edges, 7 vertices 8 faces, 18 edges, 12 vertices 58. Which is closest to the length of the diagonal of the rectangular prism with length 12 m, width 8 m, and height 6 m? 6.6 m 44 m 15.6 m 244.0 m 59. What is the distance between the points (7, 14, 8) and (9, 3, 12) to the nearest tenth? 10.9 11.9 119.0 141.0 CHALLENGE AND EXTEND 60. Multi-Step The bases of the right hexagonal prism are regular hexagons with side length a, and the height of the prism is h. Find the length of the indicated diagonal in terms of a and h. 61. Determine if the points A
(-1, 2, 4), B (1, -2, 6), and C (3, -6, 8) are collinear. 62. Algebra Write a coordinate proof of the Midpoint Formula using the Distance Formula. Given: points ), and M ( Prove: A, B, and M are collinear, and AM = MB _____ _____ _____ 2, 2, 2 63. Algebra Write a coordinate proof that the diagonals of a rectangular prism are congruent and bisect each other. Given: a rectangular prism with vertices A (0, 0, 0), B (a, 0, 0), C (a, b, 0), D (0, b, 0), E (0, 0, c), F (a, 0, c), G (a, b, c), and H (0, b, c) ̶̶ AG and ̶̶ BH are congruent and bisect each other. Prove: SPIRAL REVIEW The histogram shows the number of people by age group who attended a natural history museum opening. Find the following. (Previous course) 64. the number of people between 10 and 29 years of age that were in attendance 65. the age group that had the greatest number of people in attendance Write a formula for the area of each figure after the given change. (Lesson 9-5) 66. A parallelogram with base b and height h has its height doubled. 67. A trapezoid with height h and bases b 1 and b 2 has its base b 1 multiplied by 1 __ 2. 68. A circle with radius r has its radius tripled. Use the diagram for Exercises 69–71. (Lesson 10-1) 69. Classify the figure. 70. Name the edges. 71. Name the base. 10- 3 Formulas in Three Dimensions 677 677 �� SECTION 10A Three-Dimensional Figures Your Two Tents A manufacturer of camping gear offers two types of tents: an A-frame tent and a pyramid tent. A-frame tent Pyramid tent 1. The manufacturer’s catalog shows the top, front, and side views of each tent. It shows a two-dimensional shape for each that can be folded to form the three-dimensional
shape of the tent. Draw the catalog display for each tent. The manufacturer uses a three-dimensional coordinate system to represent the vertices of each tent. Each unit of the coordinate system represents one foot. 2. Which tent offers a greater sleeping area? 3. Compare the heights of the tents. Which tent offers more headroom? 4. A camper wants to purchase the tent that has shorter support poles so that she can fit the folded tent in ̶̶ EF in her car more easily. Find the length of pole ̶̶ TR in the the A-frame tent and the length of pole pyramid tent. Which tent should the camper buy? A-Frame Tent Vertex Coordinates A B C D E F (0, 0, 0) (0, 7, 0) (0, 3.5, 7) (8, 0, 0) (8, 7, 0) (8, 3.5, 7) Pyramid Tent Vertex Coordinates P Q R S T (0, 0, 0) (8, 0, 0) (8, 8, 0) (0, 8, 0) (4, 4, 8) 678 678 Chapter 10 Spatial Reasoning Camping at Big Bend National Park �� SECTION 10A Quiz for Lessons 10-1 Through 10-3 10-1 Solid Geometry Classify each figure. Name the vertices, edges, and bases. 1. 2. 3. Describe the three-dimensional figure that can be made from the given net. 4. Describe each cross section. 7. 5. 8. 6. 9. 10-2 Representations of Three-Dimensional Figures Use the figure made of unit cubes for Problems 10 and 11. Assume there are no hidden cubes. 10. Draw all six orthographic views. 11. Draw an isometric view. 12. Draw the block letter T in one-point perspective. 13. Draw the block letter T in two-point perspective. 10-3 Formulas in Three Dimensions Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 14. a square prism 15. a hexagonal pyramid 16. a triangular pyramid 17. A bird flies from its nest to a point that is 6 feet north, 7 feet west, and 6 feet higher in the tree than the nest. How far is the bird from the nest? Find the distance between the given points. Find
the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 19. (3, 1, -2) and (5, -5, 7) 18. (0, 0, 0) and (4, 6, 12) 20. (3, 5, 9) and (7, 2, 0) Ready to Go On? 679 679 �� 10-4 Surface Area 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 surface area of a prism. Learn and apply the formula for the surface area of a cylinder. Vocabulary lateral face lateral edge right prism oblique prism altitude surface area lateral surface axis of a cylinder right cylinder oblique cylinder Also G.5.A, G.5.B, G.6.B, G.11.D Why learn this? The surface area of ice affects how fast it will melt. If the surface exposed to the air is increased, the ice will melt faster. (See Example 5.) Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face. An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude. �� Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces. The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism. Lateral Area and Surface Area of Right Prisms The lateral area of a right prism with base perimeter P and height h is L = Ph. The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B. The surface area of a cube with edge length s is S = 6 s 2. The surface area
of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh. 680 680 Chapter 10 Spatial Reasoning �� E X A M P L E 1 Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of each right prism. Round to the nearest tenth, if necessary. A the rectangular prism L = Ph = (28) 12 = 336 cm 2 S = Ph + 2B = 336 + 2 (6) (8) = 432 cm 2 P = 2 (8) + 2 (6) = 28 cm B the regular hexagonal prism The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces. L = Ph = 36 (10) = 360 m 2 S = Ph + 2B = 360 + 2 (54 √  3 ) ≈ 547.1 m 2 P = 6 (6) = 36 m The base area is B = 1 __ aP = 54 √  3 m. 2 1. Find the lateral area and surface area of a cube with edge length 8 cm. The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis. Lateral Area and Surface Area of Right Cylinders The lateral area of a right cylinder with radius r and height h is L = 2πrh. The surface area of a right cylinder with lateral area L and base area B is S = L + 2B, or S = 2πrh + 2π r 2. 10- 4 Surface Area of Prisms and Cylinders 681 681 �� E X A M P L E 2 Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. A L = 2πrh = 2π
(1) (5) = 10π m 2 S = L + 2π r 2 = 10π + 2π (1) 2 = 12π m 2 The radius is half the diameter, or 1 m. B a cylinder with a circumference of 10π cm and a height equal to 3 times the radius Step 1 Use the circumference to find the radius. C = 2πr 10π = 2πr r = 5 Circumference of a circle Substitute 10π for C. Divide both sides by 2π. Step 2 Use the radius to find the lateral area and surface area. The height is 3 times the radius, or 15 cm. L = 2π rh = 2π (5) (15) = 150π cm 2 S = 2π rh + 2π r 2 = 150π + 2π (5) 2 = 200π cm 2 Lateral area Surface area 2. Find the lateral area and surface area of a cylinder with a base area of 49π and a height that is 2 times the radius. E X A M P L E 3 Finding Surface Areas of Composite Three-Dimensional Figures Always round at the last step of the problem. Use the value of π given by the π key on your calculator. Find the surface area of the composite figure. Round to the nearest tenth. The surface area of the right rectangular prism is S = Ph + 2B = 80 (20) + 2 (24) (16) = 2368 ft 2. A right cylinder is removed from the rectangular prism. The lateral area is L = 2π rh = 2π (4) (20) = 160π ft 2. The area of each base is B = π r 2 = π (4) 2 = 16π ft 2. The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder lateral area) - (cylinder base area) = 2368 + 160π -2 (16π) = 2368 + 128π ≈ 2770.1 ft 2 3. Find the surface area of the composite figure. Round to the nearest tenth. 682 682 Chapter 10 Spatial Reasoning �� E X A M P L E 4 Exploring Effects of Changing Dimensions The length, width, and height of the right rectangular prism are doubled. Describe the effect on the surface area.
original dimensions: length, width, and height doubled: S = Ph + 2B = 16 (3) + 2 (6) (2) = 72 in 2 S = Ph + 2B = 32 (6) + 2 (12) (4) = 288 in 2 Notice that 288 = 4 (72). If the length, width, and height are doubled, the surface area is multiplied by 2 2, or 4. 4. The height and diameter of the cylinder are multiplied by 1 __ 2. Describe the effect on the surface area. E X A M P L E 5 Chemistry Application Entertainment Ice sculptures are carved from blocks of ice that weigh hundreds of pounds. A typical ice sculpture can last 6–8 hours at room temperature (70°F). If two pieces of ice have the same volume, the one with the greater surface area will melt faster because more of it is exposed to the air. One piece of ice shown is a rectangular prism, and the other is half a cylinder. Given that the volumes are approximately equal, which will melt faster? �� rectangular prism: S = Ph + 2B = 12 (3) + 2 (8) = 52 cm 2 �� half cylinder: S = πrh + π r 2 + 2rh = π (4) (1) + π (4) 2 + 8 (1) �� = 20π + 8 ≈ 70.8 cm 2 �� The half cylinder of ice will melt faster. Use the information above to answer the following. 5. A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces above. Compare the surface areas. Which will melt faster? THINK AND DISCUSS 1. Explain how to find the surface area of a cylinder if you know the �� lateral area and the radius of the base. 2. Describe the difference between an oblique prism and a right prism. 3. GET ORGANIZED Copy and complete the graphic organizer. Write the formulas in each box. 10- 4 Surface Area of Prisms and Cylinders 683 683 �� 10-4 Exercises Exercises KEYWORD: MG7 10-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How many lateral faces does a pentagonal
prism have Find the lateral area and surface area of each right prism. p. 681 2. 3. 4. a cube with edge length 9 inches. 682 Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. 5. 6. 7. a cylinder with base area 64π m 2 and a height 3 meters less than the radius. 682 Multi-Step Find the surface area of each composite figure. Round to the nearest tenth. 8. 9 Describe the effect of each change on the surface area of the given figure. p. 683 10. The dimensions are cut in half. 11. The dimensions are multiplied by 2 __. 683 12. Consumer Application The greater the lateral area of a florescent light bulb, the more light the bulb produces. One cylindrical light bulb is 16 inches long with a 1-inch radius. Another cylindrical light bulb is 23 inches long with a 3 __ 4 -inch radius. Which bulb will produce more light? 684 684 Chapter 10 Spatial Reasoning �� Independent Practice For See Exercises Example 13–15 16–18 19–20 21–22 23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the lateral area and surface area of each right prism. Round to the nearest tenth, if necessary. 13. 14. 15. a right equilateral triangular prism with base edge length 8 ft and height 14 ft Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. 16. 17. 18. a cylinder with base circumference 16π yd 2 and a height equal to 3 times the radius Multi-Step Find the surface area of each composite figure. Round to the nearest tenth. 19. 20. Describe the effect of each change on the surface area of the given figure. 21. The dimensions are tripled. 22. The dimensions are doubled. 23. Biology Plant cells are shaped approximately like a right rectangular prism. Each cell absorbs oxygen and nutrients through its surface. Which cell can be expected to absorb at a greater rate? (Hint: 1 µm = 1 micrometer = 0.000001 meter) 10- 4 Surface Area of Prisms and C
ylinders 685 685 �� 24. Find the height of a right cylinder with surface area 160π ft 2 and radius 5 ft. 25. Find the height of a right rectangular prism with surface area 286 m 2, length 10 m, and width 8 m. 26. Find the height of a right regular hexagonal prism with lateral area 1368 m 2 and base edge length 12 m. 27. Find the surface area of the right triangular prism with vertices at (0, 0, 0), (5, 0, 0), (0, 2, 0), (0, 0, 9), (5, 0, 9), and (0, 2, 9). The dimensions of various coins are given in the table. Find the surface area of each coin. Round to the nearest hundredth. Coin Diameter (mm) Thickness (mm) Surface Area ( mm 2 ) 28. Penny 29. Nickel 30. Dime 31. Quarter 19.05 21.21 17.91 24.26 1.55 1.95 1.35 1.75 32. How can the edge lengths of a rectangular prism be changed so that the surface area is multiplied by 9? 33. How can the radius and height of a cylinder be changed so that the surface area is multiplied by 1 __ 4? 34. Landscaping Ingrid is building a shelter to protect her plants from freezing. She is planning to stretch plastic sheeting over the top and the ends of a frame. Which of the frames shown will require more plastic? 35. Critical Thinking If the length of the measurements of the net are correct to the nearest tenth of a centimeter, what is the maximum error in the surface area? 36. Write About It Explain how to use the net of a three-dimensional figure to find its surface area. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice container is a square prism with base edge length 4 in. When an 8 in. straw is inserted into the container as shown, exactly 1 in. remains outside the container. a. Find AB and BC. b. What is the height AC of the container to the nearest tenth? c. Use your result from part b to find how much material is required to manufacture the container
. Round to the nearest tenth. 686 686 Chapter 10 Spatial Reasoning 10 ftge07sec10l04004aa1st pass4/23/5cmurphy10 ft10 ftge07sec10l04005a1st pass4/12/5cmurphy10 ft10 ft�� 38. Measure the dimensions of the net of a cylinder to the nearest millimeter. Which is closest to the surface area of the cylinder? 35.8 cm 2 18.8 cm 2 16.0 cm 2 13.2 cm 2 39. The base of a triangular prism is an equilateral triangle with a perimeter of 24 inches. If the height of the prism is 5 inches, find the lateral area. 120 in 2 60 in 2 40 in 2 360 in 2 40. Gridded Response Find the surface area in square inches of a cylinder with a radius of 6 inches and a height of 5 inches. Use 3.14 for π and round your answer to the nearest tenth. CHALLENGE AND EXTEND 41. A cylinder has a radius of 8 cm and a height of 3 cm. Find the height of another cylinder that has a radius of 4 cm and the same surface area as the first cylinder. 42. If one gallon of paint covers 250 square feet, how many gallons of paint will be needed to cover the shed, not including the roof? If a gallon of paint costs $25, about how much will it cost to paint the walls of the shed? 43. The lateral area of a right rectangular prism is 144 cm 2. Its length is three times its width, and its height is twice its width. Find its surface area. SPIRAL REVIEW 44. Rebecca’s car can travel 250 miles on one tank of gas. Rebecca has traveled 154 miles. Write an inequality that models m, the number of miles farther Rebecca can travel on the tank of gas. (Previous course) 45. Blood sugar is a measure of the number of milligrams of glucose in a deciliter of blood (mg/dL). Normal fasting blood sugar levels are above 70 mg/dL and below 110 mg/dL. Write an inequality that models s, the blood sugar level of a normal patient. (Previous course) Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 46. BC 47. m∠ABC Draw the top, left, and
right views of each object. Assume there are no hidden cubes. (Lesson 10-2) 48. 49. 50. 10- 4 Surface Area of Prisms and Cylinders 687 687 �� 10-4 Model Right and Oblique Cylinders In Lesson 10-4, you learned the difference between right and oblique cylinders. In this lab, you will make models of right and oblique cylinders. Use with Lesson 10-4 TEKS G.9.D Congruence and the geometry of size: analyze the characteristics of polyhedra and other three-dimensional figures and their component parts …. Activity 1 1 Use a compass to draw at least 10 circles with a radius of 3 cm each on cardboard and then cut them out. 2 Poke a hole through the center of each circle. 3 Unbend a paper clip part way and push it through the center of each circle to model a cylinder. The stack of cardboard circles can be held straight to model a right cylinder or tilted to model an oblique cylinder. Try This 1. On each cardboard model, use string or a rubber band to outline a cross section that is parallel to the base of the cylinder. What shape is each cross section? 2. Use string or a rubber band to outline a cross section of the cardboard model of the oblique cylinder that is perpendicular to the lateral surface. What shape is the cross section? Activity 2 1 Roll a piece of paper to make a right cylinder. Tape the edges. 2 Cut along the bottom and top to approximate an oblique cylinder. 3 Untape the edge and unroll the paper. What does the net for an oblique cylinder look like? Try This 3. Cut off the curved part of the net you created in Activity 2 and translate it to the opposite side to form a rectangle. How do the side lengths of the rectangle relate to the dimensions of the cylinder? Estimate the lateral area and surface area of the oblique cylinder. 688 688 Chapter 10 Spatial Reasoning 10-5 Surface Area of Pyramids and Cones 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 surface area of a pyramid. Learn and apply the formula for the surface area of a cone. Vocabulary vertex of a pyramid regular pyramid slant height of a regular pyramid
altitude of a pyramid vertex of a cone axis of a cone right cone oblique cone slant height of a right cone altitude of a cone Also G.5.A, G.5.B, G.6.B, G.11.D Why learn this? A speaker uses part of the lateral surface of a cone to produce sound. Speaker cones are usually made of paper, plastic, or metal. (See Example 5.) �� The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. �� Lateral and Surface Area of a Regular Pyramid The lateral area of a regular pyramid with perimeter P and slant height ℓ is L = 1 __ Pℓ. 2 The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = 1 __ Pℓ + B Finding Lateral Area and Surface Area of Pyramids Find the lateral area and surface area of each pyramid. A a regular square pyramid with base edge length 5 in. Pℓ and slant height 9 in20) (9) = 90 in 2 2 S = 1 _ 2 = 90 + 25 = 115 in 2 Pℓ + B Lateral area of a regular pyramid P = 4(5) = 20 in. Surface area of a regular pyramid B = 5 2 = 25 in 2 10- 5 Surface Area of Pyramids and Cones 689 689 �� Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth. B Step 1 Find the base perimeter and apothem. The base perimeter is 6 (4) = 24 m 2. The apothem is 2 √  3 m, so the base area is 1 __ 2 aP = 1
__ 2 (2 √  3 ) (24) = 24 √  3 m 2. Step 2 Find the lateral area. Pℓ L = 1 _ 2 = 1 _ (24) (7) = 84 m 2 2 Step 3 Find the surface area. S = 1 _ 2 Pℓ + B Lateral area of a regular pyramid Substitute 24 for P and 7 for ℓ. Surface area of a regular pyramid = 84 + 24 √  3 ≈ 125.6 cm 2 Substitute 24 √  3 for B. 1. Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. Lateral and Surface Area of a Right Cone The lateral area of a right cone with radius r and slant height ℓ is L = πrℓ. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = πrℓ + π r 2. 690 690 Chapter 10 Spatial Reasoning �� E X A M P L E 2 Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of each cone. Give your answers in terms of π. A a right cone with radius 2 m and slant height 3 m L = π rℓ = π (2) (3) = 6π m 2 S = π rℓ + π r 2 = 6π + π (2) 2 = 10π m 2 Lateral area of a cone Substitute 2 for r and 3 for ℓ. Surface area of a cone Substitute 2 for r and 3 for ℓ. B Step 1 Use the Pyth
agorean Theorem to find ℓ. ℓ = √  5 2 + 12 2 = 13 ft Step 2 Find the lateral area and surface area. L = πrℓ = π (5) (13) = 65π ft 2 S = πrℓ + π r 2 = 65π + π (5) 2 = 90π ft 2 Lateral area of a right cone Substitute 5 for r and 13 for ℓ. Surface area of a right cone Substitute 5 for r and 13 for ℓ. 2. Find the lateral area and surface area of the right cone. E X A M P L E 3 Exploring Effects of Changing Dimensions The radius and slant height of the right cone are tripled. Describe the effect on the surface area. original dimensions: S = π rℓ + π r 2 = π (3) (5) + π (3) 2 = 24π cm 2 radius and slant height tripled: S = π rℓ + π r 2 = π (9) (15) + π (9) 2 = 216π cm 2 Notice that 216π = 9 (24π). If the length, width, and height are tripled, the surface area is multiplied by 3 2, or 9. 3. The base edge length and slant height of the regular square pyramid are both multiplied by 2 __ 3. Describe the effect on the surface area. 10- 5 Surface Area of Pyramids and Cones 691 691 �� E X A M P L E 4 Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. The height of the cone is 90 - 45 = 45 cm. By the Pythagorean Theorem, ℓ = √  the cone is 28 2 + 45 2 = 53 cm. The lateral area of L = πrℓ = π (28) (53) = 1484π cm 2. The lateral area of the cylinder is L = 2πrh = 2π (28) (45) = 2520π cm 2. The base area is B = π r 2 = π (28) 2 = 784π cm 2. S =
(cone lateral area) + (cylinder lateral area) + (base area) = 2520π + 784π + 1484π = 4788π cm 2 4. Find the surface area of the composite figure. E X A M P L E 5 Electronics Application Electronics The paper cones of The paper cones of antique speakers were both functional and decorative. Some had elaborate patterns or shapes. Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also 3 __ 4 of the area of the large circle, so πrℓ = 3 __ 4 π r 2. πr (10) = 3 _ π (10) 2 4 Substitute 10 for ℓ, the slant height of the cone and the radius of the large circle. r = 7.5 in. Solve for r. The diameter of the cone is 2 (7.5) = 15 in. 5. What if…? If the radius of the large circle were 12 in., what would be the radius of the cone? THINK AND DISCUSS 1. Explain why the lateral area of a regular pyramid is 1 __ 2 the base perimeter times the slant height. 2. In a right cone, which is greater, the height or the slant height? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the name of the part of the cone. 692 692 Chapter 10 Spatial Reasoning �� 10-5 Exercises Exercises GUIDED PRACTICE 1. Vocabulary Describe the endpoints of an axis of a cone Find the lateral area and surface area of each regular pyramid. p. 689 2. 3. KEYWORD: MG7 10-5 KEYWORD: MG7 Parent 4. a regular triangular pyramid with base edge length 15 in. and slant height 20 in. 691 Find the lateral area and surface area of each right cone. Give your answers in terms of π. 5. 6. 7. a cone with base area 36π ft 2 and slant height 8 ft Describe the
effect of each change on the surface area of the given figure. p. 691 8. The dimensions are cut in half. 9. The dimensions are tripled Find the surface area of each composite figure. p. 692 10. 11. 692 12. Crafts Anna is making a birthday hat from a pattern that is 3 __ 4 of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain. 10- 5 Surface Area of Pyramids and Cones 693 693 �� Independent Practice For See Exercises Example 13–15 16–18 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 lateral area and surface area of each regular pyramid. 13. 14. 15. a regular hexagonal pyramid with base edge length 7 ft and slant height 15 ft Find the lateral area and surface area of each right cone. Give your answers in terms of π. 16. 17. 18. a cone with radius 8 m and height that is 1 m less than twice the radius Describe the effect of each change on the surface area of the given figure. 19. The dimensions are divided by 3. 20. The dimensions are doubled. Find the surface area of each composite figure. 21. 22. 23. It is a tradition in England to celebrate May 1st by hanging cone-shaped baskets of flowers on neighbors’ door handles. Addy is making a basket from a piece of paper that is a semicircle with diameter 12 in. What is the diameter of the basket? �� Find the surface area of each figure. Shape Base Area Slant Height Surface Area 24. 25. 26. 27. Regular square pyramid Regular triangular pyramid Right cone Right cone 36 cm 2 √  3 m 2 16π in 2 π ft 2 5 cm √  3 m 7 in. 2 ft 694 694 Chapter 10 Spatial Reasoning �� 28. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice container
is a regular square pyramid with the dimensions shown. a. Find the surface area of the container to the nearest tenth. b. The manufacturer decides to make a container in the shape of a right cone that requires the same amount of material. The base diameter must be 9 cm. Find the slant height of the container to the nearest tenth. 29. Find the radius of a right cone with slant height 21 m and surface area 168π m 2. 30. Find the slant height of a regular square pyramid with base perimeter 32 ft and surface area 256 ft 2. 31. Find the base perimeter of a regular hexagonal pyramid with slant height 10 cm and lateral area 120 cm 2. 32. Find the surface area of a right cone with a slant height of 25 units that has its base centered at (0, 0, 0) and its vertex at (0, 0, 7). Architecture Find the surface area of each composite figure. 33. 34. The Pyramid Arena seats 21,000 people. The base of the pyramid is larger than six football fields. 35. Architecture The Pyramid Arena in Memphis, Tennessee, is a square pyramid with base edge lengths of 200 yd and a height of 32 stories. Estimate the area of the glass on the sides of the pyramid. (Hint: 1 story ≈ 10 ft) 36. Critical Thinking Explain why the slant height of a regular square pyramid must be greater than half the base edge length. 37. Write About It Explain why slant height is not defined for an oblique cone. 38. Which expressions represent the surface area of the regular square pyramid? I. t 2 _ 16 II. t 2 _ 16 + tℓ _ 2 + ts _ 2 III. t _ 2 ( t _ + ℓ) 8 I only II only I and II II and III 39. A regular square pyramid has a slant height of 18 cm and a lateral area of 216 cm 2. What is the surface area? 252 cm 2 234 cm 2 225 cm 2 240 cm 2 40. What is the lateral area of the cone? 450π cm 2 1640π cm 2 360π cm 2 369π cm 2 10- 5 Surface Area of Pyramids and Cones 695 695 �� CHALLENGE AND EXTEND 41. A frustum of a cone is a part of the cone with two parallel bases. The height of the frustum of the cone is half

the pattern below. 1, 3, 7, 13, 21, … 15. In △XYZ, ∠X and ∠Z are remote interior angles of exterior ∠XYT. If m∠X = (x + 15) °, m∠Z = (50 - 3x) °, and m∠XYT = (4x - 25) °, what is the value of x? 16. In △ABC and △DEF, ∠A ≅ ∠F. If EF = 4.5, DF = 3, ̶̶ AB would let you and AC = 1.5, what length for conclude that △ABC ∼ △FED? STANDARDIZED TEST PREP Short Response 17. A building casts a shadow that is 85 ft long when the angle of elevation to the sun is 34°. a. What is the height of the building? Round to the nearest inch and show your work. b. What is the angle of elevation to the sun when the shadow is 42 ft 6 in. long? Round to the nearest tenth of a degree and show your work. 18. Use the figure to find each of the following. Round to the nearest tenth of a centimeter and show your work. a. the length of b. the length of ̶̶ DC ̶̶ AB Extended Response 19. Tony and Paul are taking a vacation with their cousin, Greg. Tony and Paul live in the same house. Paul will go directly to the vacation spot, but Tony has to pick up Greg. Tony travels 90 miles at a bearing of N 25° E to get to his cousin’s house. He then travels due east for 50 miles to get to the vacation spot. Paul travels on one highway to get from his house to the vacation spot. For each of the following, explain in words how you found your answer and round to the nearest tenth. a. Write the vectors in component form for the route from Tony and Paul’s house to their cousin’s house and the route from their cousin’s house to the vacation spot. b. What are the direction and magnitude of Paul’s direct route from his house to the vacation spot? c. Tony and Paul leave the house at the same time and arrive at the vacation spot at the same time. If Tony traveled at an average speed of 50 mi/h, what was Paul’s average speed? Cumulative Assessment, Chapters 1

–8 581 581 �� T E X A S TAKS Grades 9–11 Obj. 10 �� Reunion Tower The 55-story Reunion Tower is one of the most recognized buildings in the Dallas skyline. Built as a part of the Hyatt Regency Hotel, the tower is topped by a geodesic dome that houses a revolving restaurant. �� The tower itself consists of four concrete cylinders arranged in a �� triangular pattern. The center cylinder contains an elevator, which �� takes visitors on a 68-second ride to the top of the tower. �� Choose one or more strategies to solve each problem. 1. The building’s observation deck, the Lookout, is on the fifty-third floor, approximately 540 feet above street level. The deck is equipped with telescopes that offer close-up views of the Dallas area. Using one of the telescopes, a visitor spots a sculpture in a nearby park. The angle of depression to the sculpture is 10°. To the nearest foot, how far is the sculpture from the base of Reunion Tower? For 2–4, use the table. 2. At noon on May 15, the shadow of Reunion Tower is 150 ft long. Find the height of the tower to the nearest foot. 3. How long is the shadow of the tower at noon on October 15? Round to the nearest foot. 4. On which of the dates shown is the tower’s shadow the longest? What is the length of the shadow to the nearest foot? 582 582 Chapter 8 Right Triangles and Trigonometry Elevation of the Sun in Dallas, Texas Date January 15 February 15 March 15 April 15 May 15 June 15 July 15 August 15 September 15 October 15 November 15 December 15 Angle of Elevation at Noon (°) 36 44 54 66 75 79 77 70 60 48 39 34 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 Lyndon B. Johnson’s Birthplace Lyndon Baines Johnson (1908–1973), the thirtysixth president of the United States, was born in Stonewall, Texas, and moved to what is now Johnson City at the age of 5. Visitors to the Lyndon B. Johnson National Historical Park can tour the home where President Johnson spent much of his childhood.

Choose one or more strategies to solve each problem. 1. In 1973, the National Park Service finished restoring President Johnson’s house to its appearance during Johnson’s childhood. The blueprint below shows the layout of the house after the restoration. Suppose there is a border of wallpaper along the edge of the ceiling around the dining room. About how long is the border? 2. During the restoration of the house, the wooden floor of the parlor, dining room, and kitchen was replaced with new wood. About how many square feet of wood was used? 3. Estimate the percentage of the square footage of the home that is occupied by porches. 4. In 1934, the Johnson family spent $666.09 for improvements to their home. The total cost consisted of materials and labor. The materials cost $416.09 more than the labor. How much did they spend on labor for the improvements? �� Problem Solving on Location 583 583 Extending Perimeter, Circumference, and Area 9A Developing Geometric Formulas 9-1 Developing Formulas for Triangles and Quadrilaterals Lab Develop π 9-2 Developing Formulas for Circles and Regular Polygons 9-3 Composite Figures Lab Develop Pick’s Theorem for Area of Lattice Polygons 9B Applying Geometric Formulas 9-4 Perimeter and Area in the Coordinate Plane 9-5 Effects of Changing Dimensions Proportionally 9-6 Geometric Probability Lab Use Geometric Probability to Estimate π KEYWORD: MG7 ChProj This map of Texas counties shows the elevations of different regions. 584 584 Chapter 9 Vocabulary Match each term on the left with a definition on the right. 1. area A. a polygon that is both equilateral and equiangular 2. kite 3. perimeter 4. regular polygon B. a quadrilateral with exactly one pair of parallel sides C. the number of nonoverlapping unit squares of a given size that exactly cover the interior of a figure D. a quadrilateral with exactly two pairs of adjacent congruent sides E. the distance around a closed plane figure Convert Units Use multiplication or division to change from one unit of measure to another. 5. 12 mi = yd Length 6. 7.3 km =

m 7. 6 in. = ft 8. 15 m = mm Metric 1 kilometer = 1000 meters 1 meter = 100 centimeters 1 centimeter = 10 millimeters Customary 1 mile = 1760 yards 1 mile = 5280 feet 1 yard = 3 feet 1 foot = 12 inches Pythagorean Theorem Find x in each right triangle. Round to the nearest tenth, if necessary. 9. 11. 10. Measure with Customary and Metric Units Measure each segment to the nearest eighth of an inch and to the nearest half of a centimeter. 12. 14. 13. Solve for a Variable Solve each equation for the indicated variable. 15. A = 1_ 2 17. A = 1_ 2(b 1 + b 2)h for b 1 bh for b 16. P = 2b + 2h for h 18. A = 1_ 2 d 1d 2 for d 1 Extending Perimeter, Circumference, and Area 585 585 �� Key Vocabulary/Vocabulario apothem apotema center of a circle centro de un circulo center of a regular polygon centro de un poligono regular central angle of a regular polygon ángulo central de un poligono circle circulo composite figure figuras compuestas geometric probability probabilidad geométrica Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. How can you use the everyday meaning of the word center to understand the term center of a circle? 2. The word composite means “of separate parts.” What do you think the term composite figure means? 3. What does the word probability mean? How do you think geometric probability differs from theoretical probability? 4. The word apothem begins with the root apo-, which means “away from.” The apothem of a regular polygon is measured “away from” the center to the midpoint of a side. What do you think is true about the apothem and the side of the polygon? Geometry TEKS G.3.C Geometric structure* use logical reasoning to prove statements are true... G.3.E Geometric structure* use deductive reasoning to prove a statement Les. 9-1 ★ ★ 9-

2 Geo. Lab Les. 9-2 Les. 9-3 9-3 Geo. Lab Les. 9-4 Les. 9-5 Les. 9-6 9-6 Geo. Lab G.5.A Geometric patterns* use... geometric patterns to develop ★ ★ ★ algebraic expressions representing geometric properties G.5.B Geometric patterns* use... geometric patterns to make generalizations about geometric properties, including... ratios in similar figures... G.7.A Dimensionality and the geometry of location* use... two- dimensional coordinate systems to represent... figures G.7.B Dimensionality and the geometry of location* use slopes... to investigate geometric relationships... ★ ★ ★ ★ G.8.A Congruence and the geometry of size* find areas of... circles, ★ ★ ★ ★ ★ ★ and composite figures G.8.C Congruence and the geometry of size*... use the ★ ★ Pythagorean Theorem G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area,... when one or more dimensions of a figure are changed... ★ * Knowledge and skills are written out completely on pages TX28–TX35. 586 586 Chapter 9 Study Strategy: Memorize Formulas Throughout a geometry course, you will learn many formulas, theorems, postulates, and corollaries. You may be required to memorize some of these. In order not to become overwhelmed by the amount of information, it helps to use flash cards. In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c. Theorem 1-6-1 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2 To create a flash card, write the name of the formula or theorem on the front of the card. Then clearly write the appropriate information on the back of the card. Be sure to include a labeled diagram. Front Back �� Try This 1. Choose a lesson from this book that you

have already studied, and make flash cards of the formulas or theorems from the lesson. 2. Review your flash cards by looking at the front of each card and trying to recall the information on the back of the card. Extending Perimeter, Circumference, and Area 587 587 �� Literal Equations Algebra A literal equation contains two or more variables. Formulas you have used to find perimeter, circumference, area, and side relationships of right triangles are examples of literal equations. See Skills Bank page S59 If you want to evaluate a formula for several different values of a given variable, it is helpful to solve for the variable first. Example Danielle plans to use 50 feet of fencing to build a dog run. Use the formula P = 2ℓ + 2w to find the length ℓ when the width w is 4, 5, 6, and 10 feet. Solve the equation for ℓ. First solve the formula for the variable. P = 2ℓ + 2w Write the original equation. P - 2w = 2ℓ P - 2w _ 2 = ℓ Subtract 2w from both sides. Divide both sides by 2. Use your result to find ℓ for each value of w. ℓ = P - 2w _ = 2 ℓ = P - 2w _ = 2 50 - 2 (4) _ 2 50 - 2 (5) _ 2 = 21 ft = 20 ft ℓ = P - 2w _ = 2 50 - 2 (6) _ 2 = 19 ft ℓ = P - 2w _ = 2 50 - 2 (10) _ 2 = 15 ft Substitute 50 for P and 4 for w. Substitute 50 for P and 5 for w. Substitute 50 for P and 6 for w. Substitute 50 for P and 10 for w. Try This TAKS Grades 9–11 Obj. 1 1. A rectangle has a perimeter of 24 cm. Use the formula P = 2ℓ + 2w to find the width when the length is 2, 3, 4, 6, and 8 cm. 2. A right triangle has a hypotenuse of length c = 65 ft. Use the Pythagorean Theorem to find the length of leg a when the length of leg b is 16, 25, 33, and 39 feet. 3. The perimeter of △ABC is 112 in. Write an expression for

a in terms of b and c, and use it to complete the following table. a b 48 36 14 c 35 36 50 588 588 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-1 Developing Formulas for Triangles and Quadrilaterals TEKS G.5.A Geometric patterns: use... geometric patterns to develop algebraic expressions representing geometric properties. Also G.1.B, G.3.C, G.3.E, G.8.C Objectives Develop and apply the formulas for the areas of triangles and special quadrilaterals. Solve problems involving perimeters and areas of triangles and special quadrilaterals. Why learn this? You can use formulas for area to help solve puzzles such as the tangram. A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. The area of a figure made with all the pieces is the sum of the areas of the pieces. Postulate 9-1-1 Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate to see that a parallelogram has the same area as a rectangle with the same base and height. � � � � A triangle is cut off one side and translated to the other side. Area Parallelogram The area of a parallelogram with base b and height h is A = bh. Remember that rectangles and squares are also parallelograms. The area of a square with side s is A = s 2, and the perimeter is P = 4s. E X A M P L E 1 Finding Measurements of Parallelograms The height of a parallelogram is measured along a segment perpendicular to a line containing the base. Find each measurement. A the area of the parallelogram Step 1 Use the Pythagorean Theorem to find the height h Step 2 Use h to find the area of the parallelogram. A = bh A = 6 (4) A = 24 in 2 Substitute 6 for b and 4 for h. Area of a parallelogram Simplify. 9- 1 Developing Formulas for Triangles and Quadrilaterals 589 589 �� Find each measurement. B the height of a rectangle in

which b = 5 cm and A = (5 x 2 - 5x) cm 2 A = bh 5 x 2 - 5x = 5h 5 ( x 2 - x) = 5h x 2 - x = h Area of a rectangle Substitute 5 x 2 - 5x for A and 5 for b. Factor 5 out of the expression for A. Divide both sides by 5. h = ( x 2 - x) cm Sym. Prop. of = C the perimeter of the rectangle, in which A = 12 x ft 2 Step 1 Use the area and the height to find the base. A = bh 12x = b (6) 2x = b Area of a rectangle Substitute 12x for A and 6 for h. Divide both sides by 6. The perimeter of a rectangle with base b and height h is P = 2b + 2h, or P = 2 (b + h). Step 2 Use the base and the height to find the perimeter. P = 2b + 2h P = 2 (2x) + 2 (6) P = (4x + 12) ft. Substitute 2x for b and 6 for h. Perimeter of a rectangle Simplify. 1. Find the base of a parallelogram in which h = 56 yd and A = 28 yd 2. To understand the formula for the area of a triangle or trapezoid, notice that two congruent triangles or two congruent trapezoids fit together to form a parallelogram. Thus the area of a triangle or trapezoid is half the area of the related parallelogram. � � �� Area Triangles and Trapezoids The area of a triangle with base b and height h is A = 1 __ 2 bh. The area of a trapezoid with bases b 1 and b 2 and height h is A = 1 __ 2 ( b 1 + b 2 ) h, or __ Finding Measurements of Triangles and Trapezoids Find each measurement. A the area of a trapezoid in which b 1 = 9 cm, b 2 = 12 cm, and h = 3 cm 9 + 12) 3 2 A = 31.5 cm 2 Area of a trapezoid Substitute 9 for b 1, 12 for b 2, and 3 for h. Simplify. 590 590 Chapter 9 Extending Perimeter, Circumference, and Area �� Find each

measurement. B the base of the triangle, in which A = x 2 in 2 bx bh 2x = b b = 2x in. Area of a triangle Substitute x 2 for A and x for h. Divide both sides by x. Multiply both sides by 2. Sym. Prop. of = C Area of a trapezoid b 2 of the trapezoid, in which A = 8 ft 3 + b 2 ) (2 ft Sym. Prop. of = Substitute 8 for A, 3 for b 1, and 2 for h. Multiply 1 __ 2 Subtract 3 from both sides. by 2. 2. Find the area of the triangle. A kite or a rhombus with diagonals d 1 and d 2 can be divided into two congruent triangles with a base of d 1 and a height of 1 __ 2 d 2. area of each triangle total area �� Area Rhombuses and Kites The area of a rhombus or kite with diagonals d 1 and d 2 is A = 1 __ Finding Measurements of Rhombuses and Kites Find each measurement. A Area of a kite d 2 of a kite in which d 1 = 16 cm and A = 48 cm 48 = 1 _ (16 cm Solve for d 2. Sym. Prop. of = Substitute 48 for A and 16 for d 1. 9- 1 Developing Formulas for Triangles and Quadrilaterals 591 591 �� Find each measurement. B the area of the rhombus 6x + 4) (10x + 10) 2 A = 1 _ (60 x 2 + 100x + 40) 2 Substitute (6x + 4) for d 1 and (10x + 10) for d 2. Multiply the binomials (FOIL). The diagonals of a rhombus or kite are perpendicular, and the diagonals of a rhombus bisect each other. A = (30 x 2 + 50x + 20) in 2 Distrib. Prop. C the area of the kite Step 1 The diagonals d 1 and d 2 form four right triangles. Use the Pythagorean Theorem to find x and y. 9 2 + x 2 = 41 2 x 2 = 1600

x = 40 9 2 + y 2 = 15 2 y 2 = 144 y = 12 Step 2 Use d 1 and d 2 to find the area. d 1 is equal to x + y, which is 52. Half of d 2 is equal to 9, so d 2 is equal to 1852) (18) 2 A = 468 ft 2 Substitute 52 for d 1 and 18 for d 2. Area of a kite Simplify. 3. Find d 2 of a rhombus in which d 1 = 3x m and A = 12xy Games Application The pieces of a tangram are arranged in a square in which s = 4 cm. Use the grid to find the perimeter and area of the red square. Perimeter: Each side of the red square is the diagonal of a square of the grid. Each grid square has a side length of 1 cm, so the diagonal is √  2 cm. The perimeter of the red square is P = 4s = 4 √  2 cm. Area: Method 1 The red square is also a rhombus. The diagonals d 1 and d 2 each measure 2 cm. So its area is 2) (2) = 2 cm 2. 2 2 Method 2 The side length of the red square is √  2 cm, so the area is cm. 2 4. In the tangram above, find the perimeter and area of the large green triangle. 592 592 Chapter 9 Extending Perimeter, Circumference, and Area �� THINK AND DISCUSS 1. Explain why the area of a triangle is half the area of a parallelogram with the same base and height. 2. Compare the formula for the area of a trapezoid with the formula for the area of a rectangle. 3. GET ORGANIZED Copy and complete the graphic organizer. Name all the shapes whose area is given by each area formula and sketch an example of each shape. 9-1 Exercises Exercises GUIDED PRACTICE Find each measurement. KEYWORD: MG7 9-1 KEYWORD: MG7 Parent. 589 1. the area of the parallelogram 2. the height of the rectangle, in which A = 10 x 2 ft 2 3. the perimeter of a square in which A = 169 cm. the area of the trapezoid 5. the base of the

triangle, in which p. 590 A = 58.5 in 2 6. b 1 of a trapezoid in which A = (48x + 68) in 2, h = 8 in., and b 2 = (9x + 12) in. the area of the rhombus p. 591 8. d 2 of the kite, in which A = 187.5 m 2 9. d 2 of a kite in which A = 12 x 2 y 3 cm 2, d 1 = 3xy cm 10. Art The stained-glass window shown is a rectangle p. 592 with a base of 4 ft and a height of 3 ft. Use the grid to find the area of each piece. 9- 1 Developing Formulas for Triangles and Quadrilaterals 593 593 �� Independent Practice For See Exercises Example 11–13 14–16 17–19 20–22 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING Find each measurement. 11. the height of the parallelogram, 12. the perimeter of the rectangle in which A = 7.5 m 2 13. the area of a parallelogram in which b = (3x + 5) ft and h = (7x - 1) ft 13. 14. the area of the triangle 15. the height of the trapezoid, in which A = 280 cm 2 16. the area of a triangle in which b = (x + 1) ft and h = 8x ft 17. the area of the kite 18. d 2 of the rhombus, in which A = (3 x 2 + 6x) m 2 19. the area of a kite in which d 1 = (6x + 5) ft and d 2 = (4x + 8) ft Crafts In origami, a square base is the starting point for the creation of many figures, such as a crane. In the pattern for the square base, ABCD is a square, and E, F, G, and H are the midpoints of the sides. If AB = 6 in., find the area of each shape. 20. rectangle ABFH 21. �

�AEJ 22. trapezoid ABFJ � � � � � � � � � Multi-Step Find the area of each figure. Round to the nearest tenth, if necessary. 23. 24. 25. Write each area in terms of x. 26. equilateral triangle 27. 30°-60°-90° triangle 28. 45°-45°-90° triangle 594 594 Chapter 9 Extending Perimeter, Circumference, and Area �� 29. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A sign manufacturer makes yield signs by cutting an equilateral triangle from a square piece of aluminum with the dimensions shown. a. Find the height of the yield sign to the nearest tenth. b. Find the area of the sign to the nearest tenth. c. How much material is left after a sign is made? Find the missing measurements for each rectangle. Base b Height h Area A Perimeter P 12 17 30. 31. 32. 33. 16 11 136 216 50 66 34. The perimeter of a rectangle is 72 in. The base is 3 times the height. Find the area of the rectangle. 35. The area of a triangle is 50 cm 2. The base of the triangle is 4 times the height. Find the height of the triangle. 36. The perimeter of an isosceles trapezoid is 40 ft. The bases of the trapezoid are 11 ft and 19 ft. Find the area of the trapezoid. Use the conversion table for Exercises 37–42. 37. 1 yd 2 = 38. 1 m 2 = 39. 1 cm 2 = 40. 1 mi 2 =? ft 2 ̶̶̶̶? mm 2 ̶̶̶̶? cm 2 ̶̶̶̶? in 2 ̶̶̶̶ History 41. A triangle has a base of 3 yd and a height of 8 yd. Find the area in square feet. 42. A rhombus has diagonals 500 yd and 800 yd in length. Find the area in square miles. Conversion Factors Metric 1 km = 1000 m 1 m = 100 cm 1 cm = 10 mm Customary 1 mi = 1760 yd 1 mi = 5280 ft 1 yd = 3 ft 1 ft =

12 in. President James Garfield was a classics professor and a major general in the Union Army. He was assassinated in 1881. Source: www.whitehouse.gov 43. The following proof of the Pythagorean Theorem was discovered by President James Garfield in 1876 while he was a member of the House of Representatives. a. Write the area of the trapezoid in terms of a and b. b. Write the areas of the three triangles in terms of a, b, and c. c. Use the Area Addition Postulate to write an equation relating your results from parts a and b. Simplify the equation to prove the Pythagorean Theorem. 44. Use the diagram to prove the formula for the area of a rectangle, given the formula for the area of a square. Given: Rectangle with base b and height h Prove: The area of the rectangle is A = bh. Plan: Use the formula for the area of a square to find the areas of the outer square and the two squares inside the figure. Write and solve an equation for the area of the rectangle. 9- 1 Developing Formulas for Triangles and Quadrilaterals 595 595 �� Prove each area formula. 45. Given: Parallelogram with area A = bh Prove: The area of the triangle is A = 1 __ 2 bh. 46. Given: Triangle with area A = 1__ 2bh Prove: The area of the trapezoid is. 2 47. Measurement Choose an appropriate unit of measurement and measure the base and height of each parallelogram. a. Find the area of each parallelogram. Give your answer with the correct precision. b. Which has the greatest area? 48. Hobbies Tina is making a kite according to the plans at right. The fabric weighs about 40 grams per square meter. The diagonal braces, or spars, weigh about 20 grams per meter. Estimate the weight of the kite. 49. Home Improvement Tom is buying tile for a 12 ft by 18 ft rectangular kitchen floor. He needs to buy 15% extra in case some of the tiles break. The tiles are squares with 4 in. sides that come in cases of 100. How many cases should he buy? �� 50. Critical Thinking If the maximum error in the given measurements of the rectangle is 0.1 cm, what is the greatest possible error in the area? Explain. ��

51. Write About It A square is also a parallelogram, a rectangle, and a rhombus. Prove that the area formula for each shape gives the same result as the formula for the area of a square. 52. Which expression best represents the area of the rectangle? 2x + 2 (x - c) x (x - c) x 2 + (x - c) 2 2x (x - c) 53. The length of a rectangle is 3 times the width. The perimeter is 48 inches. Which system of equations can be used to find the dimensions of the rectangle? ℓ = w + 3 2 (ℓ + w) = 48 ℓ = 3w 2ℓ + 6w = 48 ℓ = 3w 2 (ℓ + w) = 48 ℓ = w + 3 2ℓ + 6w = 48 596 596 Chapter 9 Extending Perimeter, Circumference, and Area �� 54. A 16- by 18-foot rectangular section of a wall will be covered by square tiles that measure 2 feet on each side. If the tiles are not cut, how many of them will be needed to cover the section of the wall? 288 144 72 17 55. The area of trapezoid HJKM is 90 square centimeters. Which is closest to the length of 10 centimeters ̶̶ JK? 11.7 centimeters 10.5 centimeters 16 centimeters 56. Gridded Response A driveway is shaped like a parallelogram with a base of 28 feet and a height of 17 feet. Covering the driveway with crushed stone will cost $2.75 per square foot. How much will it cost to cover the driveway with crushed stone? �� CHALLENGE AND EXTEND Multi-Step Find h in each parallelogram. 57. �� 58. �� 59. Algebra A rectangle has a perimeter of (26x + 16) cm and an area of (42 x 2 + 51x + 15) cm 2. Find the dimensions of the rectangle in terms of x. 60. Prove that the area of any quadrilateral with perpendicular diagonals is 1 __ 2 d 1 d 2. 61. Gardening A gardener has 24 feet of fencing to enclose a rectangular garden. a. Let x and y represent the side

lengths of the rectangle. Solve the perimeter formula 2x + 2y = 24 for y, and substitute the expression into the area formula A = xy. b. Graph the resulting function on a coordinate plane. What are the domain and range of the function? c. What are the dimensions of the rectangle that will enclose the greatest area? d. Write About It How would you find the dimensions of the rectangle with the least perimeter that would enclose a rectangular area of 100 square feet? SPIRAL REVIEW Determine the range of each function for the given domain. (Previous course) 62. f (x) = x - 3, domain: -4 ≤ x ≤ 6 63. f (x) = - x 2 + 2, domain: -2 ≤ x ≤ 2 Find the perimeter and area of each figure. Express your answers in terms of x. (Lesson 1-5) 64. �� 65. � �� Write each vector in component form. (Lesson 8-6) 66.  LM with L (4, 3) and M (5, 10) 67.  ST with S (-2, -2) and T (4, 6) 9- 1 Developing Formulas for Triangles and Quadrilaterals 597 597 9-2 Develop π The ratio of the circumference of a circle to its diameter is defined as π. All circles are similar, so this ratio is the same for all circles: π = circumference __. diameter Use with Lesson 9-2 TEKS G.5.A Geometric patterns: use numeric and geometric patterns to develop algebraic expressions representing geometric properties. Activity 1 1 Use your compass to draw a large circle on a piece of cardboard and then cut it out. 2 Use a measuring tape to measure the circle’s diameter and circumference as accurately as possible. 3 Use the results from your circle to estimate π. Compare your answers with the results of the rest of the class. Try This 1. Do you think it is possible to draw a circle whose ratio of circumference to diameter is not π? Why or why not? 2. How does knowing the relationship between circumference, diameter, and π help you determine the formula for circumference? 3. Use a ribbon to make a π measuring tape. Mark off increments of π inches or π cm on your ribbon as accurately as

possible. How could you use this π measuring tape to find the diameter of a circular object? Use your π measuring tape to measure 5 circular objects. Give the circumference and diameter of each object. 598 598 Chapter 9 Extending Perimeter, Circumference, and Area Archimedes used inscribed and circumscribed polygons to estimate the value of π. His “method of exhaustion” is considered to be an early version of calculus. In the figures below, the circumference of the circle is less than the perimeter of the larger polygon and greater than the perimeter of the smaller polygon. This fact is used to estimate π. Activity 2 1 Construct a large square. Construct the perpendicular bisectors of two adjacent sides. 2 Use your compass to draw an inscribed circle as shown. 3 Connect the midpoints of the sides to form a square that is inscribed in the circle. 4 Let P 1 represent the perimeter of the smaller square, P 2 represent the perimeter of the larger square, and C represent the circumference of the circle. Measure the squares to find P 1 and P 2 and substitute the values into the inequality below. P 1 < C < P 2 5 Divide each expression in the inequality by the diameter of the circle. Why does this give you an inequality in terms of π? Complete the inequality below.? < π < ̶̶̶̶̶? ̶̶̶̶̶ Try This 4. Use the perimeters of the inscribed and circumscribed regular hexagons to write an inequality for π. Assume the diameter of each circle is 2 units. 5. Compare the inequalities you found for π. What do you think would be true about your inequality if you used regular polygons with more sides? How could you use inscribed and circumscribed regular polygons to estimate π? 6. An alternate definition of π is the area of a circle with radius 1. How could you use this definition and the figures above to estimate the value of π? 9- 2 Geometry Lab 599 599 9-2 Developing Formulas for Circles and Regular Polygons TEKS G.8.A Congruence and the geometry of size: find areas of regular polygons, circles,... Also G.5.A, G.8.C Objectives Develop and apply the formulas for the area and circumference of a circle. Develop and apply the formula for the area of a regular polygon. Vocabulary circle center of a circle center

of a regular polygon apothem central angle of a regular polygon Who uses this? Drummers use drums of different sizes to produce different notes. The pitch is related to the area of the top of the drum. (See Example 2.) A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol ⊙ and its center. ⊙A has radius r = AB and diameter d = CD. The irrational number π is defined as the ratio of the circumference C to the diameter d, or π = C__ d Solving for C gives the formula C = πd. Also d = 2r, so C = 2πr.. You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram. The base of the parallelogram is about half the circumference, or πr, and the height is close to the radius r. So A ≅ πr · r = πr 2. The more pieces you divide the circle into, the more accurate the estimate will be. Circumference and Area Circle A circle with diameter d and radius r has circumference C = πd or C = 2πr and area Finding Measurements of Circles Find each measurement. A the area of ⊙P in terms of π A = πr 2 A = π (8) 2 A = 64π cm 2 Area of a circle Divide the diameter by 2 to find the radius, 8. Simplify. 600 600 Chapter 9 Extending Perimeter, Circumference, and Area �� Find each measurement. B the radius of ⊙X in which C = 24π in. C = 2πr 24π = 2πr r = 12 in. Circumference of a circle Substitute 24π for C. Divide both sides by 2π. C the circumference of ⊙S in which A = 9x 2 π cm 2 Step 1 Use the given area to solve for r. A = π r 2 9x 2 π = π r 2 9x 2 = r 2 3x = r Area of a circle Substitute 9x 2 π for A. Divide both sides by π. Take the square root of both sides. Step 2 Use the value of r to find

the circumference. C = 2πr C = 2π (3x) Substitute 3x for r. C = 6xπ cm Simplify. 1. Find the area of ⊙A in terms of π in which C = (4x - 6) π m. E X A M P L E 2 Music Application A drum kit contains three drums with diameters of 10 in., 12 in., and 14 in. Find the area of the top of each drum. Round to the nearest tenth. The π key gives the best possible approximation for π on your calculator. Always wait until the last step to round. 10 in. diameter A = π ( 5 2 ) r = 10 _ 2 ≅ 78.5 in 2 12 in. diameter 14 in. diameter = 5 A = π ( 6 2 ) r = 12 _ 2 ≅ 113.1 in 2 = 6 A = π (7) 2 r = 14 _ 2 = 7 ≅ 153.9 in 2 2. Use the information above to find the circumference of each drum. The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. ____ n. Each central angle measure of a regular n-gon is 360° To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles. area of each triangle: 1 _ as 2 total area of the polygon: A = n ( 1 _ as), or A = 1 _ 2 2 aP The perimeter is P = ns. Area Regular Polygon The area of a regular polygon with apothem a and perimeter P is A = 1 _ 2 aP. 9- 2 Developing Formulas for Circles and Regular Polygons 601 601 �� E X A M P L E 3 Finding the Area of a Regular Polygon Find the area of each regular polygon. Round to the nearest tenth. A a regular hexagon with side length 6 m The perimeter is 6 (6) = 36 m. The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°-90

° Triangle Theorem, the apothem is 3 √  3 m3 √  3 ) (36) 2 aP Area of a regular polygon Substitute 3 √  3 for a and 36 for P. The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525. A = 54 √  3 ≅ 93.5 m 2 Simplify. B a regular pentagon with side length 8 in. Step 1 Draw the pentagon. Draw an isosceles triangle with its vertex at the center of the ____ 5 = 72°. pentagon. The central angle is 360° Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Step 2 Use the tangent ratio to find the apothem. tan 36° = 4 _ a 4 _ a = tan 36° The tangent of an angle is opp. leg _______. adj. leg Solve for a. Step 3 Use the apothem and the given side length to find the area. aP tan 36° 2 A ≅ 110.1 in 2 ) (40) Area of a regular polygon The perimeter is 8 (5) = 40 in. Simplify. Round to the nearest tenth. 3. Find the area of a regular octagon with a side length of 4 cm. THINK AND DISCUSS 1. Describe the relationship between the circumference of a circle and π. 2. Explain how you would find the central angle of a regular polygon with n sides. 3. GET ORGANIZED Copy and complete the graphic organizer. 602 602 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-2 Exercises Exercises KEYWORD: MG7 9-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Describe how to find the apothem of a square with side length s Find each measurement. p. 600 2. the circumference of ⊙C 3. the area of ⊙A in terms of π 4. the circumference of ⊙P in which A = 36π in. Food A pizza parlor offers

pizzas with diameters of 8 in., 10 in., and 12 in. p. 601 Find the area of each size pizza. Round to the nearest tenth Find the area of each regular polygon. Round to the nearest tenth. p. 602 6. 7. Independent Practice For See Exercises Example 10–12 13 14–17 1 2 3 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 8. an equilateral triangle with an apothem of 2 ft 9. a regular dodecagon with a side length of 5 m PRACTICE AND PROBLEM SOLVING Find each measurement. Give your answers in terms of π. 10. the area of ⊙M 11. the circumference of ⊙Z 12. the diameter of ⊙G in which C = 10 ft. 13. Sports A horse trainer uses circular pens that are 35 ft, 50 ft, and 66 ft in diameter. Find the area of each pen. Round to the nearest tenth. Find the area of each regular polygon. Round to the nearest tenth, if necessary. 14. 15. 16. a regular nonagon with a perimeter of 144 in. 17. a regular pentagon with an apothem of 2 ft. 9- 2 Developing Formulas for Circles and Regular Polygons 603 603 �� Biology Dendroclimatologists study tree rings for evidence of changes in weather patterns over time. Find the central angle measure of each regular polygon. (Hint: To review polygon names, see page 382.) 18. equilateral triangle 19. square 20. pentagon 21. hexagon 22. heptagon 23. octagon 24. nonagon 25. decagon Find the area of each regular polygon. Round to the nearest tenth. 26. 29. 27. 30. 28. 31. 32. Biology You can estimate a tree’s age in years by using the formula a = r __ w, where r is the tree’s radius without bark and w is the average thickness of the tree’s rings. The circumference of a white oak tree is 100 in. The bark is 0.5 in. thick, and the average width of a ring is 0.2 in. Estimate the tree’s age. 33. /////ERROR ANALYSIS///// A circle has a

circumference of 2π in. Which calculation of the area is incorrect? Explain. Find the missing measurements for each circle. Give your answers in terms of π. Diameter d Radius r Area A Circumference C 6 34. 35. 36. 37. 100 17 36 π 38. Multi-Step Janet is designing a garden around a gazebo that is a regular hexagon with side length 6 ft. The garden will be a circle that extends 10 feet from the vertices of the hexagon. What is the area of the garden? Round to the nearest square foot. 39. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A stop sign is a regular octagon. The signs are available in two sizes: 30 in. or 36 in. a. Find the area of a 30 in. sign. Round to the nearest tenth. b. Find the area of a 36 in. sign. Round to the nearest tenth. c. Find the percent increase in metal needed to make a 36 in. sign instead of a 30 in. sign. 604 604 Chapter 9 Extending Perimeter, Circumference, and Area �� 40. Measurement A trundle wheel is used to measure distances by rolling it on the ground and counting its number of turns. If the circumference of a trundle wheel is 1 meter, what is its diameter? 41. Critical Thinking Which do you think would seat more people, a 4 ft by 6 ft rectangular table or a circular table with a diameter of 6 ft? How many people would you sit at each table? Explain your reasoning. 42. Write About It The center of each circle in the figure lies on the number line. Describe the relationship between the circumference of the largest circle and the circumferences of the four smaller circles. 43. Find the perimeter of the regular octagon to the nearest centimeter. 5 40 20 68 44. Which of the following ratios comparing a circle’s circumference C to its diameter d gives the value of π? C _ d 4C _ d 2 d _ C d _ 2C 45. Alisa has a circular tabletop with a 2-foot diameter. She wants to paint a pattern on

the table top that includes a 2-foot-by-1-foot rectangle and 4 squares with sides 0.5 foot long. Which information makes this scenario impossible? There will be no room left on the tabletop after the rectangle has been painted. A 2-foot-long rectangle will not fit on the circular tabletop. Squares cannot be painted on the circle. There will not be enough room on the table to fit all the 0.5-foot squares. CHALLENGE AND EXTEND 46. Two circles have the same center. The radius of the larger circle is 5 units longer than the radius of the smaller circle. Find the difference in the circumferences of the two circles. 47. Algebra Write the formula for the area of a circle in terms of its circumference. 48. Critical Thinking Show that the formula for the area of a regular n-gon approaches the formula for the area of a circle as n gets very large. SPIRAL REVIEW Write an equation for the linear function represented by the table. (Previous course) 49. x y -2 0 -19 -13 5 2 10 17 50. x y -3 2 0 4 9 -1 -5 -10 Find each value. (Lesson 4-8) 51. m∠B 52. AB Find each measurement. (Lesson 9-1) 53. d 2 of a kite if A = 14 cm 2 and d 1 = 20 cm 54. the area of a trapezoid in which b 1 = 3 yd, b 2 = 6 yd, and h = 4 yd 9- 2 Developing Formulas for Circles and Regular Polygons 605 605 �� 9-3 Composite Figures TEKS G.8.A Congruence and the geometry of size: find areas of... composite figures. Objectives Use the Area Addition Postulate to find the areas of composite figures. Use composite figures to estimate the areas of irregular shapes. Vocabulary composite figure Who uses this? Landscape architects must compute areas of composite figures when designing gardens. (See Example 3.) A composite figure is made up of simple shapes, such as triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, find the areas of the simple shapes and then use the Area Addition Postulate. E X A M P L E 1 Finding the Areas of Composite Figures by Adding Find the shaded area. Round to the nearest tenth, if

necessary. A B Divide the figure into rectangles. Divide the figure into parts. The base of the triangle is √  10. 2 2 - 4. 8 2 = 9 ft. area of top rectangle: A = bh = 12 (15) = 180 c m 2 area of bottom rectangle: A = bh = 9 (27) = 243 c m 2 shaded area: 180 + 243 = 423 c m 2 area of triangle: A = 1 __ 2 bh = 1 __ 2 (9) (4.8) = 21.6 ft 2 area of rectangle: A = bh = 9 (3) = 27 ft 2 area of half circle: A = 1 __ 2 π r 2 = 1 __ 2 π (4. 5 2 ) = 10.125π f t 2 shaded area: 21.6 + 27 + 10.125π ≈ 80.4 ft 2 1. Find the shaded area. Round to the nearest tenth, if necessary. 606 606 Chapter 9 Extending Perimeter, Circumference, and Area �� Sometimes you need to subtract to find the area of a composite figure. E X A M P L E 2 Finding the Areas of Composite Figures by Subtracting Find the shaded area. Round to the nearest tenth, if necessary. A B Subtract the area of the triangle from the area of the rectangle. area of rectangle: A = bh = 18 (36) = 648 m 2 area of triangle: A = 1 __ 2 bh = 1 __ 2 (36) (9) = 162 m 2 area of figure: A = 648 - 162 = 486 m 2 The two half circles have the same area as one circle. Subtract the area of the circle from the area of the rectangle. area of the rectangle: A = bh = 33 (16) = 528 f t 2 area of circle ) = 64π f t 2 area of figure: A = 528 - 64π ≈ 326.9 ft 2 2. Find the shaded area. Round to the nearest tenth, if necessary. E X A M P L E 3 Landscaping Application Katie is using the given plan to convert part of her lawn to a xeriscape garden.

A newly planted xeriscape uses 17 gallons of water per square foot per year. How much water will the garden require in one year? To find the area of the garden in square feet, divide the garden into parts. The area of the top rectangle is 28.5 (7.5) = 213.75 f t 2. The area of the center trapezoid is 1 __ 2 (12 + 18) (6) = 90 f t 2. The area of the bottom rectangle is 12 (6) = 72 f t 2. The total area of the garden is 213.75 + 90 + 72 = 375.75 f t 2. The garden will use 375.75 (17) = 6387.75 gallons of water per year. Landscaping The rainwater harvesting system at the Lady Bird Johnson Wildflower Center in Austin, Texas, collects approximately 10,200 gallons of water per inch of rain. Approximately 300,000 gallons are collected annually. 3. The lawn that Katie is replacing requires 79 gallons of water per square foot per year. How much water will Katie save by planting the xeriscape garden? 9- 3 Composite Figures 607 607 ��ge07sec09l03002aaABeckmann28.5 ft12 ft10.5 ft7.5 ft6 ft19.5 ftge07se_c09l03003aABeckmann28.5 ft12 ft6 ft7.5 ft6 ft18 ft To estimate the area of an irregular shape, you can sometimes use a composite figure. First, draw a composite figure that resembles the irregular shape. Then divide the composite figure into simple shapes. E X A M P L E 4 Estimating Areas of Irregular Shapes Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 cm. Draw a composite figure that approximates the irregular shape. Find the area of each part of the composite figure. area of triangle a: bh = 1 _ A = 1 _ (3) (1) = 1.5 c m 2 2 2 area of parallelogram b: A = bh = 3 (1) = 3 c m 2 area of trapezoid c: A = 1 _ (3 + 2) (1) = 2.5 c m 2 2 area of triangle d: A = 1 _ (2) (1) = 1 c m 2 2 area of composite figure: 1.5

+ 3 + 2.5 + 1 = 8 cm 2 The shaded area is about 8 c m 2. 4. Use a composite figure to estimate the shaded area. The grid has squares with side lengths of 1 ft. THINK AND DISCUSS 1. Describe a composite figure whose area you could find by using subtraction. 2. Explain how to find the area of an irregular shape by using a composite figure. 3. GET ORGANIZED Copy and complete the graphic organizer. Use the given composite figure. 608 608 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-3 Exercises Exercises KEYWORD: MG7 9-3 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Draw a composite figure that is made up of two rectangles Multi-Step Find the shaded area. Round to the nearest tenth, if necessary. p. 606 2. p. 607 3. 5. Interior Decorating Barbara is getting carpet p. 607 installed in her living room and hallway. The cost of installation is $6 per square yard. What is the total cost of installing the carpet. 608 Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 in. 7. 8. PRACTICE AND PROBLEM SOLVING Independent Practice Multi-Step Find the shaded area. Round to the nearest tenth, if necessary. For See Exercises Example 9. 9–10 11–12 13 14–15 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S20 Application Practice p. S36 11. 10. 12. 9- 3 Composite Figures 609 609 �� 13. Drama Pat is painting a stage backdrop for a play. The paint he is using covers 90 square feet per quart. How many quarts of paint should Pat buy? Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 m. 14. 15. Find the area of each figure first by adding and then by subtracting. Compare your answers. 16. 17. Find the area of each figure.

Give your answers in terms of π. 18. 19. 20. 21. Geography Use the grid on the map of Lake Superior to estimate the area of the surface of the lake. Each square on the grid has a side length of 100 miles. 22. Critical Thinking A trapezoid can be divided into a rectangle and two triangles. Show that the area formula for a trapezoid gives the same result as the sum of the areas of the rectangle and triangles. 23. This problem will prepare you for the Multi-Step TAKS Prep on page 614. A school crossing sign has the dimensions shown. a. Find the area of the sign. b. A manufacturer has a rectangular sheet of metal measuring 45 in. by 105 in. Draw a figure that shows how 6 school crossing signs can be cut from this sheet of metal. c. How much metal will be left after the six signs are made? 610 610 Chapter 9 Extending Perimeter, Circumference, and Area 30 ft15 ft22 ftge07sec09l03005a��Lake SuperiorWISCONSINMINN.CANADAMICHIGAN ge07se_c09l03006a�� Math History � � � � � � Hippocrates attempted to use lunes to solve a �� problem that has since been proven impossible: constructing a square with the same area as a given circle. Multi-Step Use a ruler and compass to draw each figure and then find the area. 24. A rectangle with a base length of b = 3 cm and a height of h = 4 cm has a circle with a radius of r = 1 cm removed from the interior. 25. A square with a side length of s = 4 in. shares a side with a triangle with a height of h = 5 in. and a base length of b = 4 in. and shares another side with a half circle with d = 4 in. 26. A circle with a radius of r = 5 cm has a right triangle with a base of b = 8 cm and a height of h = 6 cm removed from its interior. � 27. Multi-Step A lune is a crescent-shaped figure bounded by two intersecting circles. Find the shaded area in each of the first three diagrams, and then use your results to � find the area of the lune. � � � � �� Estimation Trace

each irregular shape and draw a composite figure that approximates it. Measure the composite figure and use it to estimate the area of the irregular shape. 28. 29. 30. Write About It Explain when you would use addition to find the area of a composite figure and when you would use subtraction. 31. Which equation can be used to find the area of the composite figure? A = bh + 1 _ (h) 2 2 A = bh + h 2 A = h + 2b + h 2 A = h + 2b + 1 _ h 2 2 32. Use a ruler to measure the dimensions of the composite figure to the nearest tenth of a centimeter. Which of the following best represents the area of the composite figure? 4 c m 2 19 cm 2 22 cm 2 42 c m 2 9- 3 Composite Figures 611 611 �� 33. Find the area of the unshaded part of the rectangle. 1800 m 2 2250 m 2 2925 m 2 4725 m 2 CHALLENGE AND EXTEND 34. An annulus is the region between two circles that have the same center. Write the formula for the area of the annulus in terms of the outer radius R and the inner radius r. 35. Draw two composite figures with the same area: one made up of two rectangles and the other made up of a rectangle and a triangle. 36. Draw a composite figure that has a total area of 10π c m 2 and is made up of a rectangle and a half circle. Label the dimensions of your figure. SPIRAL REVIEW Find each sale price. (Previous course) 37. 20% off a regular price of $19.95 38. 15% off a regular price of $34.60 Find the length of each segment. (Lesson 7-4) 39. ̶̶ BC 40. ̶̶ CD Find the area of each regular polygon. Round to the nearest tenth. (Lesson 9-2) 41. an equilateral triangle with a side length of 3 cm 42. a regular hexagon with an apothem of 4 √  3 m Q: What math classes did you take in high school? A: In high school I took Algebra 1, Geometry, Algebra 2, and Trigonometry. KEYWORD: MG7 Career Q: What math classes did you take in college? A: In college I took Precalculus, Calculus, and Statistics. Q:

What technical materials do you write? A: I write training manuals for computer software packages. Q: How do you use math? A: Some manuals I write are for math programs, so I use a lot of formulas to describe patterns and measurements. Q: What are your future plans? A: After I get a few more years experience writing manuals, I would like to train others who use these programs. Anessa Liu Technical writer 612 612 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-3 Use with Lesson 9-3 Develop Pick’s Theorem for Area of Lattice Polygons A lattice polygon is a polygon drawn on graph paper so that all its vertices are on intersections of grid lines, called lattice points. The lattice points of the grid at right are shown in red. In this lab, you will discover a formula called Pick’s Theorem, which is used to find the area of lattice polygons. Activity TEKS G.8.A Congruence and the geometry of size: find areas of... composite figures. 1 Find the area of each figure. Create a table like the one below with a row for each shape to record your answers. The first one is done for you. 2 Count the number of lattice points on the boundary of each figure. Record your answers in the table. 3 Count the number of lattice points in the interior of each figure. Record your answers in the table. Figure Area Number of Lattice Points On Boundary In Interior 2.5 5 1 A B C D E F Try This 1. Make a Conjecture What do you think is true about the relationship between the area of a figure and the number of lattice points on the boundary and in the interior of the figure? Write your conjecture as a formula in terms of the number of lattice points on the boundary B and the number of lattice points in the interior I. 2. Test your conjecture by drawing at least three different figures on graph paper and by finding their areas. 3. Estimate the area of the curved figure by using a lattice polygon. 4. Find the shaded area in the figure by subtracting. Test your formula on this figure. Does your formula work for figures with holes in them? 9- 3 Geometry Lab 613 613 �� SECTION 9A Developing Geometric Formulas Traffic Signs Traffic

signs are usually made of reflective aluminum. A manufacturer of traffic signs begins with a rectangular sheet of aluminum that measures 60 in. by 90 in. 1. A railroad crossing sign is a circle with a diameter of 30 in. The manufacturer can make 6 of these signs from the sheet of aluminum by arranging the signs as shown. How much aluminum is left over once the signs have been made? 2. A stop sign is a regular octagon. The manufacturer can use the sheet of aluminum to make 6 stop signs as shown. How much aluminum is left over in this case? 3. A yield sign is an equilateral triangle with sides 30 in. long. By arranging the triangles as shown, the manufacturer can use the sheet of aluminum to make 10 yield signs. How much aluminum is left over when yield signs are made? 4. The making of which type of sign results in the least amount of waste? 614 614 Chapter 9 Extending Perimeter, Circumference, and Area �� SECTION 9A Quiz for Lessons 9-1 Through 9-3 9-1 Developing Formulas for Triangles and Quadrilaterals Find each measurement. 1. the area of the parallelogram 2. the base of the rectangle, in which A = (24 x 2 + 8x) m 2 3. d 1 of the kite, in which A = 126 ft 2 4. the area of the rhombus 5. The tile mosaic shown is made up of 1 cm squares. Use the grid to find the perimeter and area of the green triangle, the blue trapezoid, and the yellow parallelogram. 9-2 Developing Formulas for Circles and Regular Polygons Find each measurement. 6. the circumference of ⊙R in terms of π 7. the area of ⊙E in terms of π Find the area of each regular polygon. Round to the nearest tenth. 8. a regular hexagon with apothem 6 ft 9. a regular pentagon with side length 12 m 9-3 Composite Figures Find the shaded area. Round to the nearest tenth, if necessary. 10. 11. 12. Shelby is planting grass in an irregularly shaped garden as shown. The grid has squares with side lengths of 1 yd. Estimate the area of the garden. Given that grass cost $6.50 per square yard, find the cost of the grass. Ready to Go On? 615 6

15 �� 9-4 Perimeter and Area in the Coordinate Plane TEKS G.7.A Dimensionality and the geometry of location: use... two-dimensional coordinate systems to represent... figures. Also G.7.B, G.8.A Objective Find the perimeters and areas of figures in a coordinate plane. Why learn this? You can use figures in a coordinate plane to solve puzzles like the one at right. (See Example 4.) In Lesson 9-3, you estimated the area of irregular shapes by drawing composite figures that approximated the irregular shapes and by using area formulas. Another method of estimating area is to use a grid and count the squares on the grid. E X A M P L E 1 Estimating Areas of Irregular Shapes in the Coordinate Plane Estimate the area of the irregular shape. Method 1: Draw a composite Method 2: Count the number of figure that approximates the irregular shape and find the area of the composite figure. squares inside the figure, estimating half squares. Use a ■ for a whole for a half square. square and a The area is approximately 4 + 6.5 + 5 + 4 + 5 + 3.5 + 3 + 3 + 2 = 36 unit s 2. There are approximately 31 whole squares and 13 half squares, so the area is about 31 + 1 __ 2 (13) = 37.5 unit s 2. 1. Estimate the area of the irregular shape. 616 616 Chapter 9 Extending Perimeter, Circumference, and Area �� E X A M P L E 2 Finding Perimeter and Area in the Coordinate Plane Draw and classify the polygon with vertices A (-4, 1), B (2, 4), C (4, 0), and D (-2, -3). Find the perimeter and area of the polygon. ( x 2 - x 1 ) The distance from ( x 1, y 1 ) to ( x 2, y 2 ) in a coordinate plane is d = √  2 2 + ( y 2 - y 1 ), and the slope of the line containing the y 2 - y 1 ______ x 2 - x 1. points is

m = See pages 44 and 182. Step 1 Draw the polygon. slope of Step 2 ABCD appears to be a rectangle. To verify this, use slopes to show that the sides are perpendicular. ̶̶ = 3 _ AB : - (-4) ̶̶ BC : 0 - 4 _ = -4 _ 4 - 2 2 ̶̶ = -3 _ CD : -3 - 0 _ = 1 _ -6 -2 - 4 2 1- (-3) = 4 _ _ -2 -4 - (-2) slope of slope of slope of ̶̶ DA : = -2 = -2 ̶̶ CD be the base and The consecutive sides are perpendicular, so ABCD is a rectangle. ̶̶ BC be the height of the rectangle. Step 3 Let Use the Distance Formula to find each side length. b = CD = √  h = BC = √  perimeter of ABCD: P = 2b + 2h = 2 (3 √  5 ) + 2 (2 √  5 ) = 10 √  5 units area of ABCD: A = bh = (3 √  5 ) (2 √  5 ) = 30 units 2. (-2 - 4) 2 + (-3 - 0) 2 = √  45 = 3 √  5 (4 - 2) 2 + (0 - 4) 2 = √  20 = 2 √  5 2. Draw and classify the polygon with vertices H (-3, 4), J (2, 6), K (2, 1), and L (-3, -1). Find the perimeter and area of the polygon. For a figure in a coordinate plane that does not have an area formula, it may be easier to enclose the figure in a rectangle and subtract the areas of the parts of the rectangle that are not included in the figure. E X A M P L E 3 Finding Areas in the Coordinate Plane by Subtracting Find the area of the polygon with vertices W (1, 4), X (4, 2), Y (2

, -3), and Z (-4, 0). Draw the polygon and enclose it in a rectangle. area of the rectangle: A = bh = 8 (7) = 56 units 2 area of the triangles: bh = 1 _ a: A = 1 _ (5) (4) = 10 units 2 2 2 bh = 1 _ b: A = 1 _ (3) (2) = 3 units 2 2 2 bh = 1 _ c: A = 1 _ (2) (5) = 5 units 2 2 2 bh = 1 _ d: A = 1 _ (6) (3) = 9 units 2 2 2 The area of the polygon is 56 - 10 - 3 - 5 - 9 = 29 units 2. 3. Find the area of the polygon with vertices K (-2, 4), L (6, -2), M (4, -4), and N (-6, -2). 9- 4 Perimeter and Area in the Coordinate Plane 617 617 �� E X A M P L E 4 Problem-Solving Application In the puzzle, the two figures are made up of the same pieces, but one figure appears to have a larger area. Use coordinates to show that the area does not change when the pieces are rearranged. Understand the Problem The parts of the puzzle appear to form two triangles with the same base and height that contain the same shapes, but one appears to have an area that is larger by one square unit. Make a Plan Find the areas of the shapes that make up each figure. If the corresponding areas are the same, then both figures have the same area by the Area Addition Postulate. To explain why the area appears to increase, consider the assumptions being made about the figure. Each figure is assumed to be a triangle with a base of 8 units and a height of 3 units. Both figures are divided into several smaller shapes. Solve Find the area of each shape. Top figure Bottom figure red triangle: bh = 1 _ A = 1 _ (5) (2) = 5 unit s 2 2 2 blue triangle: bh = 1 _ A = 1 _ (3) (1) = 1.5 units 2 2 2 green rectangle: A = bh = (3) (1) = 3 units 2 red triangle: bh = 1 _ A = 1 _ (5) (2) = 5 u n

its 2 2 2 blue triangle: bh = 1 _ A = 1 _ (3) (1) = 1.5 u nits 2 2 2 green rectangle: A = bh = (3) (1) = 3 units 2 yellow rectangle: A = bh = (2) (1) = 2 units 2 yellow rectangle: A = bh = (2) (1) = 2 units 2 The areas are the same. Both figures have an area of 5 + 1.5 + 3 + 2 = 11.5 units 2. If the figures were triangles, their areas would be A = 1 __ 2 (8) (3) = 12 units 2. By the Area Addition Postulate, the area is only 11.5 units 2, so the figures must not be triangles. Each figure is a quadrilateral whose shape is very close to a triangle. Look Back The slope of the hypotenuse of the red triangle is 2 __ 5. The slope of the hypotenuse of the blue triangle is 1 __ 3. Since the slopes are unequal, the hypotenuses do not form a straight line. This means the overall shapes are not triangles. 4. Create a figure and divide it into pieces so that the area of the figure appears to increase when the pieces are rearranged. 618 618 Chapter 9 Extending Perimeter, Circumference, and Area ��1234 THINK AND DISCUSS 1. Describe two ways to estimate the area of an irregular shape in a coordinate plane. 2. Explain how you could use the Distance Formula to find the area of a special quadrilateral in a coordinate plane. 3. GET ORGANIZED Copy the graph and the graphic organizer. Complete the graphic organizer by writing the steps used to find the area of the parallelogram. 9-4 Exercises Exercises GUIDED PRACTICE Estimate the area of each irregular shape. p. 616 1. 2. KEYWORD: MG7 9-4 KEYWORD: MG7 Parent. 617 Multi-Step Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 3. V (-3, 0), W (3, 0), X (0, 3) 4. F (2, 8), G (4, 4), H (2, 0) 5. P (-2, 5), Q (8, 5), R

(8, 1), S (-2, 1) 6. A (-4, 2), B (-2, 6), C (6, 6), D (8, 2 Find the area of each polygon with the given vertices. p. 617 7. S (3, 8), T (8, 3), U (2, 1) 8. L (3, 5), M (6, 8), N (9, 6), P (5, 0. 618 9. Find the area and perimeter of each polygon shown. Use your results to draw a polygon with a perimeter of 12 units and an area of 4 un its 2 and a polygon with a perimeter of 12 units and an area of 3 un its 2. 9- 4 Perimeter and Area in the Coordinate Plane 619 619 �� Independent Practice For See Exercises Example 10–11 12–15 16–17 18 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S21 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING Estimate the area of each irregular shape. 10. 11. Multi-Step Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 12. H (-3, -3), J (-3, 3), K (5, 3) 13. L (7, 5), M (5, 0), N (3, 5), P (5, 10) 14. X (2, 1), Y (5, 3), Z (7, 1) 15. A (-3, 5), B (2, 7), C (2, 1), D (-3, 3) Find the area of each polygon with the given vertices. 16. A (9, 9), B (4, -4), C (-4, 1) 17. T (-4, 4), U (5, 3), V (4, -5), W (-5, 1) 18. In which two figures do the rectangles cover the same area? Explain your reasoning. Algebra Graph each set of lines to form a triangle. Find the area and perimeter. 19. y = 2, x =

5, and y = x 20. y = -5, x = 2, and y = -2x + 7 21. Transportation The graph shows the speed of a boat versus time. a. If the base of each square on the graph represents 1 hour and the height represents 20 miles per hour, what is the area of one square on the graph? Include units in your answer. b. Estimate the shaded area in the graph. c. Critical Thinking Use your results from part a to interpret the meaning of the area you found in part b. (Hint: Look at the units.) 22. Write About It Explain how to find the perimeter of the polygon with vertices A (2, 3), B (4, 0), C (3, -2), D (-1, -1), and E (-2, 0). 23. This problem will prepare you for the Multi-Step TAKS Prep on page 638. A carnival game uses a 10-by-10 board with three targets. Each player throws a dart at the board and wins a prize if it hits a target. a. One target is a parallelogram as shown. Find its area. b. What should the coordinates be for points C and H so that the triangular target △ABC and the kite-shaped target EFGH have the same area as the parallelogram? 620 620 Chapter 9 Extending Perimeter, Circumference, and Area �� 24. A circle with center (0, 0) passes through the point (3, 4). What is the area of the circle to the nearest tenth of a square unit? 15.7 25.0 31.4 78.5 25. △ABC with vertices A (1, 1) and B (3, 5) has an area of 10 unit s 2. Which is NOT a possible location of the third vertex? C (-4, 1) C (7, 3) C (6, 1) C (3, -3) 26. Extended Response Mike estimated the area of the irregular figure to be 64 units 2. a. Explain why his answer is not very accurate. b. Explain how to use a composite figure to estimate the area. c. Explain how to estimate the area by averaging the areas of two squares. CHALLENGE AND

EXTEND Algebra Estimate the shaded area under each curve. 28. y = x 2 for 0 ≤ x ≤ 3 27. y = 2 x for 0 ≤ x ≤ 3 29. y = √  x for 0 ≤ x ≤ 9 30. Estimation Use a composite figure and the Distance Formula to estimate the perimeter of the irregular shape. 31. Graph a regular octagon on the coordinate plane with vertices on the x-and y-axes and on the lines y = x and y = -x so that the distance between opposite vertices is 2 units. Find the area and perimeter of the octagon. SPIRAL REVIEW Solve and graph each compound inequality. (Previous course) 32. -4 < x + 3 < 7 ̶̶ BC, ∠DCA ≅ ∠ACB Prove: ∠DAC ≅ ∠BAC (Lesson 4-6) 35. Given: ̶̶ DC ≅ 33. 0 < 2a + 4 < 10 34. 12 ≤ -2m + 10 ≤ 20 Find each measurement. (Lesson 9-2) 36. the area of ⊙C if the circumference is 16π cm 37. the diameter of ⊙H if the area is 121π f t 2 9- 4 Perimeter and Area in the Coordinate Plane 621 621 �� 9-5 Effects of Changing Dimensions Proportionally TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area,... when one or more dimensions of a figure are changed.... Also G.5.A, G.5.B Objectives Describe the effect on perimeter and area when one or more dimensions of a figure are changed. Apply the relationship between perimeter and area in problem solving. Why learn this? You can analyze a graph to determine whether it is misleading or to explain why it is misleading. (See Example 4.) In the graph, the height of each DVD is used to represent the number of DVDs shipped per year. However as the height of each DVD increases, the width also increases, which can create a misleading effect. E X A M P L E 1 Effects of Changing One Dimension Describe the effect of each change on the area of the given figure. A The height of the parallelogram is doubled. double the height: A = bh = 12 (18) =

216 cm 2 original dimensions: A = bh = 12 (9) = 108 cm 2 Notice that 216 = 2 (108). If the height is doubled, the area is also doubled. B The base length of the triangle with vertices A(1, 1), B(6, 1), and C (3, 5) is multiplied by 1 __. 2 Draw the triangle in a coordinate plane and find the base and height. original dimensions: bh = 1 _ A = 1 _ (5) (4) = 10 units 2 2 2 base multiplied by 1 __ 2 : bh = 1 _ A = 1 _ (2.5) (4) = 5 units 2 2 2 Notice that 5 = 1 __ 2 (10). If the base length is multiplied by 1 __ 2, the area is multiplied by 1 __ 2. 1. The height of the rectangle is tripled. Describe the effect on the area. 622 622 Chapter 9 Extending Perimeter, Circumference, and Area ge07se_c09l05001a AB�� E X A M P L E 2 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter or circumference and the area of the given figure. A The base and height of a rectangle with base 8 m and height 3 m are If the radius of a circle or the side length of a square is changed, the size of the entire figure changes proportionally. both multiplied by 5. original dimensions: P = 2 (8) + 2 (3) = 22 m A = 83 = 24 m 2 dimensions multiplied by 5: P = 2 (40) + 2 (15) = 110 m A = 40 (15) = 600 m 2 The perimeter is multiplied by 5. The area is multiplied by 5 2, or 25. B The radius of ⊙A is multiplied by 1 __. 3 original dimensions: C = 2π (9) = 18π in. A = π (9) 2 = 81π in 2 dimensions multiplied by 1 __ 3 : C = 2π (3) = 6π in. A = π (3) 2 = 9π in 2 P = 2b + 2h A = bh 5 (8) = 40; 5 (3) = 15 5 (22) = 110 25 (24) = 600 C = 2πr A = π r 2 1 __ (

9) = 3 3 The perimeter is multiplied by 1 __ 3. The area is multiplied by ( 1 __ 3 ), or 1 __ 9. 2 1 __ (18π) = 6π 3 1 __ (81π) = 9π 9 2. The base and height of the triangle with vertices P (2, 5), Q (2, 1) and R (7, 1) are tripled. Describe the effect on its area and perimeter. When all the dimensions of a figure are changed proportionally, the figure will be similar to the original figure. Effects of Changing Dimensions Proportionally Change in Dimensions Perimeter or Circumference Area All dimensions multiplied by a Changes by a factor of a Changes by a factor of Effects of Changing Area A A square has side length 5 cm. If the area is tripled, what happens to the side length? The area of the original square is A = s 2 = 5 2 = 25 cm 2. If the area is tripled, the new area is 75 cm 2. s 2 = 75 s = √  75 = 5 √  3 Set the new area equal to s 2. Take the square root of both sides and simplify. Notice that 5 √  3 = √  3 (5). The side length is multiplied by √  3. 9- 5 Effects of Changing Dimensions Proportionally 623 623 �� B A circle has a radius of 6 in. If the area is doubled, what happens to the circumference? The original area is A = π r 2 = 36π in 2, and the circumference is C = 2πr = 12π in. If the area is doubled, the new area is 72π in 2. π r 2 = 72π r 2 = 72 r 2 = √  72 = 6 √  2 C = 2πr = 2π ( 6 √  2 ) = 12 √  2 π Set the new area equal to π r 2. Divide both sides by π. Take the square root of both sides and simplify. Substitute 6 √  2 for r and simplify. Notice that 12 √  2 π = √  2 (12π). The circumference is multiplied by √  2. 3. A square has a perimeter of 36 mm. If the area

is multiplied by 1 __ 2, what happens to the side length? E X A M P L E 4 Entertainment Application The graph shows that DVD shipments totaled about 182 million in 2000, 364 million in 2001, and 685 million in 2002. The height of each DVD is used to represent the number of DVDs shipped. Explain why the graph is misleading. The height of the DVD representing shipments in 2002 is about 3.8 times the height of the DVD representing shipments in 2002. This means that the area of the DVD is multiplied by about 3.8 2, or 14.4, so the area of the larger DVD is about 14.4 times the area of the smaller DVD. The graph gives the misleading impression that the number of shipments in 2002 was more than 14 times the number in 2000, but it was actually closer to 4 times the number shipped in 2000. 4. Use the information above to create a version of the graph that is not misleading. THINK AND DISCUSS 1. Discuss how changing both dimensions of a rectangle affects the area and perimeter. 2. GET ORGANIZED Copy and complete the graphic organizer. 624 624 Chapter 9 Extending Perimeter, Circumference, and Area ��20002001YearDVDs shipped (millions)2002100200300400500600ge06se_c09105002aDVD Shipments 9-5 Exercises Exercises KEYWORD: MG7 9-5 KEYWORD: MG7 Parent GUIDED PRACTICE Describe the effect of each change on the area of the given figure. p. 622 1. The height of the triangle is doubled. 2. The height of a trapezoid with base lengths 12 cm and 18 cm and height 5 cm is multiplied by 1 __. 623 Describe the effect of each change on the perimeter or circumference and the area of the given figure. 3. The base and height of a triangle with base 12 in. and height 6 in. are both tripled. 4. The base and height of the rectangle are both multiplied by 1 __. A square has an area of 36 m 2. If the area is doubled, what happens to the p. 623 side length? 6. A circle has a diameter of 14 ft. If the area is tripled, what happens to the circumference. Business A restaurant has a weekly ad in a

local newspaper that is 2 inches wide p. 624 and 4 inches high and costs $36.75 per week. The cost of each ad is based on its area. If the owner of the restaurant decides to double the width and height of the ad, how much will the new ad cost? Independent Practice Describe the effect of each change on the area of the given figure. PRACTICE AND PROBLEM SOLVING For See Exercises Example 8–9 10–11 12–13 14 1 2 3 4 8. The height of the triangle with vertices (1, 5), (2, 3), and (-1, -6) is multiplied by 4. 9. The base of the parallelogram is multiplied by 2 __ 3. Describe the effect of each change on the perimeter or circumference and the area of the given figure. TEKS TEKS TAKS TAKS 10. The base and height of the triangle are both doubled. Skills Practice p. S21 Application Practice p. S36 11. The radius of the circle with center (0, 0) that passes through (5, 0) is multiplied by 3 __ 5. 12. A circle has a circumference of 16π mm. If you multiply the area by 1 __ 3, what happens to the radius? 13. A square has vertices (3, 2), (8, 2,) (8, 7), and (3, 7). If you triple the area, what happens to the side length? 14. Entertainment Two televisions have rectangular screens with the same ratio of base to height. One has a 32 in. diagonal, and the other has a 36 in. diagonal. a. What is the ratio of the height of the larger screen to that of the smaller screen? b. What is the ratio of the area of the larger screen to that of the smaller screen? 9- 5 Effects of Changing Dimensions Proportionally 625 625 �� Describe the effect of each change on the area of the given figure. 15. The diagonals of a rhombus are both multiplied by 8. 16. The circumference of a circle is multiplied by 2.4. 17. The base of a rectangle is multiplied by 4, and the height is multiplied by 7. 18. The apothem of a regular octagon is tripled. 19. The diagonal of a square is divided by 4. 20. One diagonal of a kite is multiplied

by 1 _. 7 21. 21. The perimeter of an equilateral triangle is doubled. 22. 22. Find the area of the trapezoid. Describe the effect of each change on the area. a. a. The length of the top base is doubled. b. The length of both bases is doubled. c. The height is doubled. d. Both bases and the height are doubled. 23. Geography A map has the scale 1 inch = 10 miles. On the map, the area of Big Bend National Park in Texas is about 12.5 square inches. Estimate the actual area of the park in acres. (Hint: 1 square mile = 640 acres) 24. Critical Thinking If you want to multiply the dimensions of a figure so that the area is 50% of the original area, what is your scale factor? Multi-Step For each figure in the coordinate plane, describe the effect on the area that results from each change. a. Only the x-coordinates of the vertices are multiplied by 3. b. Only the y-coordinates of the vertices are multiplied by 3. c. Both the x- and y-coordinates of the vertices are multiplied by 3. 25. 26. 27. Geography Geography The altitude in Big Bend National Park ranges from approximately 1800 feet along the Rio Grande to 7800 feet in the Chisos Mountains. 28. Write About It How could you change the dimensions of a parallelogram to increase the area by a factor of 5 if the parallelogram does not have to be similar to the original parallelogram? if the parallelogram does have to be similar to the original parallelogram? 29. This problem will prepare you for the Multi-Step TAKS Prep on page 638. To win a prize at a carnival, a player must toss a beanbag onto a circular disk with a diameter of 8 in. a. The organizer of the game wants players to win twice as often, so he changes the disk so that it has twice the area. What is the diameter of the new disk? b. Suppose the organizer wants players to win half as often. What should be the disk’s diameter in this case? 626 626 Chapter 9 Extending Perimeter, Circumference, and Area �� 30. Which of the following describes the effect on the area of a square when the side length is doubled? The area remains constant

. The area is reduced by a factor of 1 __. 2 The area is doubled. The area is increased by a factor of 4. 31. If the area of a circle is increased by a factor of 4, what is the change in the diameter of the circle? The diameter is 1 __ of the original diameter. 2 The diameter is 2 times the original diameter. The diameter is 4 times the original diameter. The diameter is 16 times the original diameter. 32. Tina and Kieu built rectangular play areas for their dogs. The play area for Tina’s dog is 1.5 times as long and 1.5 times as wide as the play area for Kieu’s dog. If the play area for Kieu’s dog is 60 square feet, how big is the play area for Tina’s dog? 40 ft 2 90 ft 2 135 ft 2 240 ft 2 33. Gridded Response Suppose the dimensions of a triangle with a perimeter of 18 inches are doubled. Find the perimeter of the new triangle in inches. CHALLENGE AND EXTEND 34. Algebra A square has a side length of (2x + 5) cm. If the side length is multiplied by 5, what is the area of the new square? 35. Algebra A circle has a diameter of 6 in. If the circumference is multiplied by (x + 3), what is the area of the new circle? 36. Write About It How could you change the dimensions of the composite figure to double the area if the resulting figure does not have to be similar to the original figure? if the resulting figure does have to be similar to the original figure? SPIRAL REVIEW Write an equation that can be used to determine the value of the variable in each situation. (Previous course) 37. Steve can make 2 tortillas per minute. He makes t tortillas in 36 minutes. 38. A car gets 25 mi/gal. At the beginning of a trip of m miles, the car’s gas tank contains 13 gal of gas. At the end of the trip, the car has 8 gal of gasoline left. Find the measure of each interior and each exterior angle of each regular polygon. Round to the nearest tenth, if necessary. (Lesson 6-1) 39. heptagon 40. decagon 41. 14-gon Find the area of each polygon with the given vertices. (Lesson 9-4) 42. L (-1, 1),

M (5, 2), and N (1, -5) 43. A (-4, 2), M (-2, 4), C (4, 2) and D (2, -4) 9- 5 Effects of Changing Dimensions Proportionally 627 627 �� Probability Probability An experiment is an activity in which results are observed. Each result of an experiment is called an outcome. The sample space is the set of all outcomes of an experiment. An event is any set of outcomes. See Skills Bank page S77 The probability of an event is a number from 0 to 1 that tells you how likely the event is to happen. The closer the probability is to 0, the less likely the event is to happen. The closer it is to 1, the more likely the event is to happen. An experiment is fair if all outcomes are equally likely. The theoretical probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space. P (E) = number of outcomes in event E ___ number of possible outcomes Example 1 A fair number cube has six faces, numbered 1 through 6. An experiment consists of rolling the number cube. A What is the sample space of the experiment? The sample space has 6 possible outcomes. The outcomes are 1, 2, 3, 4, 5, and 6. B What is the probability of the event “rolling a 4”? The event “rolling a 4” contains only 1 outcome. The probability is P (E) = number of outcomes in event E___ number of possible outcomes = 1_. 6 C What are the outcomes in the event “rolling an odd number”? What is the probability of rolling an odd number? The event “rolling an odd number” contains 3 outcomes. The outcomes are 1, 3, and 5. The probability is P (E) = number of outcomes in event E = 3 _ = 1 _ ___. 2 6 number of possible outcomes If two events A and B have no outcomes in common, then the probability that A or B will happen is P (A) + P (B). The complement of an event is the set of outcomes that are not in the event. If the probability of an event is p, then the probability of the complement of the event is 1 - p. 628 628 Chapter 9 Extending Perimeter, Circumference, and Area Example 2 The tiles shown

below are placed in a bag. An experiment consists of drawing a tile at random from the bag. A What is the sample space of the experiment? The sample space has 9 possible outcomes. The outcomes are 1, 2, 3, 4, A, B, C, D, E, and F. B What is the probability of choosing a 3 or a vowel? The event “choosing a 3” contains only 1 outcome. The probability is P (A) = number of outcomes in event A ___ number of possible outcomes = 1 _. 9 The event “choosing a vowel” has 2 outcomes, A and E. The probability is P (B) = number of outcomes in event B ___ number of possible outcomes = The probability of choosing a 3 or a vowel is 1 _. 3 9 9 9 C What is the probability of not choosing a letter? The event “choosing a letter” contains 5 outcomes, A, B, C, D, and E. The probability is P (E) = number of outcomes in event E ___ number of possible outcomes = 5 _. 9 The event of not choosing a letter is the complement of the event of choosing a letter. The probability of not choosing a letter is 1 - 5 __ 9 = 4 __ 9. Try This TAKS Grades 9–11 Obj. 8, 9 An experiment consists of randomly choosing one of the given shapes. 1. What is the probability of choosing a circle? 2. What is the probability of choosing a shape whose area is 36 cm 2? 3. What is the probability of choosing a quadrilateral or a triangle? 4. What is the probability of not choosing a triangle? On Track for TAKS 629 629 �� 9-6 Geometric Probability TEKS G.8.A Congruence and the geometry of size: find areas of... regular polygons, circles, and composite figures. Objectives Calculate geometric probabilities. Use geometric probability to predict results in realworld situations. Vocabulary geometric probability Why learn this? You can use geometric probability to estimate how long you may have to wait to cross a street. (See Example 2.) Remember that in probability, the set of all possible outcomes of an experiment is called the sample space. Any set of outcomes is called an event. If every outcome in the sample space is equally likely, the theoretical probability of an event is P = _____________________________ number of outcomes

in the event number of outcomes in the sample space. Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure. Three models for geometric probability are shown below. Model Length Angle Measure Area Geometric Probability Example Sample space All points on ̶̶̶ AD All points in the circle All points in the rectangle ̶̶ BC All points in the All points in the triangle All points on shaded region P = BC _ AD P = measure of angle __ 360° P = area of triangle __ area of rectangle Event Probability E X A M P L E 1 Using Length to Find Geometric Probability A point is chosen randomly on ̶̶ AD. Find the probability of each event. If an event has a probability p of occurring, the probability of the event not occurring is 1 - p. A The point is on ̶̶ AC. P = AC _ AD = 7 _ 12 B The point is not on ̶̶ AB. First find the probability that the point is on P ( ̶̶ AB ) = AB _ AD = 4 _ 12 = 1 _ 3 ̶̶ AB. Subtract from 1 to find the probability that the point is not on P (not on ̶̶ = 2 _ AB ) = 1 - 1 _ 3 3 ̶̶ AB. 630 630 Chapter 9 Extending Perimeter, Circumference, and Area �� A point is chosen randomly on Find the probability of each event. ̶̶ AD. C The point is on P ( ̶̶ AB or ̶̶ CD ) = P ( ̶̶ ̶̶ CD. AB or ̶̶ AB ) + P ( ̶̶ CD ) = 4 _ 12 + 5 _ 12 = 9 _ 12 = 3 _ 4 1. Use the figure above to find the probability that the point is ̶̶ BD. on E X A M P L E 2 Transportation Application A stoplight has the following cycle: green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. A What is the probability that the light will be yellow when you arrive? To find the probability, draw a segment to represent the number of seconds that each color light is on. P = 5 _ 60 = 1 _ 12 ≈ 0.08 The light is yellow for 5

out of every 60 seconds. B If you arrive at the light 50 times, predict about how many times you will have to stop and wait more than 10 seconds. In the model, the event of stopping and waiting more than 10 seconds is represented by a segment that starts at C and ends 10 units from D. The probability of stopping and waiting more than 10 seconds is P = 20 __ 60 = 1 __ 3. If you arrive at the light 50 times, you will probably stop and wait more than 10 seconds about 1 __ 3 (50) ≈ 17 times. 2. Use the information above. What is the probability that the light will not be red when you arrive? E X A M P L E 3 Using Angle Measures to Find Geometric Probability Use the spinner to find the probability of each event. A the pointer landing on red = 2 _ 9 P = 80 _ 360 The angle measure in the red region is 80°. B the pointer landing on purple or blue = 135 _ 360 75 +60 _ 360 = 3 _ 8 P = The angle measure in the purple region is 75°. The angle measure in the blue region is 60°. In Example 3C, you can also find the probability of the pointer landing on yellow, and subtract from 1. C the pointer not landing on yellow P = 360 - 100 _ 360 = 13 _ 18 = 260 _ 360 The angle measure in the yellow region is 100°. Substract this angle measure from 360°. 3. Use the spinner above to find the probability of the pointer landing on red or yellow. 9- 6 Geometric Probability 631 631 �� Geometric Probability I like to write a probability as a percent to see if my answer is reasonable. The probability of the pointer landing on red is 80° ____ = 2 __ ≈ 22%. 9 360° The angle measure is close to 90°, which is 25% of the circle, so the answer is reasonable. Jeremy Denton Memorial High School E X A M P L E 4 Using Area to Find Geometric Probability Find the probability that a point chosen randomly inside the rectangle is in each given shape. Round to the nearest hundredth. A the equilateral triangle The area of the triangle is A = 1 _ aP 2 = 1 _ (6) (36 √  3 ) ≈ 187 m 2. 2 The area of the rectangle is A = bh = 45 (20

) = 900 m 2. The probability is P = 187 _ 900 ≈ 0.21. B the trapezoid The area of the trapezoid is 3 + 12) (10) = 75 m 2. 2 The area of the rectangle is A = bh = 45 (20) = 900 m 2. The probability is P = 75 _ 900 ≈ 0.08. C the circle The area of the circle is ) = 36π ≈ 113.1 m 2. The area of the rectangle is A = bh = 45 (20) = 900 m 2. The probability is P = 113.1 _ ≈ 0.13. 900 4. Use the diagram above. Find the probability that a point chosen randomly inside the rectangle is not inside the triangle, circle, or trapezoid. Round to the nearest hundredth. 632 632 Chapter 9 Extending Perimeter, Circumference, and Area �� THINK AND DISCUSS 1. Explain why the ratio used in theoretical probability cannot be used to find geometric probability. 2. A spinner is one-half red and one-third blue, and the rest is yellow. How would you find the probability of the pointer landing on yellow? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, give an example of the geometric probability model. 9-6 Exercises Exercises KEYWORD: MG7 9-6 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of a model used to find geometric probability. 630 A point is chosen randomly on each event. ̶̶ WZ. Find the probability of 2. The point is on 4. The point is on ̶̶ XZ. ̶̶̶ WX or ̶̶ YZ. 3. The point is not on ̶̶̶ WY. 5. The point is on ̶̶ XY.. 631 Transportation A bus comes to a station once every 10 minutes and waits at the station for 1.5 minutes. 6. Find the probability that the bus will be at the station when you arrive. 7. If you go to the station 20 times, predict about how many times you will have to wait less than 3 minutes Use the spinner to find the probability of each event. p. 631 8. the pointer landing on green 9. the pointer landing on orange or blue 10

. the pointer not landing on red 11. the pointer landing on yellow or blue. 632 Multi-Step Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth. 12. the triangle 13. the trapezoid 14. the square 15. the part of the rectangle that does not include the square, triangle, or trapezoid 9- 6 Geometric Probability 633 633 �� Independent Practice For See Exercises Example 16–19 20–22 23–26 27–30 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S21 Application Practice p. S36 PRACTICE AND PROBLEM SOLVING A point is chosen randomly on Find the probability of each event. Round to the nearest hundredth. ̶̶̶ HM. 16. The point is on 18. The point is on ̶̶ JK. ̶̶ HJ or ̶̶ KL. 17. The point is not on 19. The point is not on ̶̶̶ LM. ̶̶ JK or ̶̶̶ LM. Communications A radio station gives a weather report every 15 minutes. Each report lasts 45 seconds. Suppose you turn on the radio at a random time. 20. Find the probability that the weather report will be on when you turn on the radio. 21. Find the probability that you will have to wait more than 5 minutes to hear the weather report. 22. If you turn on the radio at 50 random times, predict about how many times you will have to wait less than 1 minute before the start of the next weather report. Use the spinner to find the probability of each event. 23. the pointer landing on red 24. the pointer landing on yellow or blue 25. the pointer not landing on green 26. the pointer landing on red or green Multi-Step Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth, if necessary. 27. the equilateral triangle 28. the square 29. the part of the circle that does not include the square 30. the part of the rectangle that does not include the square, circle, or triangle 31. /////ERROR ANALYSIS///// In the spinner at right, the angle measure of the red region is

90°. The angle measure of the yellow region is 135°, and the angle measure of the blue region is 135°. Which value of the probability of the spinner landing on yellow is incorrect? Explain. Algebra A point is chosen randomly inside rectangle ABCD with vertices A (2, 8), B (15, 8), C (15, 1), and D (2, 1). Find the probability of each event. Round to the nearest hundredth. 32. The point lies in △KLM with vertices K (4, 3), L (5, 7), and M (9, 5). 33. The point does not lie in ⊙P with center P (2, 5) and radius 3. (Hint: draw the rectangle and circle.) 634 634 Chapter 9 Extending Perimeter, Circumference, and Area �� Algebra A point is chosen at random in the coordinate plane such that -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5. Find the probability of each event. Round to the nearest hundredth. Sports 34. The point is inside the parallelogram. 35. The point is inside the circle. 36. The point is inside the triangle or the circle. 37. The point is not inside the triangle, the parallelogram, or the circle. 38. Sports The point value of each region of an Olympic archery target is shown in the diagram. The outer diameter of each ring is 12.2 cm greater than the inner diameter. a. What is the probability of hitting the center? b. What is the probability of hitting a blue or black ring? c. What is the probability of scoring higher than five points? d. Write About It In an actual event, why might the probabilities be different from those you calculated in parts a, b, and c? Olympic archers stand 70 m from their targets. From that distance, the target appears about the size of the head of a thumbtack held at arm’s length. Source: www.olympic.org A point is chosen randomly in

each figure. Describe an event with a probability of 1 __. 2 39. 41. 40. 42. If a fly lands randomly on the tangram, what is the probability that it will land on each of the following pieces? a. the blue parallelogram b. the medium purple triangle c. the large yellow triangle d. Write About It Do the probabilities change if you arrange the tangram pieces differently? Explain. 43. Critical Thinking If a rectangle is divided into 8 congruent regions and 4 of them are shaded, what is the probability that you will randomly pick a point in the shaded area? Does it matter which four regions are shaded? Explain. 44. This problem will prepare you for the Multi-Step TAKS Prep on page 638. A carnival game board consists of balloons that are 3 inches in diameter and are attached to a rectangular board. A player who throws a dart at the board wins a prize if the dart pops a balloon. a. Find the probability of winning if there are 40 balloons on the board. b. How many balloons must be on the board for the probability of winning to be at least 0.25? 9- 6 Geometric Probability 635 635 �� 45. What is the probability that a ball thrown randomly at the backboard of the basketball goal will hit the inside rectangle? 0.14 0.21 0.26 0.27 46. Point B is between A and C. If AB = 18 inches and BC = 24 inches, what is the probability that a point chosen at random is on ̶̶ AB? 0.18 0.43 0.57 0.75 47. A skydiver jumps from an airplane and parachutes down to the 70-by-100-meter rectangular field shown. What is the probability that he will miss all three targets? 0.014 0.180 0.089 0.717 �� 48. Short Response A spinner is divided into 12 congruent regions, colored red, blue, and green. Landing on red is twice as likely as landing on blue. Landing on blue and landing on green are equally likely. a. What is the probability of landing on green? Show your work or explain in words how you got your answer. b. How many regions of the spinner are colored green? Explain your reasoning. CHALLENGE AND EXTEND 49. If you randomly choose a point

on the grid, what is the probability that it will be in a red region? 50. You are designing a target that is a square inside an 18 ft by 24 ft rectangle. What size should the square be in order for the target to have a probability of 1 __ 3? to have a probability of 3 __ 4? 51. Recreation How would you design a spinner so that 1 point is earned for landing on yellow, 3 points for landing on blue and 6 points for landing on red? Explain. SPIRAL REVIEW Simplify each expression. (Previous course) 52. (3 x 2 y) (4 x 3 y 2 ) 55. Given: A (0, 4), B (4, 6), C (4, 2), D (8, 8), and E (8, 0) 2 53. (2 m 5 ) Prove: △ABC ∼ △ADE (Lesson 7-6) 54. -8 a 4 b 6 _ 2a b 3 Find the shaded area. Round to the nearest tenth, if necessary. (Lesson 9-3) 56. 57. 636 636 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-6 Use with Lesson 9-6 Activity Use Geometric Probability to Estimate π In this lab, you will use geometric probability to estimate π. The squares in the grid below are the same width as the diameter of a penny: 0.75 in., or 19.05 mm. TEKS G.8.A Congruence and the geometry of size: find areas of... regular polygons, circles, and composite figures. 1 Toss a penny onto the grid 20 times. Let x represent the number of times the penny lands touching or covering an intersection of two grid lines. 2 Estimate π using the formula π ≈ 4 · x_. 20 Try This 1. How close is your result to π? Average the results of the entire class to get a more accurate estimate. 2. In order for a penny to touch or cover an intersection, the center of the penny can land anywhere in the shaded area. a. Find the area of the shaded region. (Hint: Each corner part is one fourth of the circle. Put the four corner parts together to form a circle with radius r.) b. Find the area of the square. c. Write

the expressions as a ratio and simplify to determine the probability of the center of the penny landing in the shaded area. 3. Explain why the formula in the activity can be used to estimate π. 9-6 Geometry Lab 637 637 � SECTION 9B Applying Geometric Formulas Step Right Up! A booster club organizes a carnival to raise money for sports uniforms. The carnival features several games that give visitors chances to win prizes. �� 1. The balloon game consists of 15 balloons attached to a vertical rectangular board with the dimensions shown. Each balloon has a diameter of 4 in. Each player throws a dart at the board and wins a prize if the dart pops a balloon. Assuming that all darts hit the board at random, what is the probability of winning a prize? �� 2. The organizers decide to make the game easier, so they double the diameter of the balloons. How does this affect the probability of winning? 3. The bean toss consists of a horizontal rectangular board that is divided into a grid. The board has coordinates (0, 0), (100, 0), (100, 60), and (0, 60). A quadrilateral on the board has coordinates A (60,0), B (100, 30), C (40, 60), and D (0, 40). Each player tosses a bean onto the board and wins a prize if the bean lands inside quadrilateral ABCD. Find the probability of winning a prize. 4. Of the three games described in Problems 1, 2, and 3, which one gives players the best chance of winning a prize? 638 638 Chapter 9 Extending Perimeter, Circumference, and Area SECTION 9B Quiz for Lessons 9-4 Through 9-6 9-4 Perimeter and Area in the Coordinate Plane Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 1. A (-2, 2), B (2, 4), C (2, -4), D (-2, -2) 2. E (-1, 5), F (3, 5), G (3, -3), H (-1, -3) Find the area of each polygon with the given vertices. 3. J (-3, 3), K (2, 2), L (-1, -3), M (-4, -1) 4

. N (-3, 1), P (3, 3), Q (5, 1), R (2, -4) 9-5 Effects of Changing Dimensions Proportionally Describe the effect of each change on the perimeter and area of the given figure. 5. The side length of the square is tripled. 6. The diagonals of a rhombus in which d 1 = 3 ft and d 2 = 9 ft are both multiplied by 1 __ 3. 7. The base and height of the rectangle are both doubled. 8. The base and height of a right triangle with base 15 in. and height 8 in. are multiplied by 1 __ 3. 9. A square has vertices (-1, 2), (3, 2), (3, -2), and (-1, -2). If you quadruple the area, what happens to the side length? 10. A restaurant sells pancakes in two sizes, silver dollar and regular. The silver-dollar pancakes have a 4-inch diameter and require 1 __ 8 cup of batter per pancake. The diameter of a regular pancake is 2.5 times the diameter of a silver-dollar pancake. About how much batter is required to make a regular pancake? 9-6 Geometric Probability Use the spinner to find the probability of each event. 11. the pointer landing on red 12. the pointer landing on red or yellow 13. the pointer not landing on green 14. the pointer landing on yellow or blue 15. A radio station plays 12 commercials per hour. Each commercial is 1 minute long. If you turn on the radio at a random time, find the probability that a commercial will be playing. Ready to Go On? 639 639 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary apothem............................. 601 circle................................ 600 center of a circle...................... 600 composite figure...................... 606 center

of a regular polygon............ 601 geometric probability................. 630 central angle of a regular polygon...... 601 Complete the sentences below with vocabulary words from the list above. 1. A(n)? is the length of a segment perpendicular to a side of a regular polygon. ̶̶̶̶ 2. The point that is equidistant from every point on a circle is the?. ̶̶̶̶ 3.? is based on a ratio of geometric measures. ̶̶̶̶ 9-1 Developing Formulas for Triangles and Quadrilaterals (pp. 589–597) E X A M P L E S EXERCISES TEKS G.1.B, G.3.C, G.3.E, G.5.A, G.8.C Find each measurement. ■ the perimeter of a square in which A = 36 in 2 Find each measurement. 4. the area of a square in which P = 36 in. A = s 2 = 36 in 2 S = √  36 = 6 in. P = 4s = 4 ⋅ 6 = 24 in. Use the Area Formula to find the side length. ■ the area of the triangle By the Pythagorean Theorem, 8 2 + b 2 = 17 2 64 + b 2 = 289 b 2 = 225, so b = 15 ft. bh = 1 _ A = 1 _ (15) (8) = 60 ft 2 2 2 ■ the diagonal d 2 of a rhombus in which A = 6 x 3 y 3 m and 4 x 2 y) d 2 2 d 2 = 3x y 2 Substitute the given values. Solve for d 2. 5. the perimeter of a rectangle in which b = 4 cm and A = 28 cm 2 6. the height of a triangle in which A = 6 x 3 y in 2 and b = 4xy in. 7. the height of the trapezoid, in which A = 48xy ft 2 8. the area of a rhombus in which d 1 = 21 yd and d 2 = 24 yd 9. the diagonal d 2 of the rhombus, in which A = 630 x 3 y 7 in 2 10. the area of a kite in which d 1 = 32 m and d

2 = 18 m 640 640 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-2 Developing Formulas for Circles and Regular Polygons (pp. 600–605) E X A M P L E S Find each measurement. ■ the circumference and area of ⊙B in terms of π C = 2πr = 2π (5xy) = 10xyπ m A = π r 2 = π (5xy) 2 = 25 x 2 y 2 π m 2 EXERCISES TEKS G.5.A, G.8.A, G.8.C Find each measurement. Round to the nearest tenth, if necessary. 11. the circumference of ⊙G 12. the area of ⊙J in which C = 14π yd ■ the area, to the nearest tenth, of a regular 13. the diameter of ⊙K in which A = 64 x 2 π m 2 hexagon with apothem 9 yd By the 30°-60°-90° Triangle Theorem, x = 9 √  3 ____ 3 = 3 √3. So s = 2x = 6 √  3, and P = 6 (6 √  3 ) = 36 √  3. aP = 1 _ A = 1 _ (9) (36 √  3 ) = 162 √  3 ≈ 280.6 yd 2 2 2 14. the area of a regular pentagon with side length 10 ft 15. the area of an equilateral triangle with side length 4 in. 16. the area of a regular octagon with side length 8 cm 17. the area of the square 9-3 Composite Figures (pp. 606–612) TEKS G.8.A E X A M P L E EXERCISES ■ Find the shaded area. Round to the nearest tenth, if necessary. Find the shaded area. Round to the nearest tenth, if necessary. 18. 19. The area of the triangle is A = 1 _ (18) (20) = 180 cm 2. 2 The area of the parallelogram is A = bh = 20 (10) = 200 cm 2. The area of the figure is the sum of the two areas. 180 + 200 = 380

cm 2 20. Study Guide: Review 641 641 �� 9-4 Perimeter and Area in the Coordinate Plane (pp. 616–621) TEKS G.7.A, G.7.B EXERCISES G.8.A Estimate the area of each irregular shape. 21. 22. Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon. 23. H (0, 3), J (3, 0), K (0, -3), L (-3, 0) 24. M (-2, 5), N (3, -2), P (-2, -2) 25. A (-2, 3), B (2, 3), C (4, -1), D (-4, -1) 26. E (-1, 3), F (3, 3), G (1, 0), H (-3, 0) Find the area of the polygon with the given vertices. 27. Q (1, 4), R (4, 3), S (2, -4), T (-3, -2) 28. V (-2, 2), W (4, 0), X (2, -3), Y (-3, 0) 29. A (1, 4), B (2, 3), C (0, -3), D (-2, -1) 30. E (-1, 2), F (2, 0), G (1, -3), H (-4, -1) E X A M P L E S ■ Estimate the area of the irregular shape. The shape has 28 approximately whole squares and 17 approximately half squares. The total area is approximately 28 + 1 _ (17) = 36.5 units 2. 2 ■ Draw and classify the polygons with vertices R (2, 4), S (3, 1), T (2, -2), and U (1, 1). Find the perimeter and area of the polygon. RSTU appears to be a rhombus. Verify this by showing that the four sides are congruent. By the Distance Formula, UR = RS = ST = TU = √  10 units.

The perimeter is 4 √  10 units. US ⋅ RT = 1 _ d 1 d 2 = 1 _ The area is A = 1 _ (2 ⋅ 6) 2 2 2 = 6 units 2. ■ Find the area of the polygon with vertices A (-3, 4), B (2, 3), C (0, -2), and D (-5, -1). area of rectangle: 7 (6) = 42 units 2 area of triangles: a: A = 1 _ (2) (5) 2 = 5 units 2 b: A = 1 _ (5) (1) 2 = 2.5 units 2 c: A = 1 _ (2) (5) = 5 units 2 2 d: A = 1 _ (5) (1) = 2.5 units 2 2 area of polygon: A = 42 - 5 - 2.5 - 5 - 2.5 = 27 units 2 642 642 Chapter 9 Extending Perimeter, Circumference, and Area �� 9-5 Effects of Changing Dimensions Proportionally (pp. 622–627) E X A M P L E EXERCISES TEKS G.5.A, G.5.B, G.11.D ■ The base and height of a rectangle with base 10 cm and height 15 cm are both doubled. Describe the effect on the area and perimeter of the figure. original: P = 2b + 2h = 2 (10) + 2 (15) = 50 cm A = bh = 10 (15) = 150 cm 2 doubled: P = 2b + 2h = 2 (20) + 2 (30) = 100 cm A = bh = 20 (30) = 600 cm 2 The perimeter increases by a factor of 2. The area increases by a factor of 4. Describe the effect of each change on the perimeter or circumference and area of the given figure. 31. The base and height of the triangle with vertices X (-1, 3), Y (-3, -2), and Z (2, -2) are tripled. 32. The side length of the square with vertices P (-1, 1), Q (3, 1), R (3, -3), and S (-1, -3) is doubled. 33. The radius of ⊙

A with radius 11 m is multiplied by 1 _. 2 34. The base and height of a triangle with base 8 ft and height 20 ft are both multiplied by 4. 9-6 Geometric Probability (pp. 630–636) TEKS G.8.A E X A M P L E S EXERCISES A point is chosen randomly on probability of each event. ̶̶ WZ. Find the A point is chosen randomly on probability of each event. ̶̶ AD. Find the ■ The point is on P (XZ) = XZ _ WZ ■ The point is on = 5 _ 6 ̶̶ XZ. = 15 _ 18 ̶̶̶ ̶̶ WX or YZ. ̶̶̶ WX ) + P ( ̶̶ YZ ) = P ( P ( = 10 _ 18 ̶̶̶ WX or = 5 _ 9 ̶̶ YZ ) = 3 _ 18 + 7 _ 18 ■ Find the probability that a point chosen randomly inside the rectangle is inside the equilateral triangle. area of rectangle A = bh = 20 (10) = 200 ft 2 area of triangle ( 5 √  3 aP = = 43.3 _ 200 ≈ 0.22 ) (30) = 25 √  3 ≈ 43.3 ft 2 35. The point is on ̶̶ AB. 36. The point is not on 37. The point is on 38. The point is on ̶̶ CD. ̶̶ CD. ̶̶ CD. ̶̶ AB or ̶̶ BC or Find the probability that a point chosen randomly inside the 40 m by 24 m rectangle is in each shape. Round to the nearest hundredth. 39. the regular hexagon 40. the triangle 41. the circle or the triangle 42. inside the rectangle but not inside the hexagon, triangle, or circle Study Guide: Review 643 643 �� Find each measurement. 1. the height h of a triangle in which A = 12 x 2 y ft 2 and b = 3x ft 2. the base b 1 of a trapezoid in which A = 161.5 cm 2, h = 17 cm, and b 2 = 13 cm 3. the area A of a kite in which d 1 = 25 in. and d 2 = 12 in. 4. Find the circumference and area of ⊙A

with diameter 12 in. Give your answers in terms of π. 5. Find the area of a regular hexagon with a side length of 14 m. Round to the nearest tenth. Find the shaded area. Round to the nearest tenth, if necessary. 6. 7. 8. The diagram shows a plan for a pond. Use a composite figure to estimate the pond’s area. The grid has squares with side lengths of 1 yd. 9. Draw and classify the polygon with vertices A (1, 5), B (2, 3), C (-2, 1), and D (-3, 3). Find the perimeter and area of the polygon. Find the area of each polygon with the given vertices. 10. E (-3, 4), F (1, 1), G (0, -4), H (-4, 1) 11. J (3, 4), K (4, -1), L (-2, -4), M (-3, 3) Describe the effect of each change on the perimeter or circumference and area of the given figure. 12. The base and height of a triangle with base 10 cm and height 12 cm are multiplied by 3. 13. The radius of a circle with radius 12 m is multiplied by 1 _. 2 14. A circular garden plot has a diameter of 21 ft. Janelle is planning a new circular plot with an area 1 __ 9 as large. How will the circumference of the new plot compare to the circumference of the old plot? A point is chosen randomly on probability of each event. ̶̶ NS. Find the 15. The point is on ̶̶̶ NQ. 16. The point is not on ̶̶̶ NQ or 17. The point is on ̶̶ RS. ̶̶ QR. 18. A shuttle bus for a festival stops at the parking lot every 18 minutes and stays at the lot for 2 minutes. If you go to the festival at a random time, what is the probability that the shuttle bus will be at the parking lot when you arrive? 644 644 Chapter 9 Extending Perimeter, Circumference, and Area �� FOCUS ON SAT STUDENT-PRODUCED RESPONSES There are two types of questions in the mathematics sections of the SAT: multiple-choice questions, where you select the correct answer from five choices, and

student-produced response questions, for which you enter the correct answer in a special grid. On the SAT, the student-produced response items do not have a penalty for incorrect answers. If you are uncertain of your answer and do not have time to rework the problem, you should still grid in the answer you have. You may want to time yourself as you take this practice test. It should take you about 9 minutes to complete. 1. A triangle has two angles with a measure of 60° and one side with a length of 12. What is the perimeter of the triangle? 4. Three overlapping squares and the coordinates of a corner of each square are shown above. What is the y-intercept of line ℓ? 2. The figure above is composed of four congruent trapezoids arranged around a shaded square. What is the area of the shaded square? 3. If △PQR ∼ △STU, m∠P = 22°, m∠Q = 57°, and m∠U = x°, what is the value of x? 5. In the figure above, what is the value of y? 6. The three angles of a triangle have measures 12x°, 3x°, and 7y°, where 7y > 60. If x and y are integers, what is the value of x? College Entrance Exam Practice 645 645 �� Any Question Type: Use a Formula Sheet When you take a standardized mathematics test, you may be given a formula sheet or a mathematics chart that accompanies the test. Although many common formulas are given on these sheets, you still need to know when the formulas are applicable, and what the variables in the formulas represent. Mathematics Chart Perimeter rectangle P = 2ℓ + 2w or P = 2 (ℓ + w) Circumference circle C = 2πr or C = πd Area rectangle A = ℓw or A = bh triangle bh or A = bh _ A = 1 _ 2 2 trapezoid or circle A = π r 2 Multiple Choice In the figure, a rectangle is inscribed in a circle. Which best represents the shaded area to the nearest tenth of a square meter? 3.4 m 2 7.6 m 2 12.6 m 2 17.2 m 2 Which formula(s) do I need? area

of a circle, area of a rectangle What do I substitute for each variable in the formulas? To use the formula for the area of a circle, I need to know the radius. The diameter of the circle is 5 m, so the radius is 2.5 m. I should substitute 2.5 for r and 3.14 for π. To use the formula for the area of a rectangle, I need to know its base and height. The base b is 4 m. To find the height, I can use the Pythagorean Theorem. 4 2 + h 2 = 5 2 16 + h 2 = 25 h 2 = 9 h = 3 What are the areas of the shapes? circle: A = π r 2 = π (2.5) 2 = 6.25π m 2 rectangle: A = bh = 4 (3) = 12 m 2 What do I do with the areas to find the answer? shaded area = area of circle - area of rectangle = 6.25π - 12 ≈ 7.6 m 2 Choice B is the correct answer. 646 646 Chapter 9 Extending Perimeter, Circumference, and Area �� Read each test problem and answer the questions that follow. Use the formula sheet below, if applicable. �� Before you begin a test, quickly review the formulas included on your formula sheet. Perimeter rectangle P = 2ℓ + 2w or P = 2 (ℓ + w) Circumference circle Area rectangle triangle trapezoid C = 2πr or C = πd A = ℓw or A = bh bh or A = bh or circle A = π r 2 Pi π π ≈ 3.14 or π ≈ 22 _ 7 Item A The circumference of a circle is 48π meters. What is the radius in meters? 6.9 meters 24 meters 12 meters 36 meters 1. Which formula would you use to solve this problem? 2. After substituting the variables in the formula, what would you need to do to find the correct answer? Item B Gridded Response The area of a trapezoid is 171 square meters. The height is 9 meters, and one base length is 23 meters. What is the other base length of the trapezoid in meters? Item C Gridded Response The area of the rectangle is 48 square miles. What is the perimeter in

miles? 5. What formula(s) would you use to solve this problem? 6. What would you substitute for each variable in the formula? 7. After substituting the variables in the formula, what would you need to do to find the correct answer? Item D Gridded Response A point is chosen randomly inside the rectangle. What is the probability that the point does not lie inside the triangle or the trapezoid? Round to the nearest hundredth. 8. Which formulas would you use to solve this problem? 9. What would you substitute for each variable 3. What formula(s) would you use to solve this in the formula(s)? problem? 4. What would you substitute for each variable in the formula? 10. After substituting the variables in the formula, what would you need to do to find the correct answer? TAKS Tackler 647 647 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–9 Multiple Choice 1. The floor of a tent is a regular hexagon. If the side length of the tent floor is 5 feet, what is the area of the floor? Round to the nearest tenth. 5. If ABCD is a rhombus in which m∠1 = (x + 15) ° and m∠2 = (2x + 12) °, what is the value of x? 32.5 square feet 65.0 square feet 75.0 square feet 129.9 square feet 2. If J is on the perpendicular bisector of ̶̶ KL, what is 3 9 18 21 the length of KL? 6. What is the area of the shaded portion of the rectangle? 34 square centimeters 36 square centimeters 38 square centimeters 50 square centimeters 7. If △XYZ is isosceles and m∠Y > 100°, which of the following must be true? m∠X < 40° m∠X > 40° ̶̶ XZ ≅ ̶̶ XY ≅ ̶̶ YZ ̶̶ XZ 8. The Eiffel Tower in Paris, France, is 300 meters tall. The first level of the tower has a height of 57 meters. A scale model of the Eiffel Tower in Shenzhen, China, is 108 meters tall. What is the height of the first level of the model? Round to the nearest tenth. 15.8 meters 20

.5 meters 56.8 meters 61.6 meters 12 18 24 36 3. What is the length of ̶̶ VY? 1.6 2 2.5 4 4. A sailor on a ship sights the light of a lighthouse at an angle of elevation of 15°. If the light in the lighthouse is 189 feet higher than the sailor’s line of sight, what is the horizontal distance between the ship and the lighthouse? Round to the nearest foot. 49 feet 51 feet 705 feet 730 feet 648 648 Chapter 9 Extending Perimeter, Circumference, and Area �� There is often more than one way to find a missing side length or angle measure in a figure. For example, you might be able to find a side length of a right triangle by using either the Pythagorean Theorem or a trigonometric ratio. Check your answer by using a different method than the one you originally used. 9. The lengths of both bases of a trapezoid are tripled. What is the effect of the change on the area of the trapezoid? The area remains the same. The area is tripled. The area increases by a factor of 6. The area increases by a factor of 9. 10. If ∠1 and ∠2 form a linear pair, which of the following must also be true about these angles? They are adjacent. They are complementary. They are congruent. They are vertical. 11. In △ABC, AB = 8, BC = 17, and AC = 2x + 1. Which of the following is a possible value of x? 3 4 9 12 12. Which line is parallel to the line with the equation y = -3x + 4? y - 3x = 8 4y - 12x = 1 3y - x = 3 2y + 6x = 5 Gridded Response 13. What is the radius of a circle in inches if the ratio of its area to its circumference is 2.5 square inches : 1 inch? 14. △JLM ∼ △RST. If JL = 5, LM = 4, RS = 3x - 1, and ST = x + 2, what is the value of x? 15. If the two diagonals of a kite measure 16 centimeters and 10 centimeters, what is the area of the kite in square centimeters? STANDARDIZED TEST PREP Short

Response 16. Two gas stations on a straight highway are 8 miles apart. If a car runs out of gas at a random point between the two gas stations, what is the probability that the car will be at least 2 miles from either gas station? Draw a diagram or write and explanation to show how you determined your answer. 17. Use the figure below to find each measure. Show your work or explain in words how you found your answers. Round the angle measure to the nearest degree. a. m∠A b. AC 18. Given that ̶̶ DE, ̶̶ DF, and ̶̶ EF are midsegments of △ABC, determine m∠C to the nearest degree. Show your work or explain in words how you determined your answer. Extended Response 19. Quadrilateral LMNP has vertices at L (1, 4), M (4, 4), N (1, 0) and P (-2, 0). a. Write a coordinate proof showing that LMNP is a parallelogram. b. Draw a rectangle with the same area as figure LMNP. Explain how you know that the figures have the same area. c. Does the rectangle you drew have the same perimeter as figure LMNP? Explain. Cumulative Assessment, Chapters 1–9 649 649 �� Spatial Reasoning 10A Three-Dimensional Figures 10-1 Solid Geometry 10-2 Representations of Three-Dimensional Figures Lab Use Nets to Create Polyhedrons 10-3 Formulas in Three Dimensions 10B Surface Area and Volume 10-4 Surface Area of Prisms and Cylinders Lab Model Right and Oblique Cylinders 10-5 Surface Area of Pyramids and Cones 10-6 Volume of Prisms and Cylinders 10-7 Volume of Pyramids and Cones 10-8 Spheres Lab Compare Surface Areas and Volumes Ext Spherical Geometry KEYWORD: MG7 ChProj Four Chromatic Gates by Herbert Bayer stands at Ervay and Federal Streets in downtown Dallas. 650 650 Chapter 10 Vocabulary Match each term on the left with a definition on the right. 1. equilateral A. the distance from the center of a regular polygon to a side of 2. parallelogram 3. apothem 4. composite figure the polygon B. a quadrilateral with four right angles C. a quadrilateral with two pairs of parallel sides D. having all sides con

gruent E. a figure made up of simple shapes, such as triangles, rectangles, trapezoids, and circles Find Area in the Coordinate Plane Find the area of each figure with the given vertices. 5. △ABC with A (0, 3), B (5, 3), and C (2, -1) 6. rectangle KLMN with K (-2, 3), L (-2, 7), M (6, 7), and N (6, 3) 7. ⊙P with center P (2, 3) that passes through the point Q (-6, 3) Circumference and Area of Circles Find the circumference and area of each circle. Give your answers in terms of π. 8. 10. 9. Distance and Midpoint Formulas Find the length and midpoint of the segment with the given endpoints. 11. A (-3, 2) and B (5, 6) 13. E (0, 1) and F (-3, 4) 12. C (-4, -4) and D (2, -3) 14. G (2, -5) and H (-2, -2) Evaluate Expressions Evaluate each expression for the given values of the variables. 15. √A_ 16. 2A_ P π for A = 121π cm 2 for A = 128 ft 2 and P = 32 ft 17. √ c 2 - a 2 for a = 8 m and c = 17 m 18. 2A_ h - b 1 for A = 60 in 2, b 1 = 8 in., and h = 6 in. Spatial Reasoning 651 651 �� Key Vocabulary/Vocabulario cone cylinder net polyhedron prism pyramid sphere cono cilindro plantilla poliedro prisma pirámide esfera surface area área total volume volumen Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following questions. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word polyhedron begins with the root poly-. List some other words that begin with poly-. What do all of these words have in common? 2. The word cone comes from the root ko-, which means “to sharpen.” Think of sharp

ening a pencil. How do you think this relates to a cone? 3. What does the word surface mean? What do you think the surface area of a three-dimensional figure is? Geometry TEKS Les. 10-1 Les. 10-2 10-3 Geo. Lab Les. 10-3 Les. 10-4 10-4 Geo. Lab Les. 10-5 Les. 10-6 Les. 10-7 Les. 10-8 G.1.C Geometric structure* compare and contrast … Euclidean and non-Euclidean geometries G.6.A Dimensionality and the geometry of location* describe and draw the intersection of a given plane with various three-dimensional … figures ★ ★ ★ G.6.B Dimensionality and the geometry of location* ★ ★ ★ ★ ★ 10-8 Tech. Lab Ext. ★ use nets to represent and construct threedimensional geometric figures G.6.C Dimensionality and the geometry of location* use orthographic and isometric views of threedimensional geometric figures … ★ G.7.C Dimensionality and the geometry of location* ★ … use formulas involving length, slope, and midpoint G.8.D Congruence and the geometry of size* find ★ ★ ★ ★ ★ surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites … G.9.D Congruence and the geometry of size* ★ ★ ★ ★ ★ analyze the characteristics of polyhedra … 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 … * Knowledge and skills are written out completely on pages TX28–TX35. 652 652 Chapter 10 Writing Strategy: Draw Three-Dimensional Figures When you encounter a three-dimensional figure such as a cylinder, cone, sphere, prism, or pyramid, it may help you to make a quick sketch so that you can visualize its shape. Use these tips to help you draw quick sketches of three-dimensional figures. ��

�� Try This 1. Explain and show how to draw a cube, a prism with equal length, width, and height. �� 2. Draw a prism, starting with two hexagons. (Hint: Draw the �� hexagons as if you were viewing them at an angle.) 3. Draw a pyramid, starting with a triangle and a point above the triangle. Spatial Reasoning 653 653 10-1 Solid Geometry TEKS G.6.A Dimensionality and the geometry of location: describe and draw the intersection of a given plane with various three-dimensional geometric figures. Objectives Classify three-dimensional figures according to their properties. Use nets and cross sections to analyze threedimensional figures. Vocabulary face edge vertex prism cylinder pyramid cone cube net cross section Why learn this? Some farmers in Japan grow cube-shaped watermelons to save space in small refrigerators. Each fruit costs about the equivalent of U.S. $80. (See Example 4.) Three-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces. Three-Dimensional Figures TERM EXAMPLE A prism is formed by two parallel congruent polygonal faces called bases connected by faces that are parallelograms. A cylinder is formed by two parallel congruent circular bases and a curved surface that connects the bases. Also G.2.B, G.6.B, G.9.D A pyramid is formed by a polygonal base and triangular faces that meet at a common vertex. A cone is formed by a circular base and a curved surface that connects the base to a vertex. A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of

their bases. Triangular prism Rectangular prism Pentagonal prism Hexagonal prism Triangular pyramid Rectangular pyramid Pentagonal pyramid Hexagonal pyramid 654 654 Chapter 10 Spatial Reasoning �� E X A M P L E 1 Classifying Three-Dimensional Figures Classify each figure. Name the vertices, edges, and bases. A � B � � � � rectangular pyramid vertices: A, B, C, D, E edges: ̶̶ AB, ̶̶ BE, ̶̶ BC, ̶̶ CE, ̶̶ CD, ̶̶ DE ̶̶ AD, ̶̶ AE, � � cylinder vertices: none edges: none base: rectangle ABCD bases: ⊙P and ⊙Q Classify each figure. Name the vertices, edges, and bases. 1a. � 1b. � � � � � � A net is a diagram of the surfaces of a three-dimensional figure that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face. E X A M P L E 2 Identifying a Three-Dimensional Figure From a Net Describe the three-dimensional figure that can be made from the given net. A B The net has two congruent triangular faces. The remaining faces are parallelograms, so the net forms a triangular prism. The net has one square face. The remaining faces are triangles, so the net forms a square pyramid. Describe the three-dimensional figure that can be made from the given net. 2a. 2b. 10- 1 Solid Geometry 655 655 A cross section is the intersection of a three-dimensional figure and a plane. E X A M P L E 3 Describing Cross Sections of Three-Dimensional Figures Describe each cross section. A B The cross section is a triangle. The cross section is a circle. Describe each cross section. 3a. 3b. E X A M P L E 4 Food Application A chef is slicing a cube-shaped watermelon for a buffet. How can the chef cut the watermelon to make a slice of each shape? A a square B a hexagon Cut parallel to the bases. Cut through the midpoints of the edges. 4. How can a chef cut a cube-shaped watermelon to make slices that are triangles? THINK AND DIS

CUSS 1. Compare prisms and cylinders. 2. GET ORGANIZED Copy and complete the graphic organizer. 656 656 Chapter 10 Spatial Reasoning �� 10-1 Exercises Exercises KEYWORD: MG7 10-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary A? has two circular bases. (prism, cylinder, or cone) ̶̶̶̶ Classify each figure. Name the vertices, edges, and bases. p. 655 2. � 3. � � � 4. 4 Describe the three-dimensional figure that can be made from the given net. p. 655 5 Describe each cross section. p. 656 8. 6. 9. 7. 10. 656 Art A sculptor has a cylindrical piece of clay. How can the sculptor slice the clay to make a slice of each given shape? 11. a circle 12. a rectangle Independent Practice For See Exercises Example 13–15 16–18 19–21 22–23 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Classify each figure. Name the vertices, edges, and bases. 13. 14. 15. Describe the three-dimensional figure that can be made from the given net. 16. 17. 18. 10- 1 Solid Geometry 657 657 �� Describe each cross section. 19. 20. 21. Architecture An architect is drawing plans for a building that is a hexagonal prism. How could the architect draw a cutaway of the building that shows a cross section in the shape of each given figure? 22. a hexagon 23. a rectangle Name a three-dimensional figure from which a cross section in the given shape can be made. 24. square 25. rectangle 26. circle 27. hexagon Write a verbal description of each figure. 28. 29. 30. Draw and label a figure that meets each description. 31. rectangular prism with length 3 cm, width 2 cm, and height 5 cm 32. regular pentagonal prism with side length 6 in. and height 8 in. 33. cylinder with radius 4 m and height 7 m Draw a net for each three-dimensional figure. 34. 35. 36. 37. This problem will prepare you for

the Multi-Step TAKS Prep on page 678. A manufacturer of camping gear makes a wall tent in the shape shown in the diagram. a. Classify the three-dimensional figure that the wall tent forms. b. What shapes make up the faces of the tent? How many of each shape are there? c. Draw a net for the wall tent. 658 658 Chapter 10 Spatial Reasoning �� 38. /////ERROR ANALYSIS///// A regular hexagonal prism is intersected by a plane as shown. Which cross section is incorrect? Explain. 39. Critical Thinking A three-dimensional figure has 5 faces. One face is adjacent to every other face. Four of the faces are congruent. Draw a figure that meets these conditions. 40. Write About It Which of the following figures is not a net for a cube? Explain. a. b. c. d. 41. Which three-dimensional figure does the net represent? 42. Which shape CANNOT be a face of a hexagonal prism? triangle hexagon parallelogram rectangle 43. What shape is the cross section formed by a cone and a plane that is perpendicular to the base and that passes through the vertex of the cone? circle triangle trapezoid rectangle 44. Which shape best represents a hexagonal prism viewed from the top? 10- 1 Solid Geometry 659 659 �� CHALLENGE AND EXTEND A double cone is formed by two cones that share the same vertex. Sketch each cross section formed by a double cone and a plane. 45. 46. 47. Crafts Elena is designing patterns for gift boxes. Draw a pattern that she can use to create each box. Be sure to include tabs for gluing the sides together. 48. a box that is a square pyramid where each triangular face is an isosceles triangle with a height equal to three times the width 49. a box that is a cylinder with the diameter equal to the height 50. a box that is a rectangular prism with a base that is twice as long as it is wide, and with a rectangular pyramid on the top base 51. A net of a prism is shown. The bases of the prism are regular hexagons, and the rectangular faces are all congruent. a. List all pairs of parallel faces in the prism. b. Draw a net of a prism with bases that are regular pentagons. How many pairs of parallel faces does the prism

have? � � � � � � � � SPIRAL REVIEW Write the equation that fits the description. (Previous course) 52. the equation of the graph that is the reflection of the graph of y = x 2 over the x-axis 53. the equation of the graph of y = x 2 after a vertical translation of 6 units upward 54. the quadratic equation of a graph that opens upward and is wider than y = x 2 Name the largest and smallest angles of each triangle. (Lesson 5-5) 55. � 56. � � � � � � �� 57. � �� Determine whether the two polygons are similar. If so, give the similarity ratio. (Lesson 7-2) 58. � � � � � �� 59. �� 660 660 Chapter 10 Spatial Reasoning 10-2 Representations of Three-Dimensional Figures TEKS G.9.D Congruence and the geometry of size: analyze the characteristics of polyhedra and other three-dimensional figures.... Also G.6.C Objectives Draw representations of three-dimensional figures. Recognize a threedimensional figure from a given representation. Vocabulary orthographic drawing isometric drawing perspective drawing vanishing point horizon Who uses this? Architects make many different kinds of drawings to represent three-dimensional figures. (See Exercise 34.) There are many ways to represent a threedimensional object. An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side. First National Bank, San Angelo E X A M P L E 1 Drawing Orthographic Views of an Object Draw all six orthographic views of the given object. Assume there are no hidden cubes. Top: Bottom: Front: Back: Left: Right: 1. Draw all six orthographic views of the given object. Assume there are no hidden cubes. 10- 2 Representations of Three-Dimensional Figures 661 661 �� Isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern. E X A M P L E 2 Drawing an Isometric View of an Object Draw an isometric view of the given object. Assume there are no

hidden cubes. 2. Draw an isometric view of the given object. Assume there are no hidden cubes. In a perspective drawing, nonvertical parallel lines are drawn so that they meet at a point called a vanishing point. Vanishing points are located on a horizontal line called the horizon. A one-point perspective drawing contains one vanishing point. A two-point perspective drawing contains two vanishing points. Perspective Drawing When making a perspective drawing, it helps me to remember that all vertical lines on the object will be vertical in the drawing. Jacob Martin MacArthur High School Three-dimensional figure One-point perspective Two-point perspective 662 662 Chapter 10 Spatial Reasoning �� E X A M P L E 3 Drawing an Object in Perspective A Draw a cube in one-point perspective. Draw a horizontal line to represent the horizon. Mark a vanishing point on the horizon. Then draw a square below the horizon. This is the front of the cube. From each corner of the square, lightly draw dashed segments to the vanishing point. Lightly draw a smaller square with vertices on the dashed segments. This is the back of the cube. Draw the edges of the cube, using dashed segments for hidden edges. Erase any segments that are not part of the cube. B Draw a rectangular prism in two-point perspective. In a one-point perspective drawing of a cube, you are looking at a face. In a two-point perspective drawing, you are looking at a corner. Draw a horizontal line to represent the horizon. Mark two vanishing points on the horizon. Then draw a vertical segment below the horizon and between the vanishing points. This is the front edge of the prism. From each endpoint of the segment, lightly draw dashed segments to each vanishing point. Draw two vertical segments connecting the dashed lines. These are other vertical edges of the prism. Lightly draw dashed segments from each endpoint of the two vertical segments to the vanishing points. Draw the edges of the prism, using dashed lines for hidden edges. Erase any lines that are not part of the prism. 3a. Draw the block letter L in one-point perspective. 3b. Draw the block letter L in two-point perspective. 10- 2 Representations of Three-Dimensional Figures 663 663 E X A M P L E 4 Relating Different Representations of an Object Relating Different Representations of an Object Determine whether each drawing represents the

given object. Assume there are no hidden cubes. B D A C Yes; the drawing is a one-point perspective view of the object. No; the cubes that share a face in the object do not share a face in the drawing. No; the figure in the drawing is made up of four cubes, and the object is made up of only three cubes. Yes; the drawing shows the six orthographic views of the object. 4. Determine whether the drawing represents the given object. Assume there are no hidden cubes. THINK AND DISCUSS 1. Describe the six orthographic views of a cube. 2. In a perspective drawing, are all parallel lines drawn so that they meet at a vanishing point? Why or why not? 3. GET ORGANIZED Copy and complete the graphic organizer. 664 664 Chapter 10 Spatial Reasoning �� 10-2 Exercises Exercises KEYWORD: MG7 10-2 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In a(n)? drawing, the vanishing points are located on the ̶̶̶̶ horizon. (orthographic, isometric, or perspective Draw all six orthographic views of each object. Assume there are no hidden cubes. p. 661 2. 3. 662 Draw an isometric view of each object. Assume there are no hidden cubes. 5. 6. 4. 7. 663 Draw each object in one-point and two-point perspectives. Assume there are no hidden cubes. 8. rectangular prism 9. block letter. 664 Determine whether each drawing represents the given object. Assume there are no hidden cubes. 11. 10. 12. 13. PRACTICE AND PROBLEM SOLVING Draw all six orthographic views of each object. Assume there are no hidden cubes. 14. 15. 16. 10- 2 Representations of Three-Dimensional Figures 665 665 �� Independent Practice Draw an isometric view of each object. Assume there are no hidden cubes. For See Exercises Example 17. 18. 19. 14–16 17–19 20–21 22–25 1 2 3 4 TE

KS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 Draw each object in one-point and two-point perspective. Assume there are no hidden cubes. 20. right triangular prism 21. block letter Determine whether each drawing represents the given object. Assume there are no hidden cubes. 22. 23. 24. 25. 26. Use the top, front, side, and isometric views to build the three-dimensional figure out of unit cubes. Then draw the figure in one-point perspective. Use the top, side, and front views to draw an isometric view of each figure. 27. 28. 29. This problem will prepare you for the Multi-Step TAKS Prep on page 678. A camping gear catalog shows the three given views of a tent. a. Draw a bottom view of the tent. b. Make a sketch of the tent. c. Each edge of the three-dimensional figure from part b represents one pole of the tent. How many poles does this tent have? 666 666 Chapter 10 Spatial Reasoning �� Draw all six orthographic views of each object. 30. 31. 32. 33. Critical Thinking Describe or draw two figures that have the same left, right, front, and back orthographic views but have different top and bottom views. 34. Architecture Perspective drawings are used by architects to show what a finished room will look like. a. Is the architect’s sketch in one-point or two-point perspective? b. Write About It How would you locate the vanishing point(s) in the architect’s sketch? 35. Which three-dimensional figure has these three views? 36. Which drawing best represents the top view of the three-dimensional figure? 37. Short Response Draw a one-point perspective view and an isometric view of a triangular prism. Explain how the two drawings are different. 10- 2 Representations of Three-Dimensional Figures 667 667 �� CHALLENGE AND EXTEND Draw each figure using one-point perspective. (Hint: First lightly draw a rectangular prism. Enclose the figure in the prism.) 38. an octagonal prism 39. a cylinder 40. a cone 41. A frustum of a cone is a part of a cone with two parallel bases. Copy the diagram of the frustum of a cone. a. Draw

the entire cone. b. Draw all six orthographic views of the frustum. c. Draw a net for the frustum. 42. Art Draw a one-point or two-point perspective drawing of the inside of a room. Include at least two pieces of furniture drawn in perspective. SPIRAL REVIEW Find the two numbers. (Previous course) 43. The sum of two numbers is 30. The difference between 2 times the first number and 2 times the second number is 20. 44. The difference between the first number and the second number is 7. When the second number is added to 4 times the first number, the result is 38. 45. The second number is 5 more than the first number. Their sum is 5. For A (4, 2), B (6, 1), C (3, 0), and D (2, 0), find the slope of each line. (Lesson 3-5) 46.  AB 48.  AD 47.  AC Describe the faces of each figure. (Lesson 10-1) 49. pentagonal prism 50. cube 51. triangular pyramid Using Technology You can use geometry software to draw figures in one- and two-point perspectives. 1. a. Draw a horizontal line to represent the horizon. Create a vanishing point on the horizon. Draw a rectangle with two sides parallel to the horizon. Draw a segment from each vertex to the vanishing point. b. Draw a smaller rectangle with vertices on the segments that intersect the horizon. Hide these segments and complete the figure. c. Drag the vanishing point to different locations on the horizon. Describe what happens to the figure. 2. Describe how you would use geometry software to draw a figure in two-point perspective. 668 668 Chapter 10 Spatial Reasoning 10-3 Use with Lesson 10-3 Activity Use Nets to Create Polyhedrons A polyhedron is formed by four or more polygons that intersect only at their edges. The faces of a regular polyhedron are all congruent regular polygons, and the same number of faces intersect at each vertex. Regular polyhedrons are also called Platonic solids. There are exactly five regular polyhedrons. TEKS G.6.B Dimensionality and the geometry of location: use nets to represent and construct three-dimensional geometric figures. Also G

.9.D Use geometry software or a compass and straightedge to create a larger version of each net on heavy paper. Fold each net into a polyhedron. NAME FACES EXAMPLE NET REGULAR POLYHEDRONS Tetrahedron 4 triangles Octahedron 8 triangles Icosahedron 20 triangles Cube 6 squares Dodecahedron 12 pentagons Try This 1. Complete the table for the number of vertices V, edges E, and faces F for each of the polyhedrons you made in Activity 1. 2. Make a Conjecture What do you think is true about the relationship between the number of vertices, edges, and faces of a polyhedron? POLYHEDRON V E F V - E + F Tetrahedron Octahedron Icosahedron Cube Dodecahedron 10-3 Geometry Lab 669 669 10-3 Formulas in Three Dimensions TEKS G.7.C Dimensionality and the geometry of location: develop and use formulas involving length, slope, and midpoint. Also G.5.A, G.8.C, G.9.D Objectives Apply Euler’s formula to find the number of vertices, edges, and faces of a polyhedron. Develop and apply the distance and midpoint formulas in three dimensions. Vocabulary polyhedron space Why learn this? Divers can use a three-dimensional coordinate system to find distances between two points under water. (See Example 5.) A polyhedron is formed by four or more polygons that intersect only at their edges. Prisms and pyramids are polyhedrons, but cylinders and cones are not. Polyhedrons Not polyhedrons In the lab before this lesson, you made a conjecture about the relationship between the vertices, edges, and faces of a polyhedron. One way to state this relationship is given below. Euler’s Formula For any polyhedron with V vertices, E edges, and F faces, V - E + F = 2. E X A M P L E 1 Using Euler’s Formula Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. A B Euler is pronounced “Oiler.” V = 4, E = 6 = 10, E = 15, F = 7 Use Euler’s formula.

Simplify. 10 - 15 + 7 ≟ 2 2 = 2 Find the number of vertices, edges, and faces of the polyhedron. Use your results to verify Euler’s formula. 1a. 1b. 670 670 Chapter 10 Spatial Reasoning A diagonal of a three-dimensional figure connects two vertices of two different faces. Diagonal d of a rectangular prism is shown in the diagram. By the Pythagorean Theorem, ℓ 2 + w 2 = x 2, and x 2 + h 2 = d 2. Using substitution. Diagonal of a Right Rectangular Prism The length of a diagonal d of a right rectangular prism with length ℓ, width w, and height h is d = √  Using the Pythagorean Theorem in Three Dimensions Find the unknown dimension in each figure. A the length of the diagonal of a 3 in. by 4 in. by 5 in. rectangular prism 3 2 + 4 2 + 5 2 d = √  = √  9 + 16 + 25 = √  50 ≈ 7.1 in. Substitute 3 for ℓ, 4 for w, and 5 for h. Simplify. B the height of a rectangular prism with an 8 ft by 12 ft base and an 18 ft diagonal 18 = √  8 2 + 12 2 + h 2 18 2 = ( √  8 2 + 12 2 + h 2 ) 324 = 64 + 144 + h 2 h 2 = 116 h = √  116 ≈ 10.8 ft 2 Substitute 18 for d, 8 for ℓ, and 12 for w. Square both sides of the equation. Simplify. Solve for h 2. Take the square root of both sides. 2. Find the length of the diagonal of a cube with edge length 5 cm. Space is the set of all points in three dimensions. Three coordinates are needed to locate a point in space. A three-dimensional coordinate system has 3 perpendicular axes: the x-axis, the y-axis, and

the z-axis. An ordered triple (x, y, z) is used to locate a point. To locate the point (3, 2, 4), start at (0, 0, 0). From there move 3 units forward, 2 units right, and then 4 units up. E X A M P L E 3 Graphing Figures in Three Dimensions Graph each figure. A a cube with edge length 4 units and one vertex at (0, 0, 0) The cube has 8 vertices: (0, 0, 0), (0, 4, 0), (0, 0, 4), (4, 0, 0), (4, 4, 0), (4, 0, 4), (0, 4, 4), (4, 4, 4). 10- 3 Formulas in Three Dimensions 671 671 �� Graph each figure. B a cylinder with radius 3 units, height 5 units, and one base centered at (0, 0, 0) Graph the center of the bottom base at (0, 0, 0). Since the height is 5, graph the center of the top base at (0, 0, 5). The radius is 3, so the bottom base will cross the x-axis at (3, 0,0) and the y-axis at (0, 3, 0). Draw the top base parallel to the bottom base and connect the bases. 3. Graph a cone with radius 5 units, height 7 units, and the base centered at (0, 0, 0). You can find the distance between the two points ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) by drawing a rectangular prism with the given points as endpoints of a diagonal. Then use the formula for the length of the diagonal. You can also use a formula related to the Distance Formula. (See Lesson 1-6.) The formula for the midpoint between ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is related to the Midpoint Formula. (See Lesson 1-6.) Distance and Midpoint Formulas in Three Dimensions The distance between the points ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is d = √ �

�� (. The midpoint of the segment with endpoints ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) is Finding Distances and Midpoints in Three Dimensions Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. A (0, 0, 0) and (3, 4, 12) distance: d = √  ( = √  (3 - 0) 2 + (4 - 0) 2 + (12 - 0) 2 = √  9 + 16 + 144 = √  169 = 13 units midpoint + 12 1.5, 2, 6) 672 672 Chapter 10 Spatial Reasoning �� Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. B (3, 8, 10) and (7, 12, 15) distance: d = √  midpoint ( ) 10 + 15 8 + 12 3 + 7 _ _ _ (7 - 3) 2 + (12 - 8) 2 + (15 - 10) 2,, 2 2 2 M (5, 10, 12.5) = √  = √ �

�� 16 + 16 + 25 = √  57 ≈ 7.5 units Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 4a. (0, 9, 5) and (6, 0, 12) 4b. (5, 8, 16) and (12, 16, 20) E X A M P L E 5 Recreation Application Two divers swam from a boat to the locations shown in the diagram. How far apart are the divers? Recreation Divers in the Comal Springs in New Braunfels can see a variety of plant and animal life. The springs, which are fed by the Edwards Aquifer, have a constant temperature of 68°F and are home to many endangered species. The location of the boat can be represented by the ordered triple (0, 0, 0), and the locations of the divers can be represented by the ordered triples (18, 9, -8) and (-15, -6, -12). d = √  ( = √  = √  (-15 - 18) 2 + (-6 - 9) 2 + (-12 + 8) 2 1330 ≈ 36.5 ft c10l03002a Use the Distance Formula to find the distance between the divers. 5. What if…? If both divers swam straight up to the surface, how far apart would they be? THINK AND DISCUSS 1. Explain how to find the distance between two points in a three-dimensional coordinate system. 2. GET ORGANIZED Copy and complete the graphic organizer. 10- 3 Formulas in Three Dimensions 673 673 Depth: 8 ft9 ft18 ft15 ft6 ftDepth: 12 ftge07se_c10l03002aAB�� 10-3 Exercises Exercises KEYWORD: MG7 10-3 KEYWORD:

MG7 Parent GUIDED PRACTICE 1. Vocabulary Explain why a cylinder is not a polyhedron. 670 Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 2. 3. 4 Find the unknown dimension in each figure. Round to the nearest tenth, if necessary. p. 671 5. the length of the diagonal of a 4 ft by 8 ft by 12 ft rectangular prism 6. the height of a rectangular prism with a 6 in. by 10 in. base and a 13 in. diagonal 7. the length of the diagonal of a square prism with a base edge length of 12 in. and a height of 1 in Graph each figure. p. 671 8. a cone with radius 8 units, height 4 units, and the base centered at (0, 0, 0) 9. a cylinder with radius 3 units, height 4 units, and one base centered at (0, 0, 0) 10. a cube with edge length 7 units and one vertex at (0, 0, 0. 672. 673 Independent Practice For See Exercises Example 15–17 18–20 21–23 24–26 27 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 11. (0, 0, 0) and (5, 9, 10) 12. (0, 3, 8) and (7, 0, 14) 13. (4, 6, 10) and (9, 12, 15) 14. Recreation 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? PRACTICE AND PROBLEM SOLVING Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 15. 16. 17. Find the unknown dimension in each figure. Round to the nearest tenth, if necessary. 18. the length of the diagonal of a 7 yd by 8 yd by 16 yd rectangular prism 19. the height of a rectangular prism with

a 15 m by 6 m base and a 17 m diagonal 20. the edge length of a cube with an 8 cm diagonal 674 674 Chapter 10 Spatial Reasoning Meteorology A typical cumulus cloud weighs about 1.4 billion pounds, which is more than 100,000 elephants. Source: usgs.gov Graph each figure. 21. a cylinder with radius 5 units, height 3 units, and one base centered at (0, 0, 0) 22. a cone with radius 2 units, height 4 units, and the base centered at (0, 0, 0) 23. a square prism with base edge length 5 units, height 3 units, and one vertex at (0, 0, 0) Find the distance between the given points. Find the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 24. (0, 0, 0) and (4, 4, 4) 25. (2, 3, 7) and (9, 10, 10) 26. (2, 5, 3) and (8, 8, 10) 27. Meteorology A cloud has an elevation of 6500 feet. A raindrop falling from the cloud was blown 700 feet south and 500 feet east before it hit the ground. How far did the raindrop travel from the cloud to the ground? 28. Multi-Step Find the length of a diagonal of the rectangular prism at right. If the length, width, and height are doubled, what happens to the length of the diagonal? For each three-dimensional figure, find the missing value and draw a figure with the correct number of vertices, edges, and faces. Vertices V Edges E Faces F Diagram 5 8 7 29. 30. 31. 8 12 9 5 5 7 32. Algebra Each base of a prism is a polygon with n sides. Write an expression for the number of vertices V, the number of edges E, and the number of faces F in terms of n. Use your results to show that Euler’s formula is true for all prisms. 33. Algebra The base of a pyramid is a polygon with n sides. Write an expression for the number of vertices V, the number of edges E, and the number of faces F in terms of n. Use your results to show that Euler’s formula is true for all pyramids. 34. This problem will prepare you for the Multi-Step TAK

S Prep on page 678. ̶̶̶ NM ≅ ̶̶ NP The tent at right is a triangular prism where ̶̶ KJ ≅ ̶̶ KL and has the given dimensions. and a. The tent manufacturer sets up the tent on a coordinate system so that J is at the origin and M has coordinates (7, 0, 0). Find the coordinates of the other vertices. b. The manufacturer wants to know the distance from K to P in order to make an extra support pole for the tent. Find KP to the nearest tenth. � �� 10- 3 Formulas in Three Dimensions 675 675 �� Find the missing dimension of each rectangular prism. Give your answers in simplest radical form. Length ℓ Width w Height h Diagonal d 6 in. 24 12 35. 36. 37. 38. 6 in. 18 2 6 in. 60 3 65 24 4 Graph each figure. 39. a cylinder with radius 4 units, height 5 units, and one base centered at (1, 2, 5) 40. a cone with radius 3 units, height 7 units, and the base centered at (3, 2, 6) 41. a cube with edge length 6 units and one vertex at (4, 2, 3) 42. a rectangular prism with vertices at (4, 2, 5), (4, 6, 5), (4, 6, 8), (8, 6, 5), (8, 2, 5), (8, 6, 8), (4, 2, 8), and (8, 2, 8) 43. a cone with radius 4 units, the vertex at (4, 7, 8), and the base centered at (4, 7, 1) 44. a cylinder with a radius of 5 units and bases centered at (2, 3, 7) and (2, 3, 15) Graph each segment with the given endpoints in a three-dimensional coordinate system. Find the length and midpoint of each segment. 46. (4, 3, 3) and (7, 4, 4) 45. (1, 2, 3) and (3, 2, 1) 47. (4, 7, 8) and (3, 1, 5) 48. (0, 0, 0) and (8, 3, 6) 49. (6, 1, 8) and

(2, 2, 6) 50. (2, 8, 5) and (3, 6, 3) 51. Multi-Step Find z if the distance between R (6, -1, -3) and S (3, 3, z) is 13. 52. Draw a figure with 6 vertices and 6 faces. 53. Estimation Measure the net for a rectangular prism and estimate the length of a diagonal. 54. Make a Conjecture What do you think is the longest segment joining two points on a rectangular prism? Test your conjecture using at least three segments whose endpoints are on the prism with vertices A (0, 0, 0), B (1, 0, 0), C (1, 2, 0), D (0, 2, 0), E (0, 0, 2), F (1, 0, 2), G (1, 2, 2), and H (0, 2, 2). 55. Critical Thinking The points A (3, 2, -3), B (5, 8, 6), and C (-3, -5, 3) form a triangle. Classify the triangle by sides and angles. 56. Write About It A cylinder has a radius of 4 and a height of 6. What is the length of the longest segment with both endpoints on the cylinder? Describe the location of the endpoints and explain why it is the longest possible segment. 676 676 Chapter 10 Spatial Reasoning 57. How many faces, edges, and vertices does a hexagonal pyramid have? 6 faces, 10 edges, 6 vertices 7 faces, 12 edges, 7 vertices 7 faces, 10 edges, 7 vertices 8 faces, 18 edges, 12 vertices 58. Which is closest to the length of the diagonal of the rectangular prism with length 12 m, width 8 m, and height 6 m? 6.6 m 44 m 15.6 m 244.0 m 59. What is the distance between the points (7, 14, 8) and (9, 3, 12) to the nearest tenth? 10.9 11.9 119.0 141.0 CHALLENGE AND EXTEND 60. Multi-Step The bases of the right hexagonal prism are regular hexagons with side length a, and the height of the prism is h. Find the length of the indicated diagonal in terms of a and h. 61. Determine if the points A

(-1, 2, 4), B (1, -2, 6), and C (3, -6, 8) are collinear. 62. Algebra Write a coordinate proof of the Midpoint Formula using the Distance Formula. Given: points ), and M ( Prove: A, B, and M are collinear, and AM = MB _____ _____ _____ 2, 2, 2 63. Algebra Write a coordinate proof that the diagonals of a rectangular prism are congruent and bisect each other. Given: a rectangular prism with vertices A (0, 0, 0), B (a, 0, 0), C (a, b, 0), D (0, b, 0), E (0, 0, c), F (a, 0, c), G (a, b, c), and H (0, b, c) ̶̶ AG and ̶̶ BH are congruent and bisect each other. Prove: SPIRAL REVIEW The histogram shows the number of people by age group who attended a natural history museum opening. Find the following. (Previous course) 64. the number of people between 10 and 29 years of age that were in attendance 65. the age group that had the greatest number of people in attendance Write a formula for the area of each figure after the given change. (Lesson 9-5) 66. A parallelogram with base b and height h has its height doubled. 67. A trapezoid with height h and bases b 1 and b 2 has its base b 1 multiplied by 1 __ 2. 68. A circle with radius r has its radius tripled. Use the diagram for Exercises 69–71. (Lesson 10-1) 69. Classify the figure. 70. Name the edges. 71. Name the base. 10- 3 Formulas in Three Dimensions 677 677 �� SECTION 10A Three-Dimensional Figures Your Two Tents A manufacturer of camping gear offers two types of tents: an A-frame tent and a pyramid tent. A-frame tent Pyramid tent 1. The manufacturer’s catalog shows the top, front, and side views of each tent. It shows a two-dimensional shape for each that can be folded to form the three-dimensional

shape of the tent. Draw the catalog display for each tent. The manufacturer uses a three-dimensional coordinate system to represent the vertices of each tent. Each unit of the coordinate system represents one foot. 2. Which tent offers a greater sleeping area? 3. Compare the heights of the tents. Which tent offers more headroom? 4. A camper wants to purchase the tent that has shorter support poles so that she can fit the folded tent in ̶̶ EF in her car more easily. Find the length of pole ̶̶ TR in the the A-frame tent and the length of pole pyramid tent. Which tent should the camper buy? A-Frame Tent Vertex Coordinates A B C D E F (0, 0, 0) (0, 7, 0) (0, 3.5, 7) (8, 0, 0) (8, 7, 0) (8, 3.5, 7) Pyramid Tent Vertex Coordinates P Q R S T (0, 0, 0) (8, 0, 0) (8, 8, 0) (0, 8, 0) (4, 4, 8) 678 678 Chapter 10 Spatial Reasoning Camping at Big Bend National Park �� SECTION 10A Quiz for Lessons 10-1 Through 10-3 10-1 Solid Geometry Classify each figure. Name the vertices, edges, and bases. 1. 2. 3. Describe the three-dimensional figure that can be made from the given net. 4. Describe each cross section. 7. 5. 8. 6. 9. 10-2 Representations of Three-Dimensional Figures Use the figure made of unit cubes for Problems 10 and 11. Assume there are no hidden cubes. 10. Draw all six orthographic views. 11. Draw an isometric view. 12. Draw the block letter T in one-point perspective. 13. Draw the block letter T in two-point perspective. 10-3 Formulas in Three Dimensions Find the number of vertices, edges, and faces of each polyhedron. Use your results to verify Euler’s formula. 14. a square prism 15. a hexagonal pyramid 16. a triangular pyramid 17. A bird flies from its nest to a point that is 6 feet north, 7 feet west, and 6 feet higher in the tree than the nest. How far is the bird from the nest? Find the distance between the given points. Find

the midpoint of the segment with the given endpoints. Round to the nearest tenth, if necessary. 19. (3, 1, -2) and (5, -5, 7) 18. (0, 0, 0) and (4, 6, 12) 20. (3, 5, 9) and (7, 2, 0) Ready to Go On? 679 679 �� 10-4 Surface Area 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 surface area of a prism. Learn and apply the formula for the surface area of a cylinder. Vocabulary lateral face lateral edge right prism oblique prism altitude surface area lateral surface axis of a cylinder right cylinder oblique cylinder Also G.5.A, G.5.B, G.6.B, G.11.D Why learn this? The surface area of ice affects how fast it will melt. If the surface exposed to the air is increased, the ice will melt faster. (See Example 5.) Prisms and cylinders have 2 congruent parallel bases. A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face. An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude. �� Surface area is the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces. The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism. Lateral Area and Surface Area of Right Prisms The lateral area of a right prism with base perimeter P and height h is L = Ph. The surface area of a right prism with lateral area L and base area B is S = L + 2B, or S = Ph + 2B. The surface area of a cube with edge length s is S = 6 s 2. The surface area

of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh. 680 680 Chapter 10 Spatial Reasoning �� E X A M P L E 1 Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of each right prism. Round to the nearest tenth, if necessary. A the rectangular prism L = Ph = (28) 12 = 336 cm 2 S = Ph + 2B = 336 + 2 (6) (8) = 432 cm 2 P = 2 (8) + 2 (6) = 28 cm B the regular hexagonal prism The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces. L = Ph = 36 (10) = 360 m 2 S = Ph + 2B = 360 + 2 (54 √  3 ) ≈ 547.1 m 2 P = 6 (6) = 36 m The base area is B = 1 __ aP = 54 √  3 m. 2 1. Find the lateral area and surface area of a cube with edge length 8 cm. The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis. Lateral Area and Surface Area of Right Cylinders The lateral area of a right cylinder with radius r and height h is L = 2πrh. The surface area of a right cylinder with lateral area L and base area B is S = L + 2B, or S = 2πrh + 2π r 2. 10- 4 Surface Area of Prisms and Cylinders 681 681 �� E X A M P L E 2 Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. A L = 2πrh = 2π

(1) (5) = 10π m 2 S = L + 2π r 2 = 10π + 2π (1) 2 = 12π m 2 The radius is half the diameter, or 1 m. B a cylinder with a circumference of 10π cm and a height equal to 3 times the radius Step 1 Use the circumference to find the radius. C = 2πr 10π = 2πr r = 5 Circumference of a circle Substitute 10π for C. Divide both sides by 2π. Step 2 Use the radius to find the lateral area and surface area. The height is 3 times the radius, or 15 cm. L = 2π rh = 2π (5) (15) = 150π cm 2 S = 2π rh + 2π r 2 = 150π + 2π (5) 2 = 200π cm 2 Lateral area Surface area 2. Find the lateral area and surface area of a cylinder with a base area of 49π and a height that is 2 times the radius. E X A M P L E 3 Finding Surface Areas of Composite Three-Dimensional Figures Always round at the last step of the problem. Use the value of π given by the π key on your calculator. Find the surface area of the composite figure. Round to the nearest tenth. The surface area of the right rectangular prism is S = Ph + 2B = 80 (20) + 2 (24) (16) = 2368 ft 2. A right cylinder is removed from the rectangular prism. The lateral area is L = 2π rh = 2π (4) (20) = 160π ft 2. The area of each base is B = π r 2 = π (4) 2 = 16π ft 2. The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure. S = (prism surface area) + (cylinder lateral area) - (cylinder base area) = 2368 + 160π -2 (16π) = 2368 + 128π ≈ 2770.1 ft 2 3. Find the surface area of the composite figure. Round to the nearest tenth. 682 682 Chapter 10 Spatial Reasoning �� E X A M P L E 4 Exploring Effects of Changing Dimensions The length, width, and height of the right rectangular prism are doubled. Describe the effect on the surface area.

original dimensions: length, width, and height doubled: S = Ph + 2B = 16 (3) + 2 (6) (2) = 72 in 2 S = Ph + 2B = 32 (6) + 2 (12) (4) = 288 in 2 Notice that 288 = 4 (72). If the length, width, and height are doubled, the surface area is multiplied by 2 2, or 4. 4. The height and diameter of the cylinder are multiplied by 1 __ 2. Describe the effect on the surface area. E X A M P L E 5 Chemistry Application Entertainment Ice sculptures are carved from blocks of ice that weigh hundreds of pounds. A typical ice sculpture can last 6–8 hours at room temperature (70°F). If two pieces of ice have the same volume, the one with the greater surface area will melt faster because more of it is exposed to the air. One piece of ice shown is a rectangular prism, and the other is half a cylinder. Given that the volumes are approximately equal, which will melt faster? �� rectangular prism: S = Ph + 2B = 12 (3) + 2 (8) = 52 cm 2 �� half cylinder: S = πrh + π r 2 + 2rh = π (4) (1) + π (4) 2 + 8 (1) �� = 20π + 8 ≈ 70.8 cm 2 �� The half cylinder of ice will melt faster. Use the information above to answer the following. 5. A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces above. Compare the surface areas. Which will melt faster? THINK AND DISCUSS 1. Explain how to find the surface area of a cylinder if you know the �� lateral area and the radius of the base. 2. Describe the difference between an oblique prism and a right prism. 3. GET ORGANIZED Copy and complete the graphic organizer. Write the formulas in each box. 10- 4 Surface Area of Prisms and Cylinders 683 683 �� 10-4 Exercises Exercises KEYWORD: MG7 10-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary How many lateral faces does a pentagonal

prism have Find the lateral area and surface area of each right prism. p. 681 2. 3. 4. a cube with edge length 9 inches. 682 Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. 5. 6. 7. a cylinder with base area 64π m 2 and a height 3 meters less than the radius. 682 Multi-Step Find the surface area of each composite figure. Round to the nearest tenth. 8. 9 Describe the effect of each change on the surface area of the given figure. p. 683 10. The dimensions are cut in half. 11. The dimensions are multiplied by 2 __. 683 12. Consumer Application The greater the lateral area of a florescent light bulb, the more light the bulb produces. One cylindrical light bulb is 16 inches long with a 1-inch radius. Another cylindrical light bulb is 23 inches long with a 3 __ 4 -inch radius. Which bulb will produce more light? 684 684 Chapter 10 Spatial Reasoning �� Independent Practice For See Exercises Example 13–15 16–18 19–20 21–22 23 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S22 Application Practice p. S37 PRACTICE AND PROBLEM SOLVING Find the lateral area and surface area of each right prism. Round to the nearest tenth, if necessary. 13. 14. 15. a right equilateral triangular prism with base edge length 8 ft and height 14 ft Find the lateral area and surface area of each right cylinder. Give your answers in terms of π. 16. 17. 18. a cylinder with base circumference 16π yd 2 and a height equal to 3 times the radius Multi-Step Find the surface area of each composite figure. Round to the nearest tenth. 19. 20. Describe the effect of each change on the surface area of the given figure. 21. The dimensions are tripled. 22. The dimensions are doubled. 23. Biology Plant cells are shaped approximately like a right rectangular prism. Each cell absorbs oxygen and nutrients through its surface. Which cell can be expected to absorb at a greater rate? (Hint: 1 µm = 1 micrometer = 0.000001 meter) 10- 4 Surface Area of Prisms and C

ylinders 685 685 �� 24. Find the height of a right cylinder with surface area 160π ft 2 and radius 5 ft. 25. Find the height of a right rectangular prism with surface area 286 m 2, length 10 m, and width 8 m. 26. Find the height of a right regular hexagonal prism with lateral area 1368 m 2 and base edge length 12 m. 27. Find the surface area of the right triangular prism with vertices at (0, 0, 0), (5, 0, 0), (0, 2, 0), (0, 0, 9), (5, 0, 9), and (0, 2, 9). The dimensions of various coins are given in the table. Find the surface area of each coin. Round to the nearest hundredth. Coin Diameter (mm) Thickness (mm) Surface Area ( mm 2 ) 28. Penny 29. Nickel 30. Dime 31. Quarter 19.05 21.21 17.91 24.26 1.55 1.95 1.35 1.75 32. How can the edge lengths of a rectangular prism be changed so that the surface area is multiplied by 9? 33. How can the radius and height of a cylinder be changed so that the surface area is multiplied by 1 __ 4? 34. Landscaping Ingrid is building a shelter to protect her plants from freezing. She is planning to stretch plastic sheeting over the top and the ends of a frame. Which of the frames shown will require more plastic? 35. Critical Thinking If the length of the measurements of the net are correct to the nearest tenth of a centimeter, what is the maximum error in the surface area? 36. Write About It Explain how to use the net of a three-dimensional figure to find its surface area. 37. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice container is a square prism with base edge length 4 in. When an 8 in. straw is inserted into the container as shown, exactly 1 in. remains outside the container. a. Find AB and BC. b. What is the height AC of the container to the nearest tenth? c. Use your result from part b to find how much material is required to manufacture the container

. Round to the nearest tenth. 686 686 Chapter 10 Spatial Reasoning 10 ftge07sec10l04004aa1st pass4/23/5cmurphy10 ft10 ftge07sec10l04005a1st pass4/12/5cmurphy10 ft10 ft�� 38. Measure the dimensions of the net of a cylinder to the nearest millimeter. Which is closest to the surface area of the cylinder? 35.8 cm 2 18.8 cm 2 16.0 cm 2 13.2 cm 2 39. The base of a triangular prism is an equilateral triangle with a perimeter of 24 inches. If the height of the prism is 5 inches, find the lateral area. 120 in 2 60 in 2 40 in 2 360 in 2 40. Gridded Response Find the surface area in square inches of a cylinder with a radius of 6 inches and a height of 5 inches. Use 3.14 for π and round your answer to the nearest tenth. CHALLENGE AND EXTEND 41. A cylinder has a radius of 8 cm and a height of 3 cm. Find the height of another cylinder that has a radius of 4 cm and the same surface area as the first cylinder. 42. If one gallon of paint covers 250 square feet, how many gallons of paint will be needed to cover the shed, not including the roof? If a gallon of paint costs $25, about how much will it cost to paint the walls of the shed? 43. The lateral area of a right rectangular prism is 144 cm 2. Its length is three times its width, and its height is twice its width. Find its surface area. SPIRAL REVIEW 44. Rebecca’s car can travel 250 miles on one tank of gas. Rebecca has traveled 154 miles. Write an inequality that models m, the number of miles farther Rebecca can travel on the tank of gas. (Previous course) 45. Blood sugar is a measure of the number of milligrams of glucose in a deciliter of blood (mg/dL). Normal fasting blood sugar levels are above 70 mg/dL and below 110 mg/dL. Write an inequality that models s, the blood sugar level of a normal patient. (Previous course) Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 46. BC 47. m∠ABC Draw the top, left, and

right views of each object. Assume there are no hidden cubes. (Lesson 10-2) 48. 49. 50. 10- 4 Surface Area of Prisms and Cylinders 687 687 �� 10-4 Model Right and Oblique Cylinders In Lesson 10-4, you learned the difference between right and oblique cylinders. In this lab, you will make models of right and oblique cylinders. Use with Lesson 10-4 TEKS G.9.D Congruence and the geometry of size: analyze the characteristics of polyhedra and other three-dimensional figures and their component parts …. Activity 1 1 Use a compass to draw at least 10 circles with a radius of 3 cm each on cardboard and then cut them out. 2 Poke a hole through the center of each circle. 3 Unbend a paper clip part way and push it through the center of each circle to model a cylinder. The stack of cardboard circles can be held straight to model a right cylinder or tilted to model an oblique cylinder. Try This 1. On each cardboard model, use string or a rubber band to outline a cross section that is parallel to the base of the cylinder. What shape is each cross section? 2. Use string or a rubber band to outline a cross section of the cardboard model of the oblique cylinder that is perpendicular to the lateral surface. What shape is the cross section? Activity 2 1 Roll a piece of paper to make a right cylinder. Tape the edges. 2 Cut along the bottom and top to approximate an oblique cylinder. 3 Untape the edge and unroll the paper. What does the net for an oblique cylinder look like? Try This 3. Cut off the curved part of the net you created in Activity 2 and translate it to the opposite side to form a rectangle. How do the side lengths of the rectangle relate to the dimensions of the cylinder? Estimate the lateral area and surface area of the oblique cylinder. 688 688 Chapter 10 Spatial Reasoning 10-5 Surface Area of Pyramids and Cones 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 surface area of a pyramid. Learn and apply the formula for the surface area of a cone. Vocabulary vertex of a pyramid regular pyramid slant height of a regular pyramid

altitude of a pyramid vertex of a cone axis of a cone right cone oblique cone slant height of a right cone altitude of a cone Also G.5.A, G.5.B, G.6.B, G.11.D Why learn this? A speaker uses part of the lateral surface of a cone to produce sound. Speaker cones are usually made of paper, plastic, or metal. (See Example 5.) �� The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid. �� Lateral and Surface Area of a Regular Pyramid The lateral area of a regular pyramid with perimeter P and slant height ℓ is L = 1 __ Pℓ. 2 The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = 1 __ Pℓ + B Finding Lateral Area and Surface Area of Pyramids Find the lateral area and surface area of each pyramid. A a regular square pyramid with base edge length 5 in. Pℓ and slant height 9 in20) (9) = 90 in 2 2 S = 1 _ 2 = 90 + 25 = 115 in 2 Pℓ + B Lateral area of a regular pyramid P = 4(5) = 20 in. Surface area of a regular pyramid B = 5 2 = 25 in 2 10- 5 Surface Area of Pyramids and Cones 689 689 �� Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth. B Step 1 Find the base perimeter and apothem. The base perimeter is 6 (4) = 24 m 2. The apothem is 2 √  3 m, so the base area is 1 __ 2 aP = 1

__ 2 (2 √  3 ) (24) = 24 √  3 m 2. Step 2 Find the lateral area. Pℓ L = 1 _ 2 = 1 _ (24) (7) = 84 m 2 2 Step 3 Find the surface area. S = 1 _ 2 Pℓ + B Lateral area of a regular pyramid Substitute 24 for P and 7 for ℓ. Surface area of a regular pyramid = 84 + 24 √  3 ≈ 125.6 cm 2 Substitute 24 √  3 for B. 1. Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base. The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base. Lateral and Surface Area of a Right Cone The lateral area of a right cone with radius r and slant height ℓ is L = πrℓ. The surface area of a right cone with lateral area L and base area B is S = L + B, or S = πrℓ + π r 2. 690 690 Chapter 10 Spatial Reasoning �� E X A M P L E 2 Finding Lateral Area and Surface Area of Right Cones Find the lateral area and surface area of each cone. Give your answers in terms of π. A a right cone with radius 2 m and slant height 3 m L = π rℓ = π (2) (3) = 6π m 2 S = π rℓ + π r 2 = 6π + π (2) 2 = 10π m 2 Lateral area of a cone Substitute 2 for r and 3 for ℓ. Surface area of a cone Substitute 2 for r and 3 for ℓ. B Step 1 Use the Pyth

agorean Theorem to find ℓ. ℓ = √  5 2 + 12 2 = 13 ft Step 2 Find the lateral area and surface area. L = πrℓ = π (5) (13) = 65π ft 2 S = πrℓ + π r 2 = 65π + π (5) 2 = 90π ft 2 Lateral area of a right cone Substitute 5 for r and 13 for ℓ. Surface area of a right cone Substitute 5 for r and 13 for ℓ. 2. Find the lateral area and surface area of the right cone. E X A M P L E 3 Exploring Effects of Changing Dimensions The radius and slant height of the right cone are tripled. Describe the effect on the surface area. original dimensions: S = π rℓ + π r 2 = π (3) (5) + π (3) 2 = 24π cm 2 radius and slant height tripled: S = π rℓ + π r 2 = π (9) (15) + π (9) 2 = 216π cm 2 Notice that 216π = 9 (24π). If the length, width, and height are tripled, the surface area is multiplied by 3 2, or 9. 3. The base edge length and slant height of the regular square pyramid are both multiplied by 2 __ 3. Describe the effect on the surface area. 10- 5 Surface Area of Pyramids and Cones 691 691 �� E X A M P L E 4 Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. The height of the cone is 90 - 45 = 45 cm. By the Pythagorean Theorem, ℓ = √  the cone is 28 2 + 45 2 = 53 cm. The lateral area of L = πrℓ = π (28) (53) = 1484π cm 2. The lateral area of the cylinder is L = 2πrh = 2π (28) (45) = 2520π cm 2. The base area is B = π r 2 = π (28) 2 = 784π cm 2. S =

(cone lateral area) + (cylinder lateral area) + (base area) = 2520π + 784π + 1484π = 4788π cm 2 4. Find the surface area of the composite figure. E X A M P L E 5 Electronics Application Electronics The paper cones of The paper cones of antique speakers were both functional and decorative. Some had elaborate patterns or shapes. Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone? The radius of the large circle used to create the pattern is the slant height of the cone. The area of the pattern is the lateral area of the cone. The area of the pattern is also 3 __ 4 of the area of the large circle, so πrℓ = 3 __ 4 π r 2. πr (10) = 3 _ π (10) 2 4 Substitute 10 for ℓ, the slant height of the cone and the radius of the large circle. r = 7.5 in. Solve for r. The diameter of the cone is 2 (7.5) = 15 in. 5. What if…? If the radius of the large circle were 12 in., what would be the radius of the cone? THINK AND DISCUSS 1. Explain why the lateral area of a regular pyramid is 1 __ 2 the base perimeter times the slant height. 2. In a right cone, which is greater, the height or the slant height? Explain. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write the name of the part of the cone. 692 692 Chapter 10 Spatial Reasoning �� 10-5 Exercises Exercises GUIDED PRACTICE 1. Vocabulary Describe the endpoints of an axis of a cone Find the lateral area and surface area of each regular pyramid. p. 689 2. 3. KEYWORD: MG7 10-5 KEYWORD: MG7 Parent 4. a regular triangular pyramid with base edge length 15 in. and slant height 20 in. 691 Find the lateral area and surface area of each right cone. Give your answers in terms of π. 5. 6. 7. a cone with base area 36π ft 2 and slant height 8 ft Describe the

effect of each change on the surface area of the given figure. p. 691 8. The dimensions are cut in half. 9. The dimensions are tripled Find the surface area of each composite figure. p. 692 10. 11. 692 12. Crafts Anna is making a birthday hat from a pattern that is 3 __ 4 of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain. 10- 5 Surface Area of Pyramids and Cones 693 693 �� Independent Practice For See Exercises Example 13–15 16–18 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 lateral area and surface area of each regular pyramid. 13. 14. 15. a regular hexagonal pyramid with base edge length 7 ft and slant height 15 ft Find the lateral area and surface area of each right cone. Give your answers in terms of π. 16. 17. 18. a cone with radius 8 m and height that is 1 m less than twice the radius Describe the effect of each change on the surface area of the given figure. 19. The dimensions are divided by 3. 20. The dimensions are doubled. Find the surface area of each composite figure. 21. 22. 23. It is a tradition in England to celebrate May 1st by hanging cone-shaped baskets of flowers on neighbors’ door handles. Addy is making a basket from a piece of paper that is a semicircle with diameter 12 in. What is the diameter of the basket? �� Find the surface area of each figure. Shape Base Area Slant Height Surface Area 24. 25. 26. 27. Regular square pyramid Regular triangular pyramid Right cone Right cone 36 cm 2 √  3 m 2 16π in 2 π ft 2 5 cm √  3 m 7 in. 2 ft 694 694 Chapter 10 Spatial Reasoning �� 28. This problem will prepare you for the Multi-Step TAKS Prep on page 724. A juice container

is a regular square pyramid with the dimensions shown. a. Find the surface area of the container to the nearest tenth. b. The manufacturer decides to make a container in the shape of a right cone that requires the same amount of material. The base diameter must be 9 cm. Find the slant height of the container to the nearest tenth. 29. Find the radius of a right cone with slant height 21 m and surface area 168π m 2. 30. Find the slant height of a regular square pyramid with base perimeter 32 ft and surface area 256 ft 2. 31. Find the base perimeter of a regular hexagonal pyramid with slant height 10 cm and lateral area 120 cm 2. 32. Find the surface area of a right cone with a slant height of 25 units that has its base centered at (0, 0, 0) and its vertex at (0, 0, 7). Architecture Find the surface area of each composite figure. 33. 34. The Pyramid Arena seats 21,000 people. The base of the pyramid is larger than six football fields. 35. Architecture The Pyramid Arena in Memphis, Tennessee, is a square pyramid with base edge lengths of 200 yd and a height of 32 stories. Estimate the area of the glass on the sides of the pyramid. (Hint: 1 story ≈ 10 ft) 36. Critical Thinking Explain why the slant height of a regular square pyramid must be greater than half the base edge length. 37. Write About It Explain why slant height is not defined for an oblique cone. 38. Which expressions represent the surface area of the regular square pyramid? I. t 2 _ 16 II. t 2 _ 16 + tℓ _ 2 + ts _ 2 III. t _ 2 ( t _ + ℓ) 8 I only II only I and II II and III 39. A regular square pyramid has a slant height of 18 cm and a lateral area of 216 cm 2. What is the surface area? 252 cm 2 234 cm 2 225 cm 2 240 cm 2 40. What is the lateral area of the cone? 450π cm 2 1640π cm 2 360π cm 2 369π cm 2 10- 5 Surface Area of Pyramids and Cones 695 695 �� CHALLENGE AND EXTEND 41. A frustum of a cone is a part of the cone with two parallel bases. The height of the frustum of the cone is half