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interior. Construct segments from this point to each vertex, forming four triangles. Measure the area of each triangle. Move the point to find a location where all four triangles have equal area. Is there more than one such location? Explain your findings. 20. Explain why x must be 48°. 21. What’s wrong with this picture? C B x A 24° D 38° E 28° 22. The 6-by-18-by-24 cm clear plastic sealed container is resting on a cylinder. It is partially filled with liquid, as shown. Sketch the container resting on its smallest face. Show the liquid level in this position. 6 cm 24 cm 18 cm IMPROVING YOUR VISUAL THINKING SKILLS Random Points What is the probability of randomly selecting from the 3-by-3 grid at right three points that form the vertices of an isosceles triangle? 436 CHAPTER 8 Area L E S S O N 8.6 Cut my pie into four pieces— I don’t think I could eat eight. YOGI BERRA Any Way You Slice It In Lesson 8.5, you discovered a formula for calculating the area of a circle. With the help of your visual thinking and problem-solving skills, you can calculate the areas of different sections of a circle. Its makers claimed this was the world’s largest slice of pizza. If you cut a slice of pizza, each slice would probably be a sector of a circle. If you could make only one straight cut with your knife, your slice would be a segment of a circle. If you don’t like the crust, you’d cut out the center of the pizza; the crust shape that would remain is called an annulus. Sector of a circle Segment of a circle Annulus A sector of a circle is the region between two radii of a circle and the included arc. A segment of a circle is the region between a chord of a circle and the included arc. An annulus is the region between two concentric circles. “Picture equations” are helpful when you try to visualize the areas of these regions. The picture equations below show you how to find the area of a sector of a circle, the area of a segment of a circle, and the area of an annulus. r a° a___ 360 a___ ( 360 ) r r 2 Asector a° b h r h b r a° a___ ( |
360 ) r 2 bh Asegment 1_ 2 r R r R R 2 r 2 Aannulus LESSON 8.6 Any Way You Slice It 437 EXAMPLE A Find the area of the shaded sector. 45° 20 cm Solution Asector, or 1 ° 5 4 The sector is, of the circle. 8 6 ° 0 3 • r2 a 60 3 • (20)2 5 4 0 6 3 • 400 1 8 50 The area is 50 cm2. The area formula for a sector. Substitute r 20 and a 45. Reduce the fraction and square 20. Multiply. EXAMPLE B Find the area of the shaded segment. 6 cm Solution According to the picture equation on page 437, the area of a segment is equivalent to the area of the sector minus the area of the triangle. You can use the method in Example A to find that the area of the sector is 1 36 cm2, 4 or 9 cm2. The area of the triangle is 1 (6)(6), or 18 cm2. So the area of the 2 segment is (9 18) cm2. EXAMPLE C The shaded area is 14 cm2, and the radius is 6 cm. Find x. 6 cm x° Solution x of the circle’s area, which is 36. The sector’s area is 36 0 (36) 14 x 36 0 14) x ( (360 ) 6 3 x 140 The central angle measures 140°. 438 CHAPTER 8 Area EXERCISES In Exercises 1–8, find the area of the shaded region. The radius of each circle is r. If two circles are shown, r is the radius of the smaller circle and R is the radius of the larger circle. You will need Construction tools for Exercise 16 1. r 6 cm 60° 2. r 8 cm 3. r 16 cm 240° 4. r 2 cm 5. r 8 cm 7. r 2 cm 10 r 8. R 12 cm r 9 cm R r 60° 6. R 7 cm r 4 cm r R 9. The shaded area is 12 cm2. Find r. 120° r 10. The shaded area is 32 cm2. Find r. 11. The shaded area is 120 cm2, and the radius is 24 cm. Find x. 12. The shaded area is 10 cm2. The radius of the large circle is 10 cm, and the |
radius of the small circle is 8 cm. Find x. 18 r x° x° 13. Suppose the pizza slice in the photo at the beginning of this lesson is a sector with a 36° arc, and the pizza has a radius of 20 ft. If one can of tomato sauce will cover 3 ft2 of pizza, how many cans would you need to cover this slice? LESSON 8.6 Any Way You Slice It 439 14. Utopia Park has just installed a circular fountain 8 meters in diameter. The Park Committee wants to pave a 1.5-meter-wide path around the fountain. If paving costs $10 per square meter, find the cost to the nearest dollar of the paved path around the fountain. This circular fountain at the Chateau de Villandry in Loire Valley, France, shares a center with the circular path around it. How many concentric arcs and circles do you see in the picture? Mathematics Attempts to solve the famous problem of rectifying a circle—finding a rectangle with the same area as a given circle—led to the creation of some special shapes made up of parts of circles. The diagrams below are based on some that Leonardo da Vinci sketched while attempting to solve this problem. 15. The illustrations below demonstrate how to rectify the pendulum. In a series of diagrams, demonstrate how to rectify each figure. a. b. c. d. 16. Construction Reverse the process you used in Exercise 15. On graph paper, draw a 12-by-6 rectangle. Use your compass to divide it into at least four parts, then rearrange the parts into a new curved figure. Draw its outline on graph paper. 440 CHAPTER 8 Area Review 17. Each set of circles is externally tangent. What is the area of the shaded region in each figure? What percentage of the area of the square is the area of the circle or circles in each figure? All given measurements are in centimeters. a. 12 b. 12 c. 12 d. 12 12 12 12 12 18. The height of a trapezoid is 15 m and the midsegment is 32 m. What is the area of the trapezoid? In Exercises 19–22, identify each statement as true or false. If true, explain why. If false, give a counterexample. 19. If the arc of a circle measures 90° and has an arc length of 24 cm, then the radius of the circle is 48 cm |
. 20. If the measure of each exterior angle of a regular polygon is 24°, then the polygon has 15 sides. 21. If the diagonals of a parallelogram bisect its angles, then the parallelogram is a square. 22. If two sides of a triangle measure 25 cm and 30 cm, then the third side must be greater than 5 cm but less than 55 cm. IMPROVING YOUR REASONING SKILLS Code Equations Each code below uses the first letters of words that will make the equation true. For example, 12M a Y is an abbreviation of the equation 12 Months a Year. Find the missing words in each code. 1. 45D an AA of an IRT 3. 90D each A of a R 2. 7 SH 4. 5 D in a P LESSON 8.6 Any Way You Slice It 441 Geometric Probability II You already know that a probability value is a number between 0 and 1 that tells you how likely something is to occur. For example, when you roll a die, three of the six rolls—namely, 2, 3, and 5—are prime numbers. Since each roll has the same or 1 chance of occurring, P(prime number) 3. 2 6 In some situations, probability depends on area. For example, suppose a meteorite is headed toward Earth. If about 1 of Earth’s surface is land, the probability that the 3 meteorite will hit land is about 1, while the probability it 3 will hit water is about 2. Because Alaska has a greater 3 area than Vermont, the probability the meteorite will land in Alaska is greater than the probability it will land in Vermont. If you knew the areas of these two states and the surface area of Earth, how could you calculate the probabilities that the meteorite would land in each state? Activity Where the Chips Fall In this activity, you will solve several probability problems that involve area. The Shape of Things At each level of a computer game, you must choose one of several shapes on a coordinate grid. The computer then randomly selects a point anywhere on the grid. If the point is outside your shape, you move to the next level. If the point is on or inside your shape, you lose the game. On the first level, a trapezoid, a pentagon, a square, and a triangle are displayed on a grid that goes from 0 to 12 on both axes. The table below gives the vertices |
of the shapes. Shape Vertices Trapezoid (1, 12), (8, 12), (7, 9), (4, 9) Pentagon (3, 1), (4, 4), (6, 4), (9, 2), (7, 0) Square Triangle (0, 6), (3, 9), (6, 6), (3, 3) (11, 0), (7, 4), (11, 12) You will need ● graph paper ● a ruler ● a penny ● a dime 442 CHAPTER 8 Area Step 1 Step 2 Step 3 Step 4 For each shape, calculate the probability the computer will choose a point on or inside that shape. Express each probability to three decimal places. What is the probability the computer will choose a point that is on or inside a quadrilateral? What is the probability it will choose a point that is outside all of the shapes? If you choose a triangle, what is the probability you will move to the next level? Which shape should you choose to have the best chance of moving to the next level? Why? Right on Target You are playing a carnival game in which you must throw one dart at the board shown at right. The score for each region is shown on the board. If your dart lands in a Bonus section, your score is tripled. The more points you get, the better your prize will be. The radii of the circles from the inside to the outside are 4 in., 8 in., 12 in., and 16 in. Assume your aim is not very good, so the dart will hit a random spot. If you miss the board completely, you get to throw again. 5 10 15 30 Bonus Bonus Step 5 Step 6 Step 7 Step 8 Step 9 What is the probability your dart will land in the red region? Blue region? Compute the probability your dart will land in a Bonus section. (The central angle measure for each Bonus section is 30°.) If you score 90 points, you will win the grand prize, a giant stuffed emu. What is the probability you will win the grand prize? If you score exactly 30 points, you win an “I ♥ Carnivals” baseball cap. What is the probability you will win the cap? Now imagine you have been practicing your dart game, and your aim has improved. Would your answers to Steps 5–8 change? Explain. The Coin Toss You own a small cafe that is popular with the mathematicians in the neighborhood. You |
devise a game in which the customer flips a coin onto a red-and-white checkered tablecloth with 1-inch squares. If it lands completely within a square, the customer wins, and doesn’t have to pay the bill. If it lands touching or crossing the boundary of a square, the customer loses. I N I N LIBERTY LIBERTY 1996 1996 Step 10 Assuming the coin stays on the table, what is the probability of the customer winning by flipping a penny? A dime? (Hint: Where must the center of the coin land in order to win?) EXPLORATION Geometric Probability II 443 Step 11 Step 12 If the customer wins only if the coin falls within a red square, what is the probability of winning with a penny? A dime? Suppose the game is always played with a penny, and a customer wins if the penny lands completely inside any square (red or white). If your daily proceeds average $300, about how much will the game cost you per day? On a Different Note Two opera stars—Rigoletto and Pollione—are auditioning for a part in an upcoming production. Since the singers have similar qualifications, the director decides to have a contest to see which man can hold a note the longest. Rigoletto has been known to hold a note for any length of time from 6 to 9 minutes. Pollione has been known to hold a note for any length of time between 5 and 7 minutes. Step 13 Draw a rectangular grid in which the bottom side represents the range of times for Rigoletto and the left side represents the range of times for Pollione. Each point in the rectangle represents one possible outcome of the contest. Step 14 On your grid, mark all the points that represent a tie. Use your diagram to find the probability that Rigoletto will win the contest. DIFFERENT DICE Understanding probability can improve your chances of winning a game. If you roll a pair of standard 6-sided dice, are you more likely to roll a sum of 6 or 12? It’s fairly common to roll a sum of 6, since many combinations of two dice add up to 6. But a 12 is only possible if you roll a 6 on each die. If you rolled a pair of standard 6-sided dice over and over again, and recorded the number of times you got each sum, the histogram would look like this: Would the distribution be different if you used different dice? What if one |
die had odd numbers and the other had even numbers? What if you used 8-sided dice? What if you rolled three 6-sided dice instead of two? Choose one of these scenarios or one that you find interesting to investigate. Make your dice and roll them 20 times. Predict what the graph will look like if you roll the dice 100 times, then check your prediction. Your project should include Your dice. Histograms of your experimental data, and your predictions and conclusions. You can use Fathom to simulate real events, such as rolling the dice that you have designed. You can obtain the results of hundreds of events very quickly, and use Fathom’s graphing capabilities to make a histogram showing the distribution of different outcomes. 444 CHAPTER 8 Area L E S S O N 8.7 No pessimist ever discovered the secrets of the stars, or sailed to an uncharted land, or opened a new doorway for the human spirit. HELEN KELLER Surface Area In Lesson 8.3, you calculated the surface areas of walls and roofs. But not all building surfaces are rectangular. How would you calculate the amount of glass necessary to cover a pyramid-shaped building? Or the number of tiles needed to cover a cone-shaped roof? In this lesson, you will learn how to find the surface areas of prisms, pyramids, cylinders, and cones. The surface area of each of these solids is the sum of the areas of all the faces or surfaces that enclose the solid. For prisms and pyramids, the faces include the solid’s bases and its remaining lateral faces. In a prism, the bases are two congruent polygons and the lateral faces are rectangles or parallelograms. In a pyramid, the base can be any polygon. The lateral faces are triangles. Lateral face Bases Base Lateral face This glass pyramid was designed by I. M. Pei for the entrance of the Louvre museum in Paris, France. This skyscraper in Chicago, Illinois, is an example of a prism. A cone is part of the roof design of this Victorian house in Massachusetts. This stone tower is a cylinder on top of a larger cylinder. LESSON 8.7 Surface Area 445 To find the surface areas of prisms and pyramids, follow these steps. Steps for Finding Surface Area 1. Draw and label each face of the solid as if you had cut the solid apart along its edges and laid it flat. Label the dimensions. 2 |
. Calculate the area of each face. If some faces are identical, you only need to find the area of one. 3. Find the total area of all the faces. EXAMPLE A Find the surface area of the rectangular prism. 2 m 4 m 5 m Solution First, draw and label all six faces. Then, find the areas of all the rectangular faces. These shipping containers are rectangular prisms. 5 4 Top 4 Bottom 5 Bases 2 2 4 Front Back 4 2 2 5 Side Side 5 Lateral faces Surface area 2(4 5) 2(2 4) 2(2 5) 40 16 20 76 The surface area of the prism is 76 m2. EXAMPLE B Find the surface area of the cylinder. Solution Imagine cutting apart the cylinder. The two bases are circular regions, so you need to find the areas of two circles. Think of the lateral surface as a wrapper. Slice it and lay it flat to get a rectangular region. You’ll need the area of this rectangle. The height of the rectangle is the height of the cylinder. The base of the rectangle is the circumference of the circular base. 12 in. 10 in. 446 CHAPTER 8 Area Top Bottom 5 5 Bases b C 2r h 12 Lateral surface Surface area 2r2 (2r)h 2 52 (2 5) 12 534 The surface area of the cylinder is about 534 in2. This ice cream plant in Burlington, Vermont, uses cylindrical containers for its milk and cream. These conservatories in Edmonton, Canada, are glass pyramids. The surface area of a pyramid is the area of the base plus the areas of the triangular faces. The height of each triangular lateral face is called the slant height. To avoid confusing slant height with the height of the pyramid, use l rather than h for slant height. Height Slant height h l In the investigation you’ll find out how to calculate the surface area of a pyramid with a regular polygon base. LESSON 8.7 Surface Area 447 Investigation 1 Surface Area of a Regular Pyramid You can cut and unfold the surface of a regular pyramid into these shapes Step 1 Step 2 Step 3 Step 4 Step 5 What is the area of each lateral face? What is the total lateral surface area? What is the total lateral surface area for any pyramid with a regular n-gon base? What is the area of the base for any regular n-gon pyramid? Use your expressions from Steps |
2 and 3 to write a formula for the surface area of a regular n-gon pyramid in terms of base length b, slant height l, and apothem a. Write another expression for the surface area of a regular n-gon pyramid in terms of height l, apothem a, and perimeter of the base, P 1_ 2 P You can find the surface area of a cone using a method similar to the one you used to find the surface area of a pyramid. Is the roof of this building in Kashan, Iran, a cone or a pyramid? What makes it hard to tell? 448 CHAPTER 8 Area Investigation 2 Surface Area of a Cone As the number of faces of a pyramid increases, it begins to look like a cone. You can think of the lateral surface as many small triangles, or as a sector of a circle. l r l 2r r r l Step 1 Step 2 What is the area of the base? What is the lateral surface area in terms of l and r? What portion is the sector of the circle? What is the area of the sector? Step 3 Write the formula for the surface area of a cone. This photograph of Sioux tepees was taken around 1902 in North or South Dakota. Is a tepee shaped more like a cone or a pyramid? EXAMPLE C Find the total surface area of the cone. Solution SA rl r2 ()(5)(10) (5)2 75 235.6 The surface area of the cone is about 236 cm2. 10 cm 5 cm LESSON 8.7 Surface Area 449 EXERCISES In Exercises 1–10, find the surface area of each solid. All quadrilaterals are rectangles, and all given measurements are in centimeters. Round your answers to the nearest 0.1 cm2. You will need Construction tools for Exercise 13 5 5 5 1. 4. 2. 37 5. 13 10 12 10 37 9 8 3 3. 10 7 6. 6 14 8 20 7. The base is a regular hexagon with apothem a 12.1, side s 14, and height h 7. 8. The base is a regular pentagon with apothem a 11 and side s 16. Each lateral edge t 17, and the height of a face l 15. a s h t l a s 9. D 8, d 4, h 9 10. l 8, w 4, h 10 11. Explain how you would find the surface |
area of this obelisk. 450 CHAPTER 8 Area 12. APPLICATION Claudette and Marie are planning to paint the exterior walls of their country farmhouse (all vertical surfaces) and to put new cedar shingles on the roof. The paint they like best costs $25 per gallon and covers 250 square feet per gallon. The wood shingles cost $65 per bundle, and each bundle covers 100 square feet. How much will this home improvement cost? All measurements are in feet. 6.5 6.5 12 15 15 12 38.5 24 End view 30 40 13. Construction The shapes of the spinning dishes in the photo are called frustums of cones. Think of them as cones with their tops cut off. Use your construction tools to draw pieces that you can cut out and tape together to form a frustum of a cone. A Sri Lankan dancer balances and spins plates. Review 14. Use patty paper, templates, or pattern blocks to create a 33.42/32.4.3.4/44 tiling. 15. Trace the figure at right. Find the lettered angle measures and arc measures. a f e c d b 150° LESSON 8.7 Surface Area 451 16. APPLICATION Suppose a circular ranch with a radius of 3 km was divided into 16 congruent sectors. In a one-year cycle, how long would the cattle graze in each sector? What would be the area of each sector? History In 1792, visiting Europeans presented horses and cattle to Hawaii’s King Kamehameha I. Cattle ranching soon developed when Mexican vaqueros came to Hawaii to train Hawaiians in ranching. Today, Hawaiian cattle ranching is big business. “Grazing geometry” is used on Hawaii’s Kahua Ranch. Ranchers divide the grazing area into sectors. The cows are rotated through each sector in turn. By the time they return to the first sector, the grass has grown back and the cycle repeats. 17. Trace the figure at right. Find the lettered angle measures. 18. If the pattern of blocks continues, what will be the surface area of the 50th solid in the pattern? (Every edge of each block has length 1 unit.) 50° IMPROVING YOUR VISUAL THINKING SKILLS Moving Coins Create a triangle of coins similar to the one shown. How can you move exactly three coins so that the triangle is pointing down rather than up? When |
you have found a solution, use a diagram to explain it 452 CHAPTER 8 Area Alternative Area Formulas In ancient Egypt, when the yearly floods of the Nile River receded, the river often followed a different course, so the shape of farmers’ fields along the banks could change from year to year. Officials then needed to measure property areas, in order to keep records and calculate taxes. Partly to keep track of land and finances, ancient Egyptians developed some of the earliest mathematics. Historians believe that ancient Egyptian tax assessors used this formula to find the area of any quadrilateral: (a c) 1 A 1 (b d) 2 2 where a, b, c, and d are the lengths, in consecutive order, of the figure’s four sides. In this activity, you will take a closer look at this ancient Egyptian formula, and another formula called Hero’s formula, named after Hero of Alexandria. Activity Calculating Area in Ancient Egypt Investigate the ancient Egyptian formula for quadrilaterals. Construct a quadrilateral and its interior. Change the labels of the sides to a, b, c, and d, consecutively. Measure the lengths of the sides and use the Sketchpad calculator to find the area according to the ancient Egyptian formula. Select the polygon interior and measure its area. How does the area given by the formula compare to the actual area? Is the ancient Egyptian formula correct? Does the ancient Egyptian formula always favor either the tax collector or the landowner, or does it favor one in some cases and the other in other cases? Explain. b a c d Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Describe the quadrilaterals for which the formula works accurately. For what kinds of quadrilaterals is it slightly inaccurate? Very inaccurate? Step 7 State the ancient Egyptian formula in words, using the word mean. EXPLORATION Alternative Area Formulas 453 Step 8 According to Hero’s formula, if s is half the perimeter of a triangle with side lengths a, b, and c, the area A is given by the formula A s(s a)(s b)(s c) Use Sketchpad to investigate Hero’s formula. Construct a triangle and its interior. Label the sides a, b, and c, and use the Sketchpad calculator to find the triangle’s area according to Hero. Compare the result to the measured area of the triangle. Does Hero’s formula work for |
all triangles? Step 9 Devise a way of calculating the area of any quadrilateral. Use Sketchpad to test your method. IMPROVING YOUR VISUAL THINKING SKILLS Cover the Square Trace each diagram below onto another sheet of paper. Cut out the four triangles in each of the two small equal squares and arrange them to exactly cover the large square. Cut out the small square and the four triangles from the square on leg EF and arrange them to exactly cover the large square. A B D E C Arrange the eight triangles to fit in here. F Arrange the five pieces to fit in here. Two squares with areas x2 and y2 are divided into the five regions as shown. Cut out the five regions and arrange them to exactly cover a larger square with an area of z2. Two squares have been divided into three right triangles and two quadrilaterals. Cut out the five regions and arrange them to exactly cover a larger square. x x y y z x 454 CHAPTER 8 Area VIEW ● CHAPTER 11 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHA CHAPTER 8 R E V I E W You should know area formulas for rectangles, parallelograms, triangles, trapezoids, regular polygons, and circles. You should also be able to show where these formulas come from and how they’re related to one another. Most importantly, you should be able to apply them to solve practical problems involving area, including the surface areas of solid figures. What occupations can you list that use area formulas? When you use area formulas for real-world applications, you have to consider units of measurement and accuracy. Should you use inches, feet, centimeters, meters, or some other unit? If you work with a circle or a regular polygon, is your answer exact or an approximation? EXERCISES For Exercises 1–10, match the area formula with the shaded area. 1. A bh 2. A 0.5bh 3. A 0.5hb1 4. A 0.5d1d2 5. A 0.5aP b2 6. A r2 r2 7. A x 36 0 8. A R2 r2 9. SA 2rl 2r2 10. LA rl A. B. C. D. E. F. G. H. I. J. For Exercises 11–13, illustrate |
each term. 11. Apothem 12. Annulus 13. Sector of a circle For Exercises 14–16, draw a diagram and explain in a paragraph how you derived the area formula for each figure. 14. Parallelogram 15. Trapezoid 16. Circle CHAPTER 8 REVIEW 455 EW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTE Solve for the unknown measures in Exercises 17–25. All measurements are in centimeters. 17. A? 18. A? a 36 s 41.6 20 40 s a 19. A? R 8 r 2 r R 20. A 576 cm2 h? 21. A 576 cm2? d1 22. A 126 cm2 a 13 cm h 9 cm b? h h 36 d1 a 36 b 23. C 18 cm A? 24. A 576 cm2 The circumference is?. r 25. Asector 16 cm2 mFAN?. N A 12 F In Exercises 26–28, find the shaded area to the nearest 0.1 cm2. In Exercises 27 and 28, the quadrilateral is a square and all arcs are arcs of a circle of radius 6 cm. 26. 27. 28. 28 cm 456 CHAPTER 8 Area ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 In Exercises 29–31, find the surface area of each prism or pyramid. All given measurements are in centimeters. All quadrilaterals are rectangles, unless otherwise labeled. 29. 8 30. The base is a trapezoid. 31. 13 5 12 35 12 10 15 5 20 24 20 30 14 For Exercises 32 and 33, plot the vertices of each figure on graph paper, then find its area. 32. Parallelogram ABCD with A(0, 0), B(14, 0), and D(6, 8) 33. Quadrilateral FOUR with F(0, 0), O(4, 3), U(9, 5), and R(4, 15) 34. The sum of the lengths of the two bases of a trapezoid is 22 cm, and its area is 66cm2. What is the height of the trapezoid? 35. Find the area of a regular pentagon to the |
nearest tenth of a square centimeter if the apothem measures 6.9 cm and each side measures 10 cm. 36. Find three noncongruent polygons, each with an area of 24 square units, on a 6-by-6 geoboard or a 6-by-6 square dot grid. 37. Lancelot wants to make a pen for his pet, Isosceles. What is the area of the largest rectangular pen that Lancelot can make with 100 meters of fencing if he uses a straight wall of the castle for one side of the pen? 38. If you have a hundred feet of rope to arrange into the perimeter of either a square or a circle, which shape will give you the maximum area? Explain. 39. Which is a better (tighter) fit: a round peg in a square hole or a square peg in a round hole? Ropes CHAPTER 8 REVIEW 457 EW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTE 40. Al Dente’s Pizzeria sells pizza by the slice, according to the sign. Which slice is the best deal (the most pizza per dollar)? 41. If you need 8 oz of dough to make a 12-inch diameter pizza, how much dough will you need to make a 16-inch pizza on a crust of the same thickness? 42. Which is the biggest slice of pie: one-fourth of a 6-inch diameter pie, one-sixth of an 8-inch diameter pie, or one-eighth of a 12-inch diameter pie? Which slice has the most crust along the curved edge? 43. The Hot-Air Balloon Club at Da Vinci High School has designed a balloon for the annual race. The panels are a regular octagon, eight squares, and sixteen isosceles trapezoids, and club members will sew them together to construct the balloon. They have built a scale model, as shown at right. The dimensions of three of the four types of panels are below, shown in feet. 3.8 ft 12 ft 80° 3.2 ft 3.2 ft 12 ft 70° 12 ft 20 ft 5 ft a. What will be the perimeter, to the nearest foot, of the balloon at its widest? What will be the perimeter, to the nearest foot, of the opening at the bottom of the balloon? b. What is the total surface area of the balloon |
to the nearest square foot? For Exercises 44–46, unless the dimensions indicate otherwise, assume each quadrilateral is a rectangle. 44. You are producing 10,000 of these metal wedges, and you must electroplate them with a thin layer of high-conducting silver. The measurements shown are in centimeters. Find the total cost for silver, if silver plating costs $1 for each 200 square centimeters. 10 8 0.5 6 458 CHAPTER 8 Area ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 45. The measurements of a chemical storage container are shown in meters. Find the cost of painting the exterior of nine of these large cylindrical containers with sealant. The sealant costs $32 per gallon. Each gallon covers 18 square meters. Do not paint the bottom faces. 7 10 46. The measurements of a copper cone are shown in inches. Find the cost of spraying an oxidizer on 100 of these copper cones. The oxidizer costs $26 per pint. Each pint covers approximately 5000 square inches. Spray only the lateral surface. 47. Hector is a very cost-conscious produce buyer. He usually buys asparagus in large bundles, each 44 cm in circumference. But today there are only small bundles that are 22 cm in circumference. Two 22 cm bundles are the same price as one 44 cm bundle. Is this a good deal or a bad deal? Why? 51 in. 48 in. TAKE ANOTHER LOOK 1. Use geometry software to construct these shapes. a. A triangle whose perimeter can vary, but whose area stays constant b. A parallelogram whose perimeter can vary, but whose area stays constant 2. True or false? The area of a triangle is equal to half the perimeter of the triangle times the radius of the inscribed circle. Support your conclusion with a convincing argument. 3. Does the area formula for a kite hold for a dart (a concave kite)? Support your conclusion with a convincing argument. 4. How can you use the Regular Polygon Area Conjecture to arrive at a formula for the area of a circle? Use a series of diagrams to help explain your reasoning. 5. Use algebra to show that the total surface area of a prism with a regular polygon base is given by the formula SA P(h a), where h is height of the prism, a is the apothem of the base, and P is the |
perimeter of the base. 6. Use algebra to show that the total surface area of a cylinder is given by the formula SA C(h r), where h is the height of the cylinder, r is the radius of the base, and C is the circumference of the base. 7. Here is a different formula for the area of a trapezoid: A mh, where m is the length of the midsegment and h is the height. Does the formula work? Use algebra or a diagram to explain why or why not. Does it work for a triangle? CHAPTER 8 REVIEW 459 EW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTER 8 REVIEW ● CHAPTE Assessing What You’ve Learned UPDATE YOUR PORTFOLIO Choose one of the more challenging problems you did in this chapter and add it to your portfolio. Write about why you chose it, what made it challenging, what strategies you used to solve it, and what you learned from it. ORGANIZE YOUR NOTEBOOK Review your notebook to be sure it’s complete and well organized. Be sure you have included all of this chapter’s area formulas in your conjecture list. Write a one-page chapter summary. WRITE IN YOUR JOURNAL Imagine yourself five or ten years from now, looking back on the influence this geometry class had on your life. How do you think you’ll be using geometry? Will this course have influenced your academic or career goals? PERFORMANCE ASSESSMENT While a classmate, a friend, a family member, or a teacher observes, demonstrate how to derive one or more of the area formulas. Explain each step, including how you arrive at the formula. WRITE TEST ITEMS Work with group members to write test items for this chapter. Include simple exercises and complex application problems. GIVE A PRESENTATION Create a poster, a model, or other visual aid, and give a presentation on how to derive one or more of the area formulas. Or present your findings from one of the Take Another Look activities. 460 CHAPTER 8 Area CHAPTER 9 The Pythagorean Theorem But serving up an action, suggesting the dynamic in the static, has become a hobby of mine....The “flowing” on that motionless plane holds my attention to such a degree that my preference is to try and make it into a cycle. M. C. ESCHER Waterfall, M |
. C. Escher, 1961 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● discover the Pythagorean Theorem, one of the most important concepts in mathematics ● use the Pythagorean Theorem to calculate the distance between any two points ● use conjectures related to the Pythagorean Theorem to solve problems L E S S O N 9.1 I am not young enough to know everything. OSCAR WILDE You will need ● scissors ● a compass ● a straightedge ● patty paper The Theorem of Pythagoras In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are called legs. In the figure at right, a and b represent the lengths of the legs, and c represents the length of the hypotenuse. In a right triangle, the side opposite the right angle is called the hypotenuse, here with length c. c b a The other two sides are legs, here with lengths a and b. FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North America Syndicate. There is a special relationship between the lengths of the legs and the length of the hypotenuse. This relationship is known today as the Pythagorean Theorem. Investigation The Three Sides of a Right Triangle The puzzle in this investigation is intended to help you recall the Pythagorean Theorem. It uses a dissection, which means you will cut apart one or more geometric figures and make the pieces fit into another figure. Construct a scalene right triangle in the middle of your paper. Label the hypotenuse c and the legs a and b. Construct a square on each side of the triangle. To locate the center of the square on the longer leg, draw its diagonals. Label the center O. a b c j O k Through point O, construct line j perpendicular to the hypotenuse and line k perpendicular to line j. Line k is parallel to the hypotenuse. Lines j and k divide the square on the longer leg into four parts. Step 1 Step 2 Step 3 Step 4 Cut out the square on the shorter leg and the four parts of the square on the longer leg. Arrange them to exactly cover the square on the hypotenuse. 462 CHAPTER 9 The Pythagorean Theorem Step 5 State the Pythagorean Theorem. The Pythagorean Theorem C-82 |
In a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. If a and b are the lengths of the legs, and c is the length of the hypotenuse, then?. History Pythagoras of Samos (ca. 569–475 B.C.E.), depicted in this statue, is often described as “the first pure mathematician.” Samos was a principal commercial center of Greece and is located on the island of Samos in the Aegean Sea. The ancient town of Samos now lies in ruins, as shown in the photo at right. Mysteriously, none of Pythagoras’s writings still exist, and we know very little about his life. He founded a mathematical society in Croton, in what is now Italy, whose members discovered irrational numbers and the five regular solids. They proved what is now called the Pythagorean Theorem, although it was discovered and used 1000 years earlier by the Chinese and Babylonians. Some math historians believe that the ancient Egyptians also used a special case of this property to construct right angles theorem is a conjecture that has been proved. Demonstrations like the one in the investigation are the first step toward proving the Pythagorean Theorem. Believe it or not, there are more than 200 proofs of the Pythagorean Theorem. Elisha Scott Loomis’s Pythagorean Proposition, first published in 1927, contains original proofs by Pythagoras, Euclid, and even Leonardo da Vinci and U. S. President James Garfield. One well-known proof of the Pythagorean Theorem is included below. You will complete another proof as an exercise. Paragraph Proof: The Pythagorean Theorem You need to show that a2 b2 equals c2 for the right triangles in the figure at left. The area of the entire square is a b2 or a2 2ab b2. The area of any triangle ab, so the sum of the areas of the four triangles is 2ab. The area of the is 1 2 quadrilateral in the center is a2 2ab b2 2ab, or a2 b2. If the quadrilateral in the center is a square then its area also equals c2. You now need to show that it is a square. You know that all the sides have length c, but you also need to show that the angles are right angles. The two |
acute angles in the right triangle, along with any angle of the quadrilateral, add up to 180°. The acute angles in a right triangle add up to 90°. Therefore the quadrilateral angle measures 90° and the quadrilateral is a square. If it is a square with side length c, then its area is c2. So, a2 b2 c2, which proves the Pythagorean Theorem. LESSON 9.1 The Theorem of Pythagoras 463 Acute triangle 7 6 8 62 72 82 Obtuse triangle 17 25 38 172 252 382 The Pythagorean Theorem works for right triangles, but does it work for all triangles? A quick check demonstrates that it doesn’t hold for other triangles. a 2 b 2 45 c 2 25 a 2 b 2 45 c 2 45 a 2 b 2 45 c 2 57. Let’s look at a few examples to see how you can use the Pythagorean Theorem to find the distance between two points. EXAMPLE A How high up on the wall will a 20-foot ladder touch if the foot of the ladder is placed 5 feet from the wall? Solution The ladder is the hypotenuse of a right triangle, so a2 b2 c2. 52 h2 202 25 h2 400 h2 375 h 375 19.4 Substitute. Multiply. Subtract 25 from both sides. Take the square root of each side. 20 ft h 5 ft The top of the ladder will touch the wall about 19.4 feet up from the ground. Notice that the exact answer in Example A is 375. However, this is a practical application, so you need to calculate the approximate answer. EXAMPLE B Find the area of the rectangular rug if the width is 12 feet and the diagonal measures 20 feet. 12 ft Solution Use the Pythagorean Theorem to find the length. a2 b2 c2 122 L2 202 144 L2 400 L2 256 L 256 L 16 The length is 16 feet. The area of the rectangle is 12 16, or 192 square feet. L 20 ft 464 CHAPTER 9 The Pythagorean Theorem EXERCISES In Exercises 1–11, find each missing length. All measurements are in centimeters. Give approximate answers accurate to the nearest tenth of a centimeter. 1. a? 5 4. d? 13 a 8 d 6 3. a? 2. c? 12 15 c 5 |
. s? 6. c? s s 24 10 6 6 8 a c 9. The base is a circle. x? 41 x 9 7. b? 8. x? 8 7 25 b 3.9 1.5 x 1.5 10. s? 11. r? 5 s y r (5, 12) (0, 0) x 12. A baseball infield is a square, each side measuring 90 feet. To the nearest foot, what is the distance from home plate to second base? Second base 13. The diagonal of a square measures 32 meters. What is the area of the square? 14. What is the length of the diagonal of a square whose area is 64 cm2? 15. The lengths of the three sides of a right triangle are consecutive integers. Find them. 16. A rectangular garden 6 meters wide has a diagonal measuring 10 meters. Find the perimeter of the garden. Home plate LESSON 9.1 The Theorem of Pythagoras 465 17. One very famous proof of the Pythagorean Theorem is by the Hindu mathematician Bhaskara. It is often called the “Behold” proof because, as the story goes, Bhaskara drew the diagram at right and offered no verbal argument other than to exclaim, “Behold.” Use algebra to fill in the steps, explaining why this diagram proves the Pythagorean Theorem. History Bhaskara (1114–1185, India) was one of the first mathematicians to gain a thorough understanding of number systems and how to solve equations, several centuries before European mathematicians. He wrote six books on mathematics and astronomy, and led the astronomical observatory at Ujjain. 18. Is ABC XYZ? Explain your reasoning. C Z 4 cm 4 cm A 5 cm B X 5 cm Y Review 19. The two quadrilaterals are squares. 20. Give the vertex arrangement for the Find x. 2-uniform tessellation. 225 cm2 36 cm2 x 21. Explain why m n 120°. m 120° n 466 CHAPTER 9 The Pythagorean Theorem 22. Calculate each lettered angle, measure, 2 are or arc. EF is a diameter; tangents. 1 and F u t 1 58 106° 2 CREATING A GEOMETRY FLIP BOOK Have you ever fanned the pages of a flip book and watched the pictures seem to move? Each page shows a picture slightly different from the |
previous one. Flip books are basic to animation technique. For more information about flip books, see www.keymath.com/DG. These five frames start off the photo series titled The Horse in Motion, by photographer, innovator, and motion picture pioneer Eadweard Muybridge (1830–1904). Here are two dissections that you can animate to demonstrate the Pythagorean Theorem. (You used another dissection in the Investigation The Three Sides of a Right Triangle.) b a a You could also animate these drawings to demonstrate area formulas. Choose one of the animations mentioned above and create a flip book that demonstrates it. Be ready to explain how your flip book demonstrates the formula you chose. Here are some practical tips. Draw your figures in the same position on each page so they don’t jump around when the pages are flipped. Use graph paper or tracing paper to help. The smaller the change from picture to picture, and the more pictures there are, the smoother the motion will be. Label each picture so that it’s clear how the process works. LESSON 9.1 The Theorem of Pythagoras 467 L E S S O N 9.2 Any time you see someone more successful than you are, they are doing something you aren’t. MALCOLM X The Converse of the Pythagorean Theorem In Lesson 9.1, you saw that if a triangle is a right triangle, then the square of the length of its hypotenuse is equal to the sum of the squares of the lengths of the two legs. What about the converse? If x, y, and z are the lengths of the three sides of a triangle and they satisfy the Pythagorean equation, a2 b2 c2, must the triangle be a right triangle? Let’s find out. You will need ● string ● a ruler ● paper clips ● a piece of patty paper Investigation Is the Converse True? Three positive integers that work in the Pythagorean equation are called Pythagorean triples. For example, 8-15-17 is a Pythagorean triple because 82 152 172. Here are nine sets of Pythagorean triples. 5-12-13 10-24-26 7-24-25 8-15-17 16-30-34 3-4-5 6-8-10 9-12-15 12-16-20 Step 1 Select one set of Pythagorean |
triples from the list above. Mark off four points, A, B, C, and D, on a string to create three consecutive lengths from your set of triples. 8 cm B A 15 cm 17 cm D 1 2 3 4 6 7 8 9 11 12 13 14 16 17 15 10 5 C 0 Step 2 Step 3 Loop three paper clips onto the string. Tie the ends together so that points A and D meet. Three group members should each pull a paper clip at point A, B, or C to stretch the string tight. 468 CHAPTER 9 The Pythagorean Theorem Step 4 Step 5 With your paper, check the largest angle. What type of triangle is formed? Select another set of triples from the list. Repeat Steps 1–4 with your new lengths. Step 6 Compare results in your group. State your results as your next conjecture. Converse of the Pythagorean Theorem C-83 If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle?. This ancient Babylonian tablet, called Plimpton 322, dates sometime between 1900 and 1600 B.C.E. It suggests several advanced Pythagorean triples, such as 1679-2400-2929. History Some historians believe Egyptian “rope stretchers” used the Converse of the Pythagorean Theorem to help reestablish land boundaries after the yearly flooding of the Nile and to help construct the pyramids. Some ancient tombs show workers carrying ropes tied with equally spaced knots. For example, 13 equally spaced knots would divide the rope into 12 equal lengths. If one person held knots 1 and 13 together, and two others held the rope at knots 4 and 8 and stretched it tight, they could have created a 3-4-5 right triangle. LESSON 9.2 The Converse of the Pythagorean Theorem 469 The proof of the Converse of the Pythagorean Theorem is very interesting because it is one of the few instances where the original theorem is used to prove the converse. Let’s take a look. One proof is started for you below. You will finish it as an exercise. Proof: Converse of the Pythagorean Theorem Conjecture: If the lengths of the three sides of a triangle work in the Pythagorean equation, then the triangle is a right triangle. C a b Given: a, b, c are the lengths of the sides of ABC and a2 |
b2 c2 Show: ABC is a right triangle Plan: Begin by constructing a second triangle, right triangle DEF (with F a right angle), with legs of lengths a and b and hypotenuse of length x. The plan is to show that x c, so that the triangles are congruent. Then show that C and F are congruent. Once you show that C is a right angle, then ABC is a right triangle and the proof is complete EXERCISES In Exercises 1–6, use the Converse of the Pythagorean Theorem to determine whether each triangle is a right triangle. 1. 8 4. 17 15 22 18 12 2. 5. 130 50 120 3. 12 6. 36 35 10 20 1.73 1.41 24 2.23 In Exercises 7 and 8, use the Converse of the Pythagorean Theorem to solve each problem. 7. Is a triangle with sides measuring 9 feet, 12 feet, and 18 feet a right triangle? 8. A window frame that seems rectangular has height 408 cm, length 306 cm, and one diagonal with length 525 cm. Is the window frame really rectangular? Explain. 470 CHAPTER 9 The Pythagorean Theorem In Exercises 9–11, find y. 9. Both quadrilaterals 10. y 11. are squares. 15 cm y 25 cm2 (–7, y) 25 x 25 m y 18 m 12. The lengths of the three sides of a right triangle are consecutive even integers. Find them. 13. Find the area of a right triangle with hypotenuse length 17 cm and one leg length 15 cm. 14. How high on a building will a 15-foot ladder touch if the foot of the ladder is 5 feet from the building? 15. The congruent sides of an isosceles triangle measure 6 cm, and the base measures 8 cm. Find the area. 16. Find the amount of fencing in linear feet needed for the perimeter of a rectangular lot with a diagonal length 39 m and a side length 36 m. 17. A rectangular piece of cardboard fits snugly on a diagonal in this box. a. What is the area of the cardboard rectangle? b. What is the length of the diagonal of the cardboard rectangle? 18. Look back at the start of the proof of the Converse of the Pythagorean Theorem. Copy the conjecture, the given, the show, the plan, and the two diagrams. Use |
the plan to complete the proof. 20 cm 40 cm 60 cm 19. What’s wrong with this picture? 20. Explain why ABC is a right triangle. 3.75 cm 2 cm 4.25 cm Review 21. Identify the point of concurrency from the construction marks. LESSON 9.2 The Converse of the Pythagorean Theorem 471 22. Line CF is tangent to circle D at C. The arc measure of CE a. why x 1 2 is a. Explain 23. What is the probability of randomly selecting three points that form an isosceles triangle from the 10 points in this isometric grid? D E C a x F 24. If the pattern of blocks continues, what will be the surface area of the 50th solid in the pattern? 25. Sketch the solid shown, but with the two blue cubes removed and the red cube moved to cover the visible face of the green cube. The outlines of stacked cubes create a visual impact in this untitled module unit sculpture by conceptual artist Sol Lewitt. IMPROVING YOUR ALGEBRA SKILLS Algebraic Sequences III Find the next three terms of this algebraic sequence. x9, 9x8y, 36x7y2, 84x6y3, 126x5y4, 126x4y5, 84x3y6,?,?,? 472 CHAPTER 9 The Pythagorean Theorem ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 8 ● USING YO USING YOUR ALGEBRA SKILLS 8 Radical Expressions When you work with the Pythagorean Theorem, you often get radical expressions, such as 50. Until now you may have left these expressions as radicals, or you may have found a decimal approximation using a calculator. Some radical expressions can be simplified. To simplify a square root means to take the square root of any perfect-square factors of the number under the radical sign. Let’s look at an example. EXAMPLE A Simplify 50. Solution One way to simplify a square root is to look for perfect-square factors. The largest perfect-square factor of 50 is 25. 50 25 2 25 2 5 2 25 is a perfect square, so you can take its square root. Another approach is to factor the number as far as possible with prime factors. Write 50 as a set |
of prime factors. Look for any square factors (factors that appear twice). 50 5 5 2 52 2 52 2 5 2 Squaring and taking the square root are inverse operations— they undo each other. So, equals 5. 52 You might argue that 52 doesn’t look any simpler than 50. However, in the days before calculators with square root buttons, mathematicians used paper-andpencil algorithms to find approximate values of square roots. Working with the smallest possible number under the radical made the algorithms easier to use. Giving an exact answer to a problem involving a square root is important in a number of situations. Some patterns are easier to discover with simplified square roots than with decimal approximations. Standardized tests often express answers in simplified form. And when you multiply radical expressions, you often have to simplify the answer. USING YOUR ALGEBRA SKILLS 8 Radical Expressions 473 ALGEBRA SKILLS 8 ● USING YOUR ALGEBRA SKILLS 8 ● USING YOUR ALGEBRA SKILLS 8 ● USING YO EXAMPLE B Multiply 36 by 52. Solution To multiply radical expressions, associate and multiply the quantities outside the radical sign, and associate and multiply the quantities inside the radical sign. 3652 3 5 6 2 15 12 15 4 3 15 23 303 EXERCISES In Exercises 1–5, express each product in its simplest form. 1. 32 2. 52 3. 3623 4. 732 5. 222 In Exercises 6–20, express each square root in its simplest form. 6. 18 7. 40 8. 75 11. 576 12. 720 13. 722 9. 85 14. 784 10. 96 15. 828 16. 2952 17. 5248 18. 8200 19. 11808 20. 16072 21. What is the next term in the pattern? 2952, 5248, 8200, 11808, 16072,... IMPROVING YOUR VISUAL THINKING SKILLS Folding Cubes II Each cube has designs on three faces. When unfolded, which figure at right could it become? 1. 2. A. A. B. B. C. C. D. D. 474 CHAPTER 9 The Pythagorean Theorem L E S S O N 9.3 In an isosceles triangle, the sum of the square roots of the two equal sides is |
equal to the square root of the third side. THE SCARECROW IN THE 1939 FILM THE WIZARD OF OZ Two Special Right Triangles In this lesson you will use the Pythagorean Theorem to discover some relationships between the sides of two special right triangles. One of these special triangles is an isosceles right triangle, also called a 45°-45°-90° triangle. Each isosceles right triangle is half a square, so they show up often in mathematics and engineering. In the next investigation, you will look for a shortcut for finding the length of an unknown side in a 45°-45°-90° triangle. 45° 45° An isosceles right triangle Step 1 Step 2 Step 3 Investigation 1 Isosceles Right Triangles Sketch an isosceles right triangle. Label the legs l and the hypotenuse h. Pick any integer for l, the length of the legs. Use the Pythagorean Theorem to find h. Simplify the square root. Repeat Step 2 with several different values for l. Share results with your group. Do you see any pattern in the relationship between l and h? 10 Step 4 State your next conjecture in terms of length l.? Isosceles Right Triangle Conjecture l h l 1? 1 10 2 2? 3 3? C-84 In an isosceles right triangle, if the legs have length l, then the hypotenuse has length?. LESSON 9.3 Two Special Right Triangles 475 You can also demonstrate this property on a geoboard or graph paper, as shown at right The other special right triangle is a 30°-60°-90° triangle. If you fold an equilateral triangle along one of its altitudes, the triangles you get are 30°-60°-90° triangles. A 30°-60°-90° triangle is half an equilateral triangle, so it also shows up often in mathematics and engineering. Let’s see if there is a shortcut for finding the lengths of its sides. 3 60° 30° A 30°-60°-90° triangle Investigation 2 30º-60º-90º Triangles Let’s start by using a little deductive thinking to find the relationships in 30°-60°-90° triangles. Triangle ABC is equilateral, and CD is an altitude. C Step 1 Step 2 Step 3 Step 4 Step 5 What are mA and |
mB? What are mACD and mBCD? What are mADC and mBDC? Is ADC BDC? Why? Is AD BD? Why? How do AC and AD compare? In a 30°-60°-90° triangle, will this relationship between the hypotenuse and the shorter leg always hold true? Explain. D A B Sketch a 30°-60°-90° triangle. Choose any integer for the length of the shorter leg. Use the relationship from Step 3 and the Pythagorean Theorem to find the length of the other leg. Simplify the square root. Repeat Step 4 with several different values for the length of the shorter leg. Share results with your group. What is the relationship between the lengths of the two legs? You should notice a pattern in your answers. 7? 14 2 1? 4? 2 Step 6 State your next conjecture in terms of the length of the shorter leg, a. 30°-60°-90° Triangle Conjecture C-85 In a 30°-60°-90° triangle, if the shorter leg has length a, then the longer leg has length? and the hypotenuse has length?. 476 CHAPTER 9 Pythagorean Theorem You can use algebra to verify that the conjecture will hold true for any 30°-60°-90° triangle. Proof: 30°-60°-90° Triangle Conjecture 2a2 a2 b2 4a2 a2 b2 3a2 b2 a3 b Start with the Pythagorean Theorem. Square 2a. Subtract a2 from both sides. Take the square root of both sides. 30° b 2a a Although you investigated only integer values, the proof shows that any number, even a non-integer, can be used for a. You can also demonstrate this property for integer values on isometric dot paper EXERCISES In Exercises 1–8, use your new conjectures to find the unknown lengths. All measurements are in centimeters. You will need Construction tools for Exercises 19 and 20 1. a? 2. b? 3. a?, b? a 72 13 2 b 60° 5 a b 4. c?, d? 5. e?, f? 6. What is the perimeter of 60° d 20 c f 17 3 30° e square SQRE? E 18 2 S R Q LESSON 9.3 Two Special Right Tri |
angles 477 7. The solid is a cube. d? H d F D 12 cm B E A G C 8. g?, h? 9. What is the area of the triangle? 30° h g 120 130 10. Find the coordinates of P. 11. What’s wrong with this picture? y P (?,?) 45° x (1, 0) (0, –1) 8 60° 15 30° 17 12. Sketch and label a figure to demonstrate that 27 is equivalent to 33. (Use isometric dot paper to aid your sketch.) 13. Sketch and label a figure to demonstrate that 32 is equivalent to 42. (Use square dot paper or graph paper.) 14. In equilateral triangle ABC, AE, BF, and CD are all angle bisectors, medians, and altitudes simultaneously. These three segments divide the equilateral triangle into six overlapping 30°-60°-90° triangles and six smaller, non-overlapping 30°-60°-90° triangles. a. One of the overlapping triangles is CDB. Name the other five triangles that are congruent to it. b. One of the non-overlapping triangles is MDA. Name the other A five triangles congruent to it. 15. Use algebra and deductive reasoning to show that the Isosceles Right Triangle Conjecture holds true for any isosceles right triangle. Use the figure at right. 16. Find the area of an equilateral triangle whose sides measure 26 meters. 17. An equilateral triangle has an altitude that measures 26 meters. Find the area of the triangle to the nearest square meter. 18. Sketch the largest 45°-45°-90° triangle that fits in a 30°-60°-90° triangle. What is the ratio of the area of the 30°-60°-90° triangle to the area of the 45°-45°-90° triangle? 478 CHAPTER 9 The Pythagorean Theorem Review Construction In Exercises 19 and 20, choose either patty paper or a compass and straightedge and perform the constructions. 19. Given the segment with length a below, construct segments with lengths a2, a3, and a5. a 20. Mini-Investigation Draw a right triangle with sides of lengths 6 cm, 8 cm, and 10 cm. Locate the midpoint of each side. Construct a semicircle on each side with the midpoints |
of the sides as centers. Find the area of each semicircle. What relationship do you notice among the three areas? 21. The Jiuzhang suanshu is an ancient Chinese mathematics text of 246 problems. Some solutions use the gou gu, the Chinese name for what we call the Pythagorean Theorem. The gou gu reads gou2 gu2 (xian)2. Here is a gou gu problem translated from the ninth chapter of Jiuzhang. A rope hangs from the top of a pole with three chih of it lying on the ground. When it is tightly stretched so that its end just touches the ground, it is eight chih from the base of the pole. How long is the rope? 22. Explain why m1 m2 90°. 23. The lateral surface area of the cone below is unwrapped into a sector. What is the angle at the vertex of the sector? 1 2 l? l r l 27 cm, r 6 cm IMPROVING YOUR VISUAL THINKING SKILLS Mudville Monsters The 11 starting members of the Mudville Monsters football team and their coach, Osgood Gipper, have been invited to compete in the Smallville Punt, Pass, and Kick Competition. To get there, they must cross the deep Smallville River. The only way across is with a small boat owned by two very small Smallville football players. The boat holds just one Monster visitor or the two Smallville players. The Smallville players agree to help the Mudville players across if the visitors agree to pay $5 each time the boat crosses the river. If the Monsters have a total of $100 among them, do they have enough money to get all players and the coach to the other side of the river? LESSON 9.3 Two Special Right Triangles 479 A Pythagorean Fractal If you wanted to draw a picture to state the Pythagorean Theorem without words, you’d probably draw a right triangle with squares on each of the three sides. This is the way you first explored the Pythagorean Theorem in Lesson 9.1. Another picture of the theorem is even simpler: a right triangle divided into two right triangles. Here, a right triangle with hypotenuse c is divided into two smaller triangles, the smaller with hypotenuse a and the larger with hypotenuse b. Clearly, their areas add up to the area of the whole triangle. What’s surprising is that all three triangles have the |
same angle measures. Why? Though different in size, the three triangles all have the same shape. Figures that have the same shape but not necessarily the same size are called similar figures. You’ll use these similar triangles to prove the Pythagorean Theorem in a later chapter. a b c a b A beautifully complex fractal combines both of these pictorial representations of the Pythagorean Theorem. The fractal starts with a right triangle with squares on each side. Then similar triangles are built onto the squares. Then squares are built onto the new triangles, and so on. In this exploration, you’ll create this fractal. 480 CHAPTER 9 The Pythagorean Theorem You will need ● the worksheet The Right Triangle Fractal (optional) Activity The Right Triangle Fractal The Geometer’s Sketchpad software uses custom tools to save the steps of repeated constructions. They are very helpful for fractals like this one. Step 1 Use The Geometer’s Sketchpad to create the fractal on page 480. Follow the Procedure Note. Notice that each square has two congruent triangles on two opposite sides. Use a reflection to guarantee that the triangles are congruent. 1. Use the diameter of a circle and an inscribed angle to make a triangle that always remains a right triangle. 2. It is important to construct the altitude to the hypotenuse in each triangle in order to divide it into similar triangles. 3. Create custom tools to make squares and similar triangles repeatedly. Step 2 Step 3 Step 4 After you successfully make the Pythagorean fractal, you’re ready to investigate its fascinating patterns. First, try dragging a vertex of the original triangle. Does the Pythagorean Theorem still apply to the branches of this figure? That is, does the sum of the areas of the branches on the legs equal the area of the branch on the hypotenuse? See if you can answer without actually measuring all the areas. Consider your original sketch to be a single right triangle with a square built on each side. Call this sketch Stage 0 of your fractal. Explore these questions. a. At Stage 1, you add three triangles and six squares to your construction. On a piece of paper, draw a rough sketch of Stage 1. How much area do you add to this fractal between Stage 0 and Stage 1? (Don’t measure any areas to answer this.) b. Draw a rough sketch of Stage 2. How much area |
do you add between Stage 1 and Stage 2? c. How much area is added at any new stage? d. A true fractal exists only after an infinite number of stages. If you could build a true fractal based on the construction in this activity, what would be its total area? Step 5 Give the same color and shade to sets of squares that are congruent. What do you notice about these sets of squares other than their equal area? Describe any patterns you find in sets of congruent squares. Step 6 Describe any other patterns you can find in the Pythagorean fractal. EXPLORATION A Pythagorean Fractal 481 Story Problems You have learned that drawing a diagram will help you to solve difficult problems. By now you know to look for many special relationships in your diagrams, such as congruent polygons, parallel lines, and right triangles. L E S S O N 9.4 You may be disappointed if you fail, but you are doomed if you don’t try. BEVERLY SILLS FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North America Syndicate. EXAMPLE What is the longest stick that will fit inside a 24-by-30-by-18-inch box? Solution Draw a diagram. You can lay a stick with length d diagonally at the bottom of the box. But you can position an even longer stick with length x along the diagonal of the box, as shown. How long is this stick? 18 in. x 24 in. d 30 in. Both d and x are the hypotenuses of right triangles, but finding d 2 will help you find x. 302 242 d2 900 576 d2 d2 1476 d2 182 x2 1476 182 x2 1476 324 x2 1800 x2 x 42.4 The longest possible stick is about 42.4 in. EXERCISES 1. A giant California redwood tree 36 meters tall cracked in a violent storm and fell as if hinged. The tip of the once beautiful tree hit the ground 24 meters from the base. Researcher Red Woods wishes to investigate the crack. How many meters up from the base of the tree does he have to climb? x 24 m 2. Amir’s sister is away at college, and he wants to mail her a 34 in. baseball bat. The packing service sells only one kind of box, which measures 24 in. by 2 |
in. by 18 in. Will the box be big enough? 482 CHAPTER 9 The Pythagorean Theorem 3. Meteorologist Paul Windward and geologist Rhaina Stone are rushing to a paleontology conference in Pecos Gulch. Paul lifts off in his balloon at noon from Lost Wages, heading east for Pecos Gulch Conference Center. With the wind blowing west to east, he averages a land speed of 30 km/hr. This will allow him to arrive in 4 hours, just as the conference begins. Meanwhile, Rhaina is 160 km north of Lost Wages. At the moment of Paul’s lift off, Rhaina hops into an off-roading vehicle and heads directly for the conference center. At what average speed must she travel to arrive at the same time Paul does? Career Meteorologists study the weather and the atmosphere. They also look at air quality, oceanic influence on weather, changes in climate over time, and even other planetary climates. They make forecasts using satellite photographs, weather balloons, contour maps, and mathematics to calculate wind speed or the arrival of a storm. 4. A 25-foot ladder is placed against a building. The bottom of the ladder is 7 feet from the building. If the top of the ladder slips down 4 feet, how many feet will the bottom slide out? (It is not 4 feet.) 5. The front and back walls of an A-frame cabin are isosceles triangles, each with a base measuring 10 m and legs measuring 13 m. The entire front wall is made of glass 1 cm thick that cost $120/m2. What did the glass for the front wall cost? 6. A regular hexagonal prism fits perfectly inside a cylindrical box with diameter 6 cm and height 10 cm. What is the surface area of the prism? What is the surface area of the cylinder? 7. Find the perimeter of an equilateral triangle whose median measures 6 cm. 8. APPLICATION According to the Americans with Disabilities Act, the slope of a wheelchair ramp must be no greater than 1. What is the length of ramp needed to gain a height of 2 1 4 feet? Read the Science Connection on the top of page 484 and then figure out how much force is required to go up the ramp if a person and a wheelchair together weigh 200 pounds. LESSON 9.4 Story Problems 483 Science It takes less effort to roll objects up an inclined plane, or ramp, than to lift |
them straight up. Work is a measure of force applied over distance, and you calculate it as a product of force and distance. For example, a force of 100 pounds is required to hold up a 100-pound object. The work required to lift it 2 feet is 200 foot-pounds. But if you use a 4-foot-long ramp to roll it up, you’ll do the 200 foot-pounds of work over a 4-foot distance. So you need to apply only 50 pounds of force at any given moment. For Exercises 9 and 10, refer to the above Science Connection about inclined planes. 9. Compare what it would take to lift an object these three different ways. a. How much work, in foot-pounds, is necessary to lift 80 pounds straight up 2 feet? b. If a ramp 4 feet long is used to raise the 80 pounds up 2 feet, how much force, in pounds, will it take? c. If a ramp 8 feet long is used to raise the 80 pounds up 2 feet, how much force, in pounds, will it take? 10. If you can exert only 70 pounds of force and you need to lift a 160-pound steel drum up 2 feet, what is the minimum length of ramp you should set up? Review Recreation This set of enameled porcelain qi qiao bowls can be arranged to form a 37-by-37 cm square (as shown) or other shapes, or used separately. Each bowl is 10 cm deep. Dishes of this type are usually used to serve candies, nuts, dried fruits, and other snacks on special occasions. The qi qiao, or tangram puzzle, originated in China and consists of seven pieces—five isosceles right triangles, a square, and a parallelogram. The puzzle involves rearranging the pieces into a square, or hundreds of other shapes (a few are shown below). Private collection, Berkeley, California. Photo by Cheryl Fenton. Swan Cat Rabbit Horse with Rider 11. If the area of the red square piece is 4 cm2, what are the dimensions of the other six pieces? 484 CHAPTER 9 The Pythagorean Theorem 12. Make a set of your own seven tangram pieces and create the Cat, Rabbit, Swan, and Horse with Rider as shown on page 484. 13. Find the radius of circle Q. 14. Find the length of AC. (?, 6) r y 150° Q |
x 45° A C 36 cm 30° B 15. The two rays are tangent to the circle. What’s wrong with this picture? A B 54° C D 226° 16. In the figure below, point A is the image of point A after a reflection over OT. What are the coordinates of A? 17. Which congruence shortcut can you use to show that ABP DCP? 18. Identify the point of concurrency in QUO from the construction marks. y A T 30° O A (8, 0 19. In parallelogram QUID, mQ 2x 5° and mI 4x 55°. What is mU? 20. In PRO, mP 70° and mR 45°. Which side of the triangle is the shortest? IMPROVING YOUR VISUAL THINKING SKILLS Fold, Punch, and Snip A square sheet of paper is folded vertically, a hole is punched out of the center, and then one of the corners is snipped off. When the paper is unfolded it will look like the figure at right. Sketch what a square sheet of paper will look like when it is unfolded after the following sequence of folds, punches, and snips. Fold once. Fold twice. Snip double-fold corner. Punch opposite corner. LESSON 9.4 Story Problems 485 L E S S O N 9.5 We talk too much; we should talk less and draw more. JOHANN WOLFGANG VON GOETHE Distance in Coordinate Geometry Viki is standing on the corner of Seventh Street and 8th Avenue, and her brother Scott is on the corner of Second Street and 3rd Avenue. To find her shortest sidewalk route to Scott, Viki can simply count blocks. But if Viki wants to know her diagonal distance to Scott, she would need the Pythagorean Theorem to measure across blocks. V 8th Ave 7th Ave 6th Ave 5th Ave 4th Ave 3rd Ave 2nd Ave 1st Ave You can think of a coordinate plane as a grid of streets with two sets of parallel lines running perpendicular to each other. Every segment in the plane that is not in the x- or y-direction is the hypotenuse of a right triangle whose legs are in the x- and y-directions. So you can use the Pythagorean Theorem to find the distance between any two points on a coordinate plane. y x You will need ● |
graph paper Investigation 1 The Distance Formula In Steps 1 and 2, find the length of each segment by using the segment as the hypotenuse of a right triangle. Simply count the squares on the horizontal and vertical legs, then use the Pythagorean Theorem to find the length of the hypotenuse. Step 1 Copy graphs a–d from the next page onto your own graph paper. Use each segment as the hypotenuse of a right triangle. Draw the legs along the grid lines. Find the length of each segment. 486 CHAPTER 9 The Pythagorean Theorem a. y 5 5 x c. y 5 b. y 5 d. x 5 y –5 x x 5 –4 Step 2 Graph each pair of points, then find the distances between them. a. (1, 2), (11, 7) b. (9,6), (3, 10) What if the points are so far apart that it’s not practical to plot them? For example, what is the distance between the points A(15, 34) and B(42, 70)? A formula that uses the coordinates of the given points would be helpful. To find this formula, you first need to find the lengths of the legs in terms of the x- and y-coordinates. From your work with slope triangles, you know how to calculate horizontal and vertical distances. B (42, 70)? A (15, 34)? Step 3 Step 4 Step 5 Step 6 Write an expression for the length of the horizontal leg using the x-coordinates. Write a similar expression for the length of the vertical leg using the y-coordinates. Use your expressions from Steps 3 and 4, and the Pythagorean Theorem, to find the distance between points A(15, 34) and B(42, 70). Generalize what you have learned about the distance between two points in a coordinate plane. Copy and complete the conjecture below. Distance Formula The distance between points Ax1, y1 and Bx2, y2 is given by C-86 AB2? 2? 2 or AB? 2? 2 Let’s look at an example to see how you can apply the distance formula. LESSON 9.5 Distance in Coordinate Geometry 487 EXAMPLE A Find the distance between A(8, 15) and B(7, 23). Solution AB2 x2 x1 y1 2 y2 2 7 82 23 152 152 82 AB2 |
289 AB 17 The distance formula. Substitute 8 for x1, 15 for y1, 7 for x2, and 23 for y2. Subtract. Square 15 and 8 and add. Take the square root of both sides. The distance formula is also used to write the equation of a circle. EXAMPLE B Write an equation for the circle with center (5, 4) and radius 7 units. Solution Let (x, y) represent any point on the circle. The distance from (x, y) to the circle’s center, (5, 4), is 7. Substitute this information in the distance formula. (x 5)2 (y 4)2 72 y (x, y) x (5, 4) Substitute (x, y) for (x2, y2). Substitute (5, 4) for (x1, y1). Substitute 7 as the distance. So, the equation in standard form is x 52 y 42 72. You will need ● graph paper Investigation 2 The Equation of a Circle Find equations for a few more circles and then generalize the equation for any circle with radius r and center (h, k). Step 1 Given its center and radius, graph each circle on graph paper. a. Center (1, 2), r 8 c. Center (3, 4), r 10 b. Center (0, 2), r 6 Step 2 Select any point on each circle; label it (x, y). Use the distance formula to write an equation expressing the distance between the center of each circle and (x, y). Step 3 Copy and complete the conjecture for the equation of a circle. Equation of a Circle The equation of a circle with radius r and center (h, k) is -87 r (h, k) (x, y) x 488 CHAPTER 9 The Pythagorean Theorem Let’s look at an example that uses the equation of a circle in reverse. EXAMPLE C Find the center and radius of the circle x 22 y 52 36. Solution Rewrite the equation of the circle in the standard form. x h2 y k2 r 2 x (2)2 y 52 62 Identify the values of h, k, and r. The center is (2, 5) and the radius is 6. EXERCISES In Exercises 1–3, find the distance between each pair of points. 1. (10, 20), (13 |
, 16) 2. (15, 37), (42, 73) 3. (19, 16), (3, 14) 4. Look back at the diagram of Viki’s and Scott’s locations on page 486. Assume each block is approximately 50 meters long. What is the shortest distance from Viki to Scott to the nearest meter? 5. Find the perimeter of ABC with vertices A(2, 4), B(8, 12), and C(24, 0). 6. Determine whether DEF with vertices D(6, 6), E(39, 12), and F(24, 18) is scalene, isosceles, or equilateral. For Exercises 7 and 8, find the equation of the circle. 7. Center (0, 0), r 4 8. Center (2, 0), r 5 For Exercises 9 and 10, find the radius and center of the circle. 9. x 22 y 52 62 10. x2 y 12 81 11. The center of a circle is (3, 1). One point on the circle is (6, 2). Find the equation of the circle. 12. Mini-Investigation How would you find the distance between two points in a three-dimensional coordinate system? Investigate and make a conjecture. a. What is the distance from the origin (0, 0, 0) to (2, 1, 3)? b. What is the distance between P(1, 2, 3) and Q(5, 6, 15)? c. Complete this conjecture: are two If Ax1, y1, z1 and Bx2, y2, z2 points in a three-dimensional coordinate system, then the distance AB is? 2? 2?2. This point is the graph of the ordered triple (2, 1, 3). z 5 –4 y 5 –5 x 4 –5 LESSON 9.5 Distance in Coordinate Geometry 489 Review 13. Find the coordinates of A. 14. k?, m? A (?,?) y 150° (0, –1) x (1, 0) 15. The large triangle is equilateral. Find x and y. 60° k m 3 12 y x 16. Antonio is a biologist studying life in a pond. He needs to know how deep the water is. He notices a water lily sticking straight up from the water, whose blossom is 8 cm above the water |
’s surface. Antonio pulls the lily to one side, keeping the stem straight, until the blossom touches the water at a spot 40 cm from where the stem first broke the water’s surface. How is Antonio able to calculate the depth of the water? What is the depth? 17. CURT is the image of CURT under a R' rotation transformation. Copy the polygon and its image onto patty paper. Find the center of rotation and the measure of the angle of rotation. Explain your method. C' U T R C T' U' IMPROVING YOUR VISUAL THINKING SKILLS The Spider and the Fly (attributed to the British puzzlist Henry E. Dudeney, 1857–1930) In a rectangular room, measuring 30 by 12 by 12 feet, a spider is at point A on the middle of one of the end walls, 1 foot from the ceiling. A fly is at point B on the center of the opposite wall, 1 foot from the floor. What is the shortest distance that the spider must crawl to reach the fly, which remains stationary? The spider never drops or uses its web, but crawls fairly. 12 ft 12 ft 30 ft 490 CHAPTER 9 The Pythagorean Theorem Ladder Climb Suppose a house painter rests a 20-foot ladder against a building, then decides the ladder needs to rest 1 foot higher against the building. Will moving the ladder 1 foot toward the building do the job? If it needs to be 2 feet lower, will moving the ladder 2 feet away from the building do the trick? Let’s investigate. You will need ● a graphing calculator Step 1 Step 2 Step 3 Step 4 Step 5 Activity Climbing the Wall Sketch a ladder leaning against a vertical wall, with the foot of the ladder resting on horizontal ground. Label the sketch using y for height reached by the ladder and x for the distance from the base of the wall to the foot of the ladder. Write an equation relating x, y, and the length of the ladder and solve it for y. You now have a function for the height reached by the ladder in terms of the distance from the wall to the foot of the ladder. Enter this equation into your calculator. Before you graph the equation, think about the settings you’ll want for the graph window. What are the greatest and least values possible for x and y? Enter reasonable settings, then graph the equation. Describe the shape of the graph |
. Trace along the graph, starting at x 0. Record values (rounded to the nearest 0.1 unit) for the height reached by the ladder when x 3, 6, 9, and 12. If you move the foot of the ladder away from the wall 3 feet at a time, will each move result in the same change in the height reached by the ladder? Explain. Find the value for x that gives a y-value approximately equal to x. How is this value related to the length of the ladder? Sketch the ladder in this position. What angle does the ladder make with the ground? Step 6 Should you lean a ladder against a wall in such a way that x is greater than y? Explain. How does your graph support your explanation? EXPLORATION Ladder Climb 491 L E S S O N 9.6 You must do things you think you cannot do. ELEANOR ROOSEVELT Circles and the Pythagorean Theorem In Chapter 6, you discovered a number of properties that involved right angles in and around circles. In this lesson you will use the conjectures you made, along with the Pythagorean Theorem, to solve some challenging problems. Let’s review two of the most useful conjectures. Tangent Conjecture: A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Angles Inscribed in a Semicircle Conjecture: Angles inscribed in a semicircle are right angles. Here are two examples that use these conjectures along with the Pythagorean Theorem. EXAMPLE A TA is tangent to circle N at A. TATA 123 cm. Find the area of the shaded region. T 30° G A N Solution TA is tangent at A, so TAN is a right angle and TAN is a 30°-60°-90° triangle. The longer leg is 123 cm, so the shorter leg (also the radius of the circle) is 12 cm. The area of the entire circle is 144 cm2. The area of the shaded region is 360 60, or 5, of the area of the circle. Therefore the shaded area is 6 36 0 5 (144), or 120 cm2. 6 EXAMPLE B AB 6 cm and BC 8 cm. Find the area of the circle. C B A Solution Inscribed angle ABC is a right angle, so ABC diameter. By the Pythagorean Theorem, if AB 6 |
cm and BC 8 cm, then AC 10 cm. Therefore the radius of the circle is 5 cm and the area of the circle is 25 cm2. is a semicircle and AC is a 492 CHAPTER 9 The Pythagorean Theorem EXERCISES In Exercises 1–4, find the area of the shaded region in each figure. Assume lines that appear tangent are tangent at the labeled points. You will need Construction tools for Exercise 20 1. OD 24 cm 2. HT 83 cm 3. HA 83 cm 4. HO 83 cm O B 105° Y D T 60° R A H H 120° R T 120° O H 5. A 3-meter-wide circular track is shown at right. The radius of the inner circle is 12 meters. What is the longest straight path that stays on the track? (In other words, find AB.) 6. An annulus has a 36 cm chord of the outer circle that is also tangent to the inner concentric circle. Find the area of the annulus. A T O B 7. In her latest expedition, Ertha Diggs has uncovered a portion of circular, terra-cotta pipe that she believes is part of an early water drainage system. To find the diameter of the original pipe, she lays a meterstick across the portion and measures the length of the chord at 48 cm. The depth of the portion from the midpoint of the chord is 6 cm. What was the pipe’s original diameter? 8. APPLICATION A machinery belt needs to be replaced. The belt runs around two wheels, crossing between them so that the larger wheel turns the smaller wheel in the opposite direction. The diameter of the larger wheel is 36 cm, and the diameter of the smaller is 24 cm. The distance between the centers of the two wheels is 60 cm. The belt crosses 24 cm from the center of the smaller wheel. What is the length of the belt? 24 36 cm 24 cm 60 cm 9. A circle of radius 6 has chord AB of length 6. If point C is selected randomly on the circle, what is the probability that ABC is obtuse? LESSON 9.6 Circles and the Pythagorean Theorem 493 In Exercises 10 and 11, each triangle is equilateral. Find the area of the inscribed circle and the area of the circumscribed circle. How many times greater is the area of the circumscribed circle than the area of the inscribed circle? |
10. AB 6 cm C 11. DE 23 cm F A B D E 12. The Gothic arch is based on the equilateral triangle. If the base of the arch measures 80 cm, what is the area of the shaded region? 13. Each of three circles of radius 6 cm is tangent to the other two, and they are inscribed in a rectangle, as shown. What is the height of the rectangle? 80 cm 80 cm 6 6 14. Sector ARC has a radius of 9 cm and an angle that measures 80°. When sector ARC is cut out and AR and RC are taped together, they form a cone. The length of AC becomes the circumference of the base of the cone. What is the height of the cone? 15. APPLICATION Will plans to use a circular cross section of wood to make a square table. The cross section has a circumference of 336 cm. To the nearest centimeter, what is the side length of the largest square that he can cut from it? C 9 cm h A 80° R x 16. Find the coordinates of point M. 17. Find the coordinates of point K. M y 135° x (1, 0) (0, –1) 494 CHAPTER 9 The Pythagorean Theorem y 210° x (1, 0) K (0, –1) Review 18. Find the equation of a circle with center (3, 3) and radius 6. 19. Find the radius and center of a circle with the equation x2 y2 2x 1 100. 20. Construction Construct a circle and a chord in a circle. With compass and straightedge, construct a second chord parallel and congruent to the first chord. Explain your method. 21. Explain why the opposite sides of a regular hexagon are parallel. 22. Find the rule for this number pattern???? 23. APPLICATION Felice wants to determine the diameter of a large heating duct. She places a carpenter’s square up to the surface of the cylinder, and the length of each tangent segment is 10 inches. a. What is the diameter? Explain your reasoning. b. Describe another way she can find the diameter of the duct. IMPROVING YOUR REASONING SKILLS Reasonable ’rithmetic I Each letter in these problems represents a different digit. 1. What is the value of B? 2. What is the value of J LESSON 9.6 Circles and the Pythagorean Theorem |
495 ● CHAPTER 11 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER CHAPTER 9 R E V I E W If 50 years from now you’ve forgotten everything else you learned in geometry, you’ll probably still remember the Pythagorean Theorem. (Though let’s hope you don’t really forget everything else!) That’s because it has practical applications in the mathematics and science that you encounter throughout your education. It’s one thing to remember the equation a2 b2 c2. It’s another to know what it means and to be able to apply it. Review your work from this chapter to be sure you understand how to use special triangle shortcuts and how to find the distance between two points in a coordinate plane. EXERCISES For Exercises 1–4, measurements are given in centimeters. You will need Construction tools for Exercise 30 1. x? 15 25 x 2. AB? C 3. Is ABC an acute, obtuse, or right triangle? C 70 A 240 260 B 13 12 A B 4. The solid is a rectangular prism. AB? A 6 8 B 24 5. Find the coordinates of 6. Find the coordinates of point U. y (0, 1) 30° U (?,?) x (1, 0) point V. y (0, 1) 225° V (?,?) x (1, 0) 7. What is the area of the 8. The area of this square is triangle? 144 cm2. Find d. 9. What is the area of trapezoid ABCD? 40 cm 30° d D 12 cm A C 20 cm 15 cm B 496 CHAPTER 9 The Pythagorean Theorem ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 10. The arc is a semicircle. What is the area of the shaded region? 11. Rays TA and TB are tangent to circle O at A and B respectively, and BT 63 cm. What is the area of the shaded region? 12. The quadrilateral is a square, and QE 22 cm. What is the area of the shaded region? 7 in. 25 in. T B 120° O A 13. The area of circle Q is 350 cm2. Find the area of square ABCD to the nearest 0. |
1 cm2. 14. Determine whether ABC with vertices A(3, 5), B(11, 3), and C(8, 8) is an equilateral, isosceles, or isosceles right triangle 15. Sagebrush Sally leaves camp on her dirt bike traveling east at 60 km/hr with a full tank of gas. After 2 hours, she stops and does a little prospecting—with no luck. So she heads north for 2 hours at 45 km/hr. She stops again, and this time hits pay dirt. Sally knows that she can travel at most 350 km on one tank of gas. Does she have enough fuel to get back to camp? If not, how close can she get? 16. A parallelogram has sides measuring 8.5 cm and 12 cm, and a diagonal measuring 15 cm. Is the parallelogram a rectangle? If not, is the 15 cm diagonal the longer or shorter diagonal? 17. After an argument, Peter and Paul walk away from each other on separate paths at a right angle to each other. Peter is walking 2 km/hr, and Paul is walking 3 km/hr. After 20 min, Paul sits down to think. After 30 min, Peter stops. Both decide to apologize. How far apart are they? How long will it take them to reach each other if they both start running straight toward each other at 5 km/hr? 18. Flora is away at camp and wants to mail her flute back home. The flute is 24 inches long. Will it fit diagonally within a box whose inside dimensions are 12 by 16 by 14 inches? 19. To the nearest foot, find the original height of a fallen flagpole that cracked and fell as if hinged, forming an angle of 45 degrees with the ground. The tip of the pole hit the ground 12 feet from its base. CHAPTER 9 REVIEW 497 EW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTE 20. You are standing 12 feet from a cylindrical corn-syrup storage tank. The distance from you to a point of tangency on the tank is 35 feet. What is the radius of the tank? Technology Radio and TV stations broadcast from high towers. Their signals are picked up by radios and TVs in homes within a certain radius. Because Earth is spherical, these signals don’t get picked up beyond the point of |
tangency. r r 35 feet 12 feet 21. APPLICATION Read the Technology Connection above. What is the maximum broadcasting radius from a radio tower 1800 feet tall (approximately 0.34 mile)? The radius of Earth is approximately 3960 miles, and you can assume the ground around the tower is nearly flat. Round your answer to the nearest 10 miles. 20 m 25 m 22. A diver hooked to a 25-meter line is searching for the remains of a Spanish galleon in the Caribbean Sea. The sea is 20 meters deep and the bottom is flat. What is the area of circular region that the diver can explore? 23. What are the lengths of the two legs of a 30°-60°-90° triangle if the length of the hypotenuse is 123? 24. Find the side length of an equilateral triangle with an area of 363 m2. 25. Find the perimeter of an equilateral triangle with a height of 73. 26. Al baked brownies for himself and his two sisters. He divided the square pan of brownies into three parts. He measured three 30° angles at one of the corners so that two pieces formed right triangles and the middle piece formed a kite. Did he divide the pan of brownies equally? Draw a sketch and explain your reasoning. 27. A circle has a central angle AOB that measures 80°. If point C is selected randomly on the circle, what is the probability that ABC is obtuse? 498 CHAPTER 9 The Pythagorean Theorem ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 28. One of the sketches below shows the greatest area that you can enclose in a right- angled corner with a rope of length s. Which one? Explain your reasoning. s s1_ 2 s1_ 2 s A triangle A square A quarter-circle 29. A wire is attached to a block of wood at point A. 1.4 m The wire is pulled over a pulley as shown. How far will the block move if the wire is pulled 1.4 meters in the direction of the arrow? 3.9 m 1.5 m A? MIXED REVIEW 30. Construction Construct an isosceles triangle that has a base length equal to half the length of one leg. 31. In a regular octagon inscribed in a circle, how many diagonals pass through the center of the circle? In a |
regular nonagon? a regular 20-gon? What is the general rule? 32. A bug clings to a point two inches from the center of a spinning fan blade. The blade spins around once per second. How fast does the bug travel in inches per second? In Exercises 33–40, identify the statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 33. The area of a rectangle and the area of a parallelogram are both given by the formula A bh, where A is the area, b is the length of the base, and h is the height. 34. When a figure is reflected over a line, the line of reflection is perpendicular to every segment joining a point on the original figure with its image. 35. In an isosceles right triangle, if the legs have length x, then the hypotenuse has length x3. 36. The area of a kite or a rhombus can be found by using the formula A (0.5)d1d2, where A is the area and d1 and d2 are the lengths of the diagonals. 37. If the coordinates of points A and B are x1, y1 2 x2 AB x1 y2 y1 2. and x2, y2, respectively, then 38. A glide reflection is a combination of a translation and a rotation. CHAPTER 9 REVIEW 499 EW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTE 39. Equilateral triangles, squares, and regular octagons can be used to create monohedral tessellations. 40. In a 30°-60°-90° triangle, if the shorter leg has length x, then the longer leg has length x3 and the hypotenuse has length 2x. In Exercises 41–46, select the correct answer. 41. The hypotenuse of a right triangle is always?. A. opposite the smallest angle and is the shortest side. B. opposite the largest angle and is the shortest side. C. opposite the smallest angle and is the longest side. D. opposite the largest angle and is the longest side. 42. The area of a triangle is given by the formula?, where A is the area, b is the length of the base, and h is the height. A. A b |
h C. A 2bh B. A 1 bh 2 D. A b2h 43. If the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle must be a(n)? triangle. A. right C. obtuse B. acute D. scalene 44. The ordered pair rule (x, y) → (y, x) is a?. A. reflection over the x-axis C. reflection over the line y x B. reflection over the y-axis D. rotation 90° about the origin 45. The composition of two reflections over two intersecting lines is equivalent to?. A. a single reflection C. a rotation B. a translation D. no transformation 46. The total surface area of a cone is equal to?, where r is the radius of the circular base and l is the slant height. A. r2 2r C. rl 2r B. rl D. rl r2 47. Create a flowchart proof to show that the diagonal of a rectangle divides the rectangle into two congruent triangles. A D B C 500 CHAPTER 9 The Pythagorean Theorem ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 48. Copy the ball positions onto patty paper. a. At what point on the S cushion should a player aim so that the cue ball bounces off and strikes the 8-ball? Mark the point with the letter A. b. At what point on the W cushion should a player aim so that the cue ball bounces off and strikes the 8-ball? Mark the point with the letter B. W N S E 49. Find the area and the perimeter of 50. Find the area of the shaded region. the trapezoid. 12 cm 4 cm 5 cm 5 cm 3 cm 120° 4 cm 51. An Olympic swimming pool has length 50 meters and width 25 meters. What is the diagonal distance across the pool? 52. The side length of a regular pentagon is 6 cm, and the apothem measures about 4.1 cm. What is the area of the pentagon? 53. The box below has dimensions 25 cm, 36 cm, and x cm. The diagonal shown has length 65 cm. Find the value of x. 54. The cylindrical container below has an open top. Find the surface area of the container (inside and out) |
to the nearest square foot. 65 cm 25 cm 36 cm x cm TAKE ANOTHER LOOK 1. Use geometry software to demonstrate the Pythagorean Theorem. Does your demonstration still work if you use a shape other than a square—for example, an equilateral triangle or a semicircle? 2. Find Elisha Scott Loomis’s Pythagorean Proposition and demonstrate one of the proofs of the Pythagorean Theorem from the book. 9 ft 5 ft A C B CHAPTER 9 REVIEW 501 EW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTER 9 REVIEW ● CHAPTE 3. The Zhoubi Suanjing, one of the oldest sources of Chinese mathematics and astronomy, contains the diagram at right demonstrating the Pythagorean Theorem (called gou gu in China). Find out how the Chinese used and proved the gou gu, and present your findings. 4. Use the SSS Congruence Conjecture to verify the converse of the 30°-60°-90° Triangle Conjecture. That is, show that if a triangle has sides with lengths x, x3, and 2x, then it is a 30°60°-90° triangle. 5. Starting with an isosceles right triangle, use geometry software or a compass and straightedge to start a right triangle like the one shown. Continue constructing right triangles on the hypotenuse of the previous triangle at least five more times. Calculate the length of each hypotenuse and leave them in radical form. Assessing What You’ve Learned 1 4 3 1 1 2 1 UPDATE YOUR PORTFOLIO Choose a challenging project, Take Another Look activity, or exercise you did in this chapter and add it to your portfolio. Explain the strategies you used. ORGANIZE YOUR NOTEBOOK Review your notebook and your conjecture list to be sure they are complete. Write a one-page chapter summary. WRITE IN YOUR JOURNAL Why do you think the Pythagorean Theorem is considered one of the most important theorems in mathematics? WRITE TEST ITEMS Work with group members to write test items for this chapter. Try to demonstrate more than one way to solve each problem. GIVE A PRESENTATION Create a visual aid and give a presentation about the Pythagorean Theorem. 502 CHAPTER 9 The Pythagorean Theorem CHAPTER 10 Volume Perhaps all I pursue is astonishment and so I try to |
awaken only astonishment in my viewers. Sometimes “beauty” is a nasty business. M. C. ESCHER Verblifa tin, M. C. Escher, 1963 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● explore and define many three-dimensional solids ● discover formulas for finding the volumes of prisms, pyramids, cylinders, cones, and spheres ● learn how density is related to volume ● derive a formula for the surface area of a sphere L E S S O N 10.1 Everything in nature adheres to the cone, the cylinder, and the cube. PAUL CEZANNE The Geometry of Solids Most of the geometric figures you have worked with so far have been flat plane figures with two dimensions—base and height. In this chapter you will work with solid figures with three dimensions—length, width, and height. Most real-world solids, like rocks and plants, are very irregular, but many others are geometric. Some real-world geometric solids occur in nature: viruses, oranges, crystals, the earth itself. Others are human-made: books, buildings, baseballs, soup cans, ice cream cones. This amethyst crystal is an irregular solid, but parts of it have familiar shapes. The geometry of diamonds Still Life With a Basket (1888–1890) by French post-impressionist painter Paul Cézanne (1839–1906) uses geometric solids to portray everyday objects. Science Three-dimensional geometry plays an important role in the structure of molecules. For example, when carbon atoms are arranged in a very rigid network, they form diamonds, one of the earth’s hardest materials. But when carbon atoms are arranged in planes of hexagonal rings, they form graphite, a soft material used in pencil lead. Carbon atoms can also bond into very large molecules. Named fullerenes, after U.S. engineer Buckminster Fuller (1895–1983), these carbon molecules have the same symmetry as a soccer ball, as shown at left. They are popularly called buckyballs. The geometry of graphite 504 CHAPTER 10 Volume A solid formed by polygons that enclose a single region of space is called a polyhedron. The flat polygonal surfaces of a polyhedron are called its faces. Although a face of a polyhedron includes the polygon and its interior region, we identify |
the face by naming the polygon that encloses it. A segment where two faces intersect is called an edge. The point of intersection of three or more edges is called a vertex of the polyhedron. Edge Face Vertex Just as a polygon is classified by its number of sides, a polyhedron is classified by its number of faces. The prefixes for polyhedrons are the same as they are for polygons with one exception: A polyhedron with four faces is called a tetrahedron. Here are some examples of polyhedrons. Hexahedrons Heptahedrons Decahedrons If each face of a polyhedron is enclosed by a regular polygon, and each face is congruent to the other faces, and the faces meet at each vertex in exactly the same way, then the polyhedron is called a regular polyhedron. The regular polyhedron shown at right is called a regular dodecahedron because it has 12 faces. Regular dodecahedron The Ramat Polin housing complex in Jerusalem, Israel, has many polyhedral shapes. LESSON 10.1 The Geometry of Solids 505 A prism is a special type of polyhedron, with two faces called bases, that are congruent, parallel polygons. The other faces of the polyhedron, called lateral faces, are parallelograms that connect the corresponding sides of the bases. The lateral faces meet to form the lateral edges. Each solid shown below is a prism. Bases Lateral face Lateral edge Rectangular prism Triangular prism Hexagonal prism Prisms are classified by their bases. For example, a prism with triangular bases is a triangular prism, and a prism with hexagonal bases is a hexagonal prism. A prism whose lateral faces are rectangles is called a right prism. Its lateral edges are perpendicular to its bases. A prism that is not a right prism is called an oblique prism. The altitude of a prism is any perpendicular segment from one base to the plane of the other base. The length of an altitude is the height of the prism. Altitude Altitude Right pentagonal prism Oblique triangular prism A pyramid is another special type of polyhedron. Pyramids have only one base. Like a prism, the other faces are called the lateral faces, and they meet to form the lateral edges. The common vertex of the lateral faces is the vertex of the pyramid. Lateral face Vertex Lateral edge Base Altitude Triangular pyramid |
Trapezoidal pyramid Hexagonal pyramid Square pyramid Like prisms, pyramids are also classified by their bases. The pyramids of Egypt are square pyramids because they have square bases. The altitude of the pyramid is the perpendicular segment from its vertex to the plane of its base. The length of the altitude is the height of the pyramid. Polyhedrons are geometric solids with flat surfaces. There are also geometric solids that have curved surfaces. One that all sports fans know well is the ball, or sphere—you can think of it as a threedimensional circle. An orange is one example of a sphere found in nature. What are some others? 506 CHAPTER 10 Volume 100 Cans (1962 oil on canvas), by pop art artist Andy Warhol (1925–1987), repeatedly uses the cylindrical shape of a soup can to make an artistic statement with a popular image. A sphere is the set of all points in space at a given distance from a given point. The given distance is called the radius of the sphere, and the given point is the center of the sphere. A hemisphere is half a sphere and its circular base. The circle that encloses the base of a hemisphere is called a great circle of the sphere. Every plane that passes through the center of a sphere determines a great circle. All the longitude lines on a globe of Earth are great circles. The equator is the only latitude line that is a great circle. Radius Radius Center Center Sphere Hemisphere Great circle Another solid with a curved surface is a cylinder. Soup cans, compact discs (CDs), and plumbing pipes are shaped like cylinders. Like a prism, a cylinder has two bases that are both parallel and congruent. Instead of polygons, however, the bases of cylinders are circles and their interiors. The segment connecting the centers of the bases is called the axis of the cylinder. The radius of the cylinder is the radius of a base. If the axis of a cylinder is perpendicular to the bases, then the cylinder is a right cylinder. A cylinder that is not a right cylinder is an oblique cylinder. The altitude of a cylinder is any perpendicular segment from the plane of one base to the plane of the other. The height of a cylinder is the length of an altitude. Radius Axis Bases Altitude Right cylinder Oblique cylinder LESSON 10.1 The Geometry of Solids 507 A third type of solid with a curved surface is a cone. Funnels and ice cream cones are |
shaped like cones. Like a pyramid, a cone has a base and a vertex. The base of a cone is a circle and its interior. The radius of a cone is the radius of the base. The vertex of a cone is the point that is the greatest perpendicular distance from the base. The altitude of a cone is the perpendicular segment from the vertex to the plane of the base. The length of the altitude is the height of a cone. If the line segment connecting the vertex of a cone with the center of its base is perpendicular to the base, then it is a right cone. Vertex Altitude Altitude Radius Base Right cone Oblique cone EXERCISES 1. Complete this definition: A pyramid is a? with one? face (called the base) and whose other faces (lateral faces) are? formed by segments connecting the vertices of the base to a common point (the vertex) not on the base. For Exercises 2–9, refer to the figures below. All measurements are in centimeters. 2. Name the bases of the prism. 3. Name all the lateral faces of the prism. 4. Name all the lateral edges of the prism. 5. What is the height of the prism? 6. Name the base of the pyramid. 7. Name the vertex of the pyramid. 8. Name all the lateral edges of the pyramid. 9. What is the height of the pyramid? T 9 5 Q 15 N Y P 6 G 7 E 10 8 U R A P S 6 1 G 13 T 508 CHAPTER 10 Volume For Exercises 10–22, match each real object with a geometry term. You may use a geometry term more than once or not at all. 10. Die 17. Honeycomb 11. Tomb of Egyptian rulers 18. Stop sign 12. Holder for a scoop of ice cream 19. Moon 13. Wedge or doorstop 20. Can of tuna fish 14. Box of breakfast cereal 15. Plastic bowl with lid 16. Ingot of silver 21. Book 22. Pup tent A. Cylinder B. Cone C. Square prism D. Square pyramid E. Sphere F. Triangular pyramid G. Octagonal prism H. Triangular prism I. Trapezoidal prism J. Rectangular prism K. Heptagonal pyramid L. Hexagonal prism M. Hemisphere For Exercises 23–26, draw and label each solid. Use dashed lines to show the hidden edges. 23. A triangular pyramid whose base is an |
equilateral triangular region (use the proper marks to show that the base is equilateral) 24. A hexahedron with two trapezoidal faces 25. A cylinder with a height that is twice the diameter of the base (use x and 2x to indicate the height and the diameter) 26. A right cone with a height that is half the diameter of the base For Exercises 27–35, identify each statement as true or false. Sketch a counterexample for each false statement or explain why it is false. 27. A lateral face of a pyramid is always a triangular region. 28. A lateral edge of a pyramid is always perpendicular to the base. 29. Every slice of a prism cut parallel to the bases is congruent to the bases. 30. When the lateral surface of a right cylinder is unwrapped and laid flat, it is a rectangle. 31. When the lateral surface of a right circular cone is unwrapped and laid flat, it is a triangle. 32. Every section of a cylinder, parallel to the base, is congruent to the base. 33. The length of a segment from the vertex of a cone to the circular base is the height of the cone. 34. The length of the axis of a right cylinder is the height of the cylinder. 35. All slices of a sphere passing through the sphere’s center are congruent. LESSON 10.1 The Geometry of Solids 509 36. Write a paragraph describing the visual tricks that Belgian artist René Magritte (1898–1967) plays in his painting at right. The Promenades of Euclid (1935 oil on canvas), René Magritte 37. An antiprism is a polyhedron with two congruent bases and lateral faces that are triangles. Complete the tables below for prisms and antiprisms. Describe any relationships you see between the number of lateral faces, total faces, edges, and vertices of related prisms and antiprisms. Triangular prism Rectangular prism Pentagonal prism Hexagonal prism n-gonal prism 3 6............ 18 10 Triangular antiprism Rectangular antiprism Pentagonal antiprism Hexagonal antiprism n-gonal antiprism 6 10............ 24 10 Lateral faces Total faces Edges Vertices Lateral faces Total faces Edges Vertices |
510 CHAPTER 10 Volume Review For Exercises 38 and 39, how many cubes measuring 1 cm on each edge will fit into the container? 38. A box measuring 2 cm on each inside edge 39. A box measuring 3 cm by 4 cm by 5 cm on the inside edges 40. What is the maximum number of boxes measuring 1 cm by 1 cm by 2 cm that can fit within a box whose inside dimensions are 3 cm by 4 cm by 5 cm? 41. For each net, decide whether it folds to make a box. If it does, copy the net and mark each pair of opposite faces with the same symbol. a. b. c. d. IMPROVING YOUR VISUAL THINKING SKILLS Piet Hein’s Puzzle In 1936, while listening to a lecture on quantum physics, the Danish mathematician Piet Hein (1905–1996) devised the following visual thinking puzzle: What are all the possible nonconvex solids that can be created by joining four or fewer cubes face-to-face? A nonconvex polyhedron is a solid that has at least one diagonal that is exterior to the solid. For example, four cubes in a row, joined face-to-face, form a convex polyhedron. But four cubes joined face-to-face into an L-shape form a nonconvex polyhedron. Use isometric dot paper to sketch the nonconvex solids that solve Piet Hein’s puzzle. There are seven solids. LESSON 10.1 The Geometry of Solids 511 Euler’s Formula for Polyhedrons In this activity you will discover a relationship among the vertices, edges, and faces of a polyhedron. This relationship is called Euler’s Formula for Polyhedrons, named after Leonhard Euler. Let’s first build some of the polyhedrons you learned about in Lesson 10.1. Activity Toothpick Polyhedrons First, you’ll model polyhedrons using toothpicks as edges and using small balls of clay, gumdrops, or dried peas as connectors. Step 1 Build and save the polyhedrons shown in parts a–d below and described in parts e–i on the top of page 513. You may have to cut or break some sticks. Share the tasks among the group. a. c. b. d. You will need ● toothpicks ● modeling clay, gumdrops, |
or dried peas 512 CHAPTER 10 Volume e. Build a tetrahedron. f. Build an octahedron. g. Build a nonahedron. h. Build at least two different-shaped decahedrons. i. Build at least two different-shaped dodecahedrons. Step 2 Classify all the different polyhedrons your class built as prisms, pyramids, regular polyhedrons, or just polyhedrons. Next, you’ll look for a relationship among the vertices, faces, and edges of the polyhedrons. Step 3 Count the number of vertices (V), edges (E), and faces (F) of each polyhedron model. Copy and complete a chart like this one. Polyhedron Vertices (V) Faces (F) Edges (E)............ Step 4 Look for patterns in the table. By adding, subtracting, or multiplying V, F, and E (or a combination of two or three of these operations), you can discover a formula that is commonly known as Euler’s Formula for Polyhedrons. Step 5 Now that you have discovered the formula relating the number of vertices, edges, and faces of a polyhedron, use it to answer each of these questions. a. Which polyhedron has 4 vertices and 6 edges? Can you build another polyhedron with a different number of faces that also has 4 vertices and 6 edges? b. Which polyhedron has 6 vertices and 12 edges? Can you build another polyhedron with a different number of faces that also has 6 vertices and 12 edges? c. If a solid has 8 faces and 12 vertices, how many edges will it have? d. If a solid has 7 faces and 12 edges, how many vertices will it have? e. If a solid has 6 faces, what are all the possible combinations of vertices and edges it can have? EXPLORATION Euler’s Formula for Polyhedrons 513 L E S S O N 10.2 How much deeper would oceans be if sponges didn’t live there? STEVEN WRIGHT Volume of Prisms and Cylinders In real life you encounter many volume problems. For example, when you shop for groceries, it’s a good idea to compare the volumes and the prices of different items to find the best buy. When you |
fill a car’s gas tank or when you fit last night’s leftovers into a freezer dish, you fill the volume of an empty container. Many occupations also require familiarity with volume. An engineer must calculate the volume and the weight of sections of a bridge to avoid too much stress on any one section. Chemists, biologists, physicists, and geologists must all make careful volume measurements in their research. Carpenters, plumbers, and painters also know and use volume relationships. A chef must measure the correct volume of each ingredient in a cake to ensure a tasty success. American artist Wayne Thiebaud (b 1920) painted Bakery Counter in 1962. Volume is the measure of the amount of space contained in a solid. You use cubic units to measure volume: cubic inches in.3, cubic feet ft3, cubic yards yd3, cubic centimeters cm3, cubic meters m3, and so on. The volume of an object is the number of unit cubes that completely fill the space within the object. 1 1 1 1 Length: 1 unit Volume: 1 cubic unit Volume: 20 cubic units 514 CHAPTER 10 Volume Step 1 Investigation The Volume Formula for Prisms and Cylinders Find the volume of each right rectangular prism below in cubic centimeters. That is, how many 1 cm-by-1 cm-by-1 cm cubes will fit into each solid? Within your group, discuss different strategies for finding each volume. How could you find the volume of any right rectangular prism? a. 3 cm b. c. 2 cm 4 cm 8 cm 3 cm 12 cm 10 cm 10 cm 30 cm Notice that the number of cubes resting on the base equals the number of square units in the area of the base. The number of layers of cubes equals the number of units in the height of the prism. So you can multiply the area of the base by the height of the prism to calculate the volume. Step 2 Complete the conjecture. Conjecture A C-88a If B is the area of the base of a right rectangular prism and H is the height of the solid, then the formula for the volume is V?. In Chapter 8, you discovered that you can reshape parallelograms, triangles, trapezoids, and circles into rectangles to find their area. You can use the same method to find the areas of bases that have these shapes. Then you can multiply the area of the base by the height of the prism to find its volume. For |
example, to find the volume of a right triangular prism, find the area of the triangular base (the number of cubes resting on the base) and multiply it by the height (the number of layers of cubes). So you can extend Conjecture A (the volume of right rectangular prisms) to all right prisms and right cylinders. Step 3 Complete the conjecture. Conjecture B C-88b If B is the area of the base of a right prism (or cylinder) and H is the height of the solid, then the formula for the volume is V?. LESSON 10.2 Volume of Prisms and Cylinders 515 What about the volume of an oblique prism or cylinder? You can approximate the shape of this oblique rectangular prism with a staggered stack of three reams of 8.5-by-11-inch paper. If you nudge the individual pieces of paper into a slanted stack, then your approximation can be even better. 8.5 in. 6 in. 11 in. Oblique rectangular prism Stacked reams of 8.5-by-11-inch paper Stacked sheets of paper Sheets of paper stacked straight Rearranging the paper into a right rectangular prism changes the shape, but certainly the volume of paper hasn’t changed. The area of the base, 8.5 by 11 inches, didn’t change and the height, 6 inches, didn’t change, either. In the same way, you can use coffee filters, coins, candies, or chemistry filter papers to show that an oblique cylinder has the same volume as a right cylinder with the same base and height. Use the stacking model to extend Conjecture B (the volume of right prisms and cylinders) to oblique prisms and cylinders. Complete the conjecture. Step 4 Conjecture C C-88c The volume of an oblique prism (or cylinder) is the same as the volume of a right prism (or cylinder) that has the same? and the same?. Finally, you can combine Conjectures A, B, and C into one conjecture for finding the volume of any prism or cylinder, whether it’s right or oblique. Step 5 Copy and complete the conjecture. Prism-Cylinder Volume Conjecture The volume of a prism or a cylinder is the? multiplied by the?. C-88 If you successfully completed the investigation, you saw that the same volume formula applies to all prisms and |
cylinders, regardless of the shapes of their bases. To calculate the volume of a prism or cylinder, first calculate the area of the base using the formula appropriate to its shape. Then multiply the area of the base by the height of the solid. In oblique prisms and cylinders, the lateral edges are no longer at right angles to the bases, so you do not use the length of the lateral edge as the height. 516 CHAPTER 10 Volume EXAMPLE A Find the volume of a right trapezoidal prism that has a height of 15 cm. The two bases of the trapezoid measure 4 cm and 8 cm, and its height is 5 cm. 15 Solution Use B 1 2 base. hb1 V BH b2 for the area of the trapezoidal 8 4 5 The formula for volume of a prism, where B is the area of its base and H is its height. Substitute 1 (5)(4 8) for B, applying the formula 2 for area of a trapezoid. Substitute 15 for H, the height of the prism. Simplify. 1 (5)(4 8) (15) 2 (30)(15) 450 The volume is 450 cm3. EXAMPLE B Find the volume of an oblique cylinder that has a base with a radius of 6 inches and a height of 7 inches. 6 7 Solution Use B r2 for the area of the circular base. V BH 62(7) 36(7) 252 791.68 The formula for volume of a cylinder. Substitute 62 for B, applying the formula for area of a circle. Simplify. Use the key on your calculator to get an approximate answer. The volume is 252 in.3, or about 791.68 in.3. EXERCISES Find the volume of each solid in Exercises 1–6. All measurements are in centimeters. Round approximate answers to two decimal places. 1. Oblique rectangular prism 2. Right triangular prism 3. Right trapezoidal prism LESSON 10.2 Volume of Prisms and Cylinders 517 4. Right cylinder 4 10 5. Right semicircular cylinder 6. Right cylinder with a 90° slice removed 6 8 6 12 90° 7. Use the information about the base and height of each solid to find the volume. All measurements are given in centimeters. Information about Height of solid base of solid Right triangular Right rectangular prism h b H prism h H b d. V e. V f. |
V Right trapezoidal prism b2 H h b Right cylinder r H g. V h. V i. V j. V k. V l. V 7, b 6, b2 h 8, r 3 b 9, b2 h 12, r 6 b 8, b2 h 18, r 8 19, 12, H 20 a. V H 20 b. V H 23 c. V For Exercises 8–9, sketch and label each solid described, then find the volume. 8. An oblique trapezoidal prism. The trapezoidal base has a height of 4 in. and bases that measure 8 in. and 12 in. The height of the prism is 24 in. 9. A right circular cylinder with a height of T. The radius of the base is Q. 10. Sketch and label two different rectangular prisms each with a volume of 288 cm3. In Exercises 11–13, express the volume of each solid with the help of algebra. 11. Right rectangular prism 12. Oblique cylinder 2x x x 2r 3r 13. Right rectangular prism with a rectangular hole 2x x x 3x 5x 14. APPLICATION A cord of firewood is 128 cubic feet. Margaretta has three storage boxes for firewood that each measure 2 feet by 3 feet by 4 feet. Does she have enough space to order a full cord of firewood? A half cord? A quarter cord? Explain. 518 CHAPTER 10 Volume Career In construction and landscaping, sand, rocks, gravel, and fill dirt are often sold by the “yard,” which actually means a cubic yard. 15. APPLICATION A contractor needs to build up a ramp as shown at right from the street to the front of a garage door. How many cubic yards of fill will she need? 2 yards 17 yards 10 yards 16. If an average rectangular block of limestone used to build the Great Pyramid of Khufu at Giza is approximately 2.5 feet by 3 feet by 4 feet and limestone weighs approximately 170 pounds per cubic foot, what is the weight of one of the nearly 2,300,000 limestone blocks used to build the pyramid? 2.5 ft 3 ft 4 ft 17. Although the Exxon Valdez oil spill (11 million gallons of oil) is one of the most notorious oil spills, it was small compared to the 250 million gallons of crude oil that were spilled during the 1991 Persian Gulf War. A gallon occupies 0.13368 |
cubic foot. How many rectangular swimming pools, each 20 feet by 30 feet by 5 feet, could be filled with 250 million gallons of crude oil? The Great Pyramid of Khufu at Giza, Egypt, was built around 2500 B.C.E. 18. When folded, a 12-by-12-foot section of the AIDS Memorial Quilt requires about 1 cubic foot of storage. In 1996, the quilt consisted of 32,000 3-by-6-foot panels. What was the quilt’s volume in 1996? If the storage facility had a floor area of 1,500 square feet, how high did the quilt panels need to be stacked? The NAMES Project AIDS Memorial Quilt memorializes persons all around the world who have died of AIDS. In 1996, the 32,000 panels represented less than 10% of the AIDS deaths in the United States alone, yet the quilt could cover about 19 football fields. LESSON 10.2 Volume of Prisms and Cylinders 519 Review For Exercises 19 and 20, draw and label each solid. Use dashed lines to show the hidden edges. 19. An octahedron with all triangular faces and another octahedron with at least one nontriangular face 20. A cylinder with both radius and height r, a cone with both radius and height r resting flush on one base of the cylinder, and a hemisphere with radius r resting flush on the other base of the cylinder For Exercises 21 and 22, identify each statement as true or false. Sketch a counterexample for each false statement or explain why it is false. 21. A prism always has an even number of vertices. 22. A section of a cube is either a square or a rectangle. 23. The tower below is an unusual shape. It’s neither a cylinder nor a cone. Sketch a two-dimensional figure and an axis such that if you spin your figure about the axis, it will create a solid of revolution shaped like the tower. The Sugong Tower Mosque in Turpan, China 24. Do research to find a photo or drawing of a chemical model of a crystal. Sketch it. What type of polyhedral structure does it exhibit? You will find helpful Internet links at www.keymath.com/DG. Science Ice is a well-known crystal structure. If ice were denser than water, it would sink to the bottom of the ocean, away from heat sources. Eventually the oceans |
would fill from the bottom up with ice, and we would have an ice planet. What a cold thought! 520 CHAPTER 10 Volume 25. Six points are equally spaced around a circular track with a 20 m radius. Ben runs around the track from one point, past the second, to the third. Al runs straight from the first point to the second, and then straight to the third. How much farther does Ben run than Al? S 26. AS and AT are tangent to circle O at S and T, respectively. mSMO 90°, mSAT 90°, SM 6. Find the exact value of PA. O M A P T THE SOMA CUBE If you solved Piet Hein’s puzzle at the end of the previous lesson, you now have sketches of the seven nonconvex polyhedrons that can be assembled using four or fewer cubes. These seven polyhedrons consist of a total of 27 cubes: 6 sets of 4 cubes and 1 set of 3 cubes. You are ready for the next rather amazing discovery. These pieces can be arranged to form a 3-by-3-by-3 cube. The puzzle of how to put them together in a perfect cube is known as the Soma Cube puzzle. Use cubes (wood, plastic, or sugar cubes) to build one set of the seven pieces of the Soma Cube. Use glue, tape, or putty to connect the cubes. Solve the Soma Cube puzzle. Put the pieces together to make a 3-by-3-by-3 cube. Now build these other shapes. How do you build the sofa? The tunnel? The castle? The winners’ podium? Sofa Tunnel Castle The Winners' Podium Create a shape of your own that uses all the pieces. Go to more about the Soma Cube. www.keymath.com/DG to learn LESSON 10.2 Volume of Prisms and Cylinders 521 Volume of Pyramids and Cones There is a simple relationship between the volumes of prisms and pyramids with congruent bases and the same height, and between cylinders and cones with congruent bases and the same height. You’ll discover this relationship in the investigation. Same height Congruent bases Same volume? Investigation The Volume Formula for Pyramids and Cones L E S S O N 10.3 If I had influence with the good fairy... I should ask that her gift to each child in the world be |
a sense of wonder so indestructible that it would last throughout life. RACHEL CARSON You will need ● container pairs of prisms and pyramids ● container pairs of cylinders and cones ● sand, rice, birdseed, or water Step 1 Step 2 Step 3 Step 4 Step 5 Choose a prism and a pyramid that have congruent bases and the same height. Fill the pyramid, then pour the contents into the prism. About what fraction of the prism is filled by the volume of one pyramid? Check your answer by repeating Step 2 until the prism is filled. Choose a cone and a cylinder that have congruent bases and the same height and repeat Steps 2 and 3. Compare your results with the results of others. Did you get similar results with both your pyramid-prism pair and the cone-cylinder pair? You should be ready to make a conjecture. Pyramid-Cone Volume Conjecture C-89 If B is the area of the base of a pyramid or a cone and H is the height of the solid, then the formula for the volume is V?. 522 CHAPTER 10 Volume If you successfully completed the investigation, you probably noticed that the volume formula is the same for all pyramids and cones, regardless of the type of base they have. To calculate the volume of a pyramid or cone, first find the area of its base. Then find the product of that area and the height of the solid, and multiply by the fraction you discovered in the investigation. EXAMPLE A Find the volume of a regular hexagonal pyramid with a height of 8 cm. Each side of its base is 6 cm. 8 cm Solution First find the area of the base. To find the area of a regular hexagon, you need the apothem. By the 30°-60°-90° Triangle Conjecture, the apothem is 33 cm. 6 cm Base 6 cm 6 cm 3 3 3 cm 3 cm 30° 60° ap B 1 2 33(36) B 1 2 B 543 The area of a regular polygon is one-half the apothem times the perimeter. Substitute 33 for a and 36 for p. Multiply. The base has an area of 543 cm2. Now find the volume of the pyramid. BH V 1 3 543(8) V 1 3 V 1443 The volume of a pyramid is one-third the area of the base times the height. Substitute 543 for B and 8 for |
H. Multiply. The volume is 1443 cm3 or approximately 249.4 cm3. EXAMPLE B A cone has a base radius of 3 in. and a volume of 24 in.3. Find the height. 3 in. LESSON 10.3 Volume of Pyramids and Cones 523 Solution Start with the volume formula and solve for H. V 1 BH 3 V 1 r2(H) 3 24 1 32(H) 3 24 3H 8 H The height of the cone is 8 in. Volume formula for pyramids and cones. The base of a cone is a circle. Substitute 24 for the volume and 3 for the radius. Square the 3 and multiply by 1. 3 Solve for H. EXERCISES Find the volume of each solid named in Exercises 1–6. All measurements are in centimeters. 1. Square pyramid 2. Cone 3. Trapezoidal pyramid 9 8 8 7 6 15 5 8 4 4. Triangular pyramid 5. Semicircular cone 6. Cylinder with cone 12 removed 12 14 16 6 13 5 In Exercises 7–9, express the total volume of each solid with the help of algebra. In Exercise 9, what percentage of the volume is filled with the liquid? All measurements are in centimeters. 7. Square pyramid 8. Cone 9. Cone m m m 2b b 9x 6x 10x 12x 8x 524 CHAPTER 10 Volume 10. Use the information about the base and height of each solid to find the volume. All measurements are given in centimeters. Information about Height of solid base of solid Triangular pyramid Rectangular pyramid Trapezoidal pyramid Cone H h b H 20 a. V H 20 b. V H 24 c. V H b h d. V e. V f. V b2 H h H r b g. V h. V i. V j. V k. V l. V 7, b 6, b2 h 6, r 3 b 9, b2 h 8, r 6 b 13, b2 h 17, r 8 22, 29, For Exercises 11 and 12, sketch and label the solids described. 11. Sketch and label a square pyramid with height H feet and each side of the base M feet. The altitude meets the square base at the intersection of the two diagonals. Find the volume in terms of H and M. 12. Sketch two different circular cones |
each with a volume of 2304 cm3. 13. Mount Fuji, the active volcano in Honshu, Japan, is 3776 m high and has a slope of approximately 30°. Mount Etna, in Sicily, is 3350 m high and approximately 50 km across the base. If you assume they both can be approximated by cones, which volcano is larger? Mount Fuji is Japan’s highest mountain. Legend claims that an earthquake created it. 14. Bretislav has designed a crystal glass sculpture. Part of the piece is in the shape of a large regular pentagonal pyramid, shown at right. The apothem of the base measures 27.5 cm. How much will this part weigh if the glass he plans to use weighs 2.85 grams per cubic centimeter? 15. Jamala has designed a container that she claims will hold 50 in.3. The net is shown at right. Check her calculations. What is the volume of the solid formed by this net? 30 cm a 40 cm 6 in. 6 in. 8 in. 8 in. LESSON 10.3 Volume of Pyramids and Cones 525 16. Find the volume of the solid formed by rotating the shaded figure about the x-axis. y 3 2 1 Review 17. Find the volume of the liquid in this right rectangular prism. All measurements are given in centimeters. 1 2 3 x 18. APPLICATION A swimming pool is in the shape of this prism. A cubic foot of water is about 7.5 gallons. How many gallons of water can the pool hold? If a pump is able to pump water into the pool at a rate of 15 gallons per minute, how long will it take to fill the pool? 19. APPLICATION A landscape architect is building a stone retaining wall as sketched at right. How many cubic feet of stone will she need? 6x 12x 4x 30 ft 4 ft 18 in. 13 ft 120 in. 14 ft 16 ft 31 in. 48 in. 20. As bad as tanker oil spills are, they are only about 12% of the 3.5 million tons of oil that enters the oceans each year. The rest comes from routine tanker operations, sewage treatment plants’ runoff, natural sources, and offshore oil rigs. One month’s maintenance and routine operation of a single supertanker produces up to 17,000 gallons of oil sludge that gets into the ocean! If a cylindrical barrel is about 1.6 feet in diameter and |
2.8 feet tall, how many barrels are needed to hold 17,000 gallons of oil sludge? Recall that a cubic foot of water is about 7.5 gallons. 21. Find the surface area of each of the following polyhedrons. (See the shapes on page 528.) Give exact answers. a. A regular tetrahedron with an edge of 4 cm b. A regular hexahedron with an edge of 4 cm c. A regular icosahedron with an edge of 4 cm d. The dodecahedron shown at right, made of four congruent rectangles and eight congruent triangles 9 5 6 526 CHAPTER 10 Volume 22. Given the triangle at right, reflect D over AC to D. Then reflect D over BC to D. Explain why D, C, D are collinear. 23. In each diagram, WXYZ is a parallelogram. Find the coordinates of Y. c. b. a. y y D A C y Z (c, d) Z (c, d) Y Z (c, d) Y W X (a, 0) x W X (a, b) x W (e, f ) B Y X (a, b) x THE WORLD’S LARGEST PYRAMID The pyramid at Cholula, Mexico, shown at right, was built between the second and eighth centuries C.E. Like most pyramids of the Americas, it has a flat top. In fact, it is really two flat-topped pyramids. Some people claim it is the world’s largest pyramid—even larger than the Great Pyramid of Khufu at Giza (shown on page 519) erected around 2500 B.C.E. Is it? Which of the two has the greater volume? Your project should include Volume calculations for both pyramids. Scale models of both pyramids. 1020 ft 520 ft 270 ft 1400 ft 200 ft 90 ft 110 ft 1400 ft This church in Cholula appears to be on a hill, but it is actually built on top of an ancient pyramid! Side view of the two pyramids at Cholula 1020 ft 756 ft 481 ft 756 ft Dimensions of the Great Pyramid at Giza 1400 ft 110 ft 190 ft 190 ft Dissected view of bottom pyramid at Cholula LESSON 10.3 Volume of Pyramids and Cones 527 The Five Platonic Solids Regular poly |
hedrons have intrigued mathematicians for thousands of years. Greek philosophers saw the principles of mathematics and science as the guiding forces of the universe. Plato (429–347 B.C.E.) reasoned that because all objects are three-dimensional, their smallest parts must be in the shape of regular polyhedrons. There are only five regular polyhedrons, and they are commonly called the Platonic solids. Plato assigned each regular solid to one of the five “atoms”: the tetrahedron to fire, the icosahedron to water, the octahedron to air, the cube or hexahedron to earth, and the dodecahedron to the cosmos. Plato Fire Water Air Earth Cosmos Regular tetrahedron (4 faces) Regular icosahedron (20 faces) Regular octahedron (8 faces) Regular hexahedron (6 faces) Regular dodecahedron (12 faces) Activity Modeling the Platonic Solids What would each of the five Platonic solids look like when unfolded? There is more than one way to unfold each polyhedron. Recall that a flat figure that you can fold into a polyhedron is called its net. You will need ● poster board or cardboard ● a compass and straightedge ● scissors ● glue, paste, or cellophane tape ● colored pens or pencils for decorating the solids (optional ) 528 CHAPTER 10 Volume Step 1 Step 2 Step 3 One face is missing in the net at right. Complete the net to show what the regular tetrahedron would look like if it were cut open along the three lateral edges and unfolded into one piece. Two faces are missing in the net at right. Complete the net to show what the regular hexahedron would look like if it were cut open along the lateral edges and three top edges, then unfolded. Here is one possible net for the regular icosahedron. When the net is folded together, the five top triangles meet at one top point. Which edge—a, b, or c—does the edge labeled x line up with? x Step 4 The regular octahedron is similar to the icosahedron but has only eight equilateral triangles. Complete the octahedron net at right. Two faces are missing. Step 5 The regular dodecahedron has 12 regular pentagons as faces. If you cut the dodecahedron into two equal parts, they would look like two flowers |
, each having five pentagon-shaped petals around a center pentagon. Complete the net for half a dodecahedron. a b c Now you know what the nets of the five Platonic solids could look like. Let’s use the nets to construct and assemble models of the five Platonic solids. See the Procedure Note for some tips. 1. To save time, build solids with the same shape faces at the same time. 2. Construct a regular polygon template for each shape face. 3. Erase the unnecessary line segments. 4. Leave tabs on some edges for gluing. Tabs Tabs Tabs 5. Decorate each solid before you cut it out. 6. Score on both sides of the net by running a pen or compass point over the fold lines. Step 6 Construct the nets for icosahedron, octahedron, and tetrahedron with equilateral triangles. EXPLORATION The Five Platonic Solids 529 Step 7 Step 8 Construct a net for the hexahedron, or cube, with squares. You will construct a net for the dodecahedron with regular pentagons. To construct a regular pentagon, follow Steps A–F below. Construct a circle. Construct two perpendicular diameters. Find M, the midpoint of OA. Swing an arc with radius BM intersecting OC at point D. Step A B Step B B Step C B C O A C MO A C D MO A BD is the length of each side of the pentagon. Starting at point B, mark off BD on the circumference five times. Connect the points to form a pentagon. Step D B Step E B Step F B C D MO A C D MO A C O D M A Step 9 Follow Steps A–D below to construct half the net for the dodecahedron. Construct a large regular pentagon. Lightly draw all the diagonals. The smaller regular pentagon will be one of the 12 faces of the dodecahedron. Step A Step B Draw the diagonals of the central pentagon and extend them to the sides of the larger pentagon. Find the five pentagons that encircle the central pentagon. Step C Step D Step 10 Can you explain from looking at the nets why there are only five Platonic solids? 530 CHAPTER 10 Volume L E S S O N 10.4 If you have made mistakes... there is always another chance for you. |
.. for this thing we call “failure” is not the falling down, but the staying down. MARY PICKFORD Volume Problems Volume has applications in science, medicine, engineering, and construction. For example, a chemist needs to accurately measure the volume of reactive substances. A doctor may need to calculate the volume of a cancerous tumor based on a body scan. Engineers and construction personnel need to determine the volume of building supplies such as concrete or asphalt. The volume of the rooms in a completed building will ultimately determine the size of mechanical devices such as air conditioning units. Sometimes, if you know the volume of a solid, you can calculate an unknown length of a base or the solid’s height. Here are two examples. EXAMPLE A The volume of this right triangular prism is 1440 cm3. Find the height of the prism. 8 15 H Solution V BH V 1 (bh)H 2 1440 1 (8)(15)H 2 1440 60H 24 H The height of the prism is 24 cm. Volume formula for prisms and cylinders. The base of the prism is a triangle. Substitute 1440 for the volume, 8 for the base of the triangle, and 15 for the height of the triangle. Multiply. Solve for H. EXAMPLE B The volume of this sector of a right cylinder is 2814 m3. Find the radius of the base of the cylinder to the nearest m. Solution The volume is the area of the base times the height. To find the area of the sector, you first find what fraction 1 4 0. the sector is of the whole circle: 9 0 6 3 40° r 14 V BH r2H V 1 9 2814 1 r2(14) 9 9 814 r2 2 14 575.8 r2 24 r The radius is about 24 m. LESSON 10.4 Volume Problems 531 EXERCISES 1. If you cut a 1-inch square out of each corner of an 8.5-by-11-inch piece of paper and fold it into a box without a lid, what is the volume of the container? You will need Construction tools for Exercise 18 2. The prism at right has equilateral triangle bases with side lengths of 4 cm. The height of the prism is 8 cm. Find the volume. 3. A triangular pyramid has a volume of 180 cm3 and a height of 12 cm. Find the length of a side of the triangular base if |
the triangle’s height from that side is 6 cm. 4. A trapezoidal pyramid has a volume of 3168 cm3, and its height is 36 cm. The lengths of the two bases of the trapezoidal base are 20 cm and 28 cm. What is the height of the trapezoidal base? 5. The volume of a cylinder is 628 cm3. Find the radius of the base if the cylinder has a height of 8 cm. Round your answer to the nearest 0.1 cm. 6. If you roll an 8.5-by-11-inch piece of paper into a cylinder by bringing the two longer sides together, you get a tall, thin cylinder. If you roll an 8.5-by-11-inch piece of paper into a cylinder by bringing the two shorter sides together, you get a short, fat cylinder. Which of the two cylinders has the greater volume? 7. Sylvia has just discovered that the valve on her cement truck failed during the night and that all the contents ran out to form a giant cone of hardened cement. To make an insurance claim, she needs to figure out how much cement is in the cone. The circumference of its base is 44 feet, and it is 5 feet high. Calculate the volume to the nearest cubic foot. 8. A sealed rectangular container 6 cm by 12 cm by 15 cm is sitting on its smallest face. It is filled with water up to 5 cm from the top. How many centimeters from the bottom will the water level reach if the container is placed on its largest face? 9. To test his assistant, noted adventurer Dakota Davis states that the volume of the regular hexagonal ring at right is equal to the volume of the regular hexagonal hole in its center. The assistant must confirm or refute this, using dimensions shown in the figure. What should he say to Dakota? 532 CHAPTER 10 Volume 2 cm 4 cm 6 cm Use this information to solve Exercises 10–12: Water weighs about 63 pounds per cubic foot, and a cubic foot of water is about 7.5 gallons. 10. APPLICATION A king-size waterbed mattress measures 5.5 feet by 6.5 feet by 8 inches deep. To the nearest pound, how much does the water in this waterbed weigh? 5.5 ft 8 in. 6.5 ft 11. A child’s wading pool has a diameter of 7 feet 7 ft and is 8 inches deep. How many gallons of water can the |
pool hold? Round your answer to the nearest 0.1 gallon. 8 in. 12. Madeleine’s hot tub has the shape of a regular hexagonal prism. The chart on the hot-tub heater tells how long it takes to warm different amounts of water by 10°F. Help Madeleine determine how long it will take to raise the water temperature from 93°F to 103°F. Minutes to Raise Temperature 10°F Gallons 350 400 450 500 550 600 650 700 Minutes 9 10 11 12 14 15 16 18 3 ft 13. A standard juice box holds 8 fluid ounces. A fluid ounce of liquid occupies 1.8 in.3. Design a cylindrical can that will hold about the same volume as one juice box. What are some possible dimensions of the can? 3 ft 14. The photo at right shows an ice tray that is designed for a person who has the use of only one hand—each piece of ice will rotate out of the tray when pushed with one finger. Suppose the tray has a length of 12 inches and a height of 1 inch. Approximate the volume of water the tray holds if it is filled to the top. (Ignore the thickness of the plastic.) Review 15. Find the height of this right square pyramid. Give your answer to the nearest 0.1 cm. 16. EC is tangent at C. ED is tangent at D. Find x. 10 cm 8 cm 80° A B 120° O D C E x LESSON 10.4 Volume Problems 533 17. In the figure at right, ABCE is a parallelogram and BCDE is a rectangle. Write a paragraph proof showing that ABD is isosceles. D E 18. Construction Use your compass and straightedge to construct an isosceles trapezoid with a base angle of 45° and the length of one base three times the length of the other base. 19. M is the midpoint of AC and BD. For each statement, select always (A), sometimes (S), or never (N). a. ABCD is a parallelogram. b. ABCD is a rhombus. c. ABCD is a kite. d. AMD AMB e. DAM BCM A D C B C M A B IMPROVING YOUR REASONING SKILLS Bert’s Magic Hexagram Bert is the queen’s favorite jester. He entertains himself with puzzles. Bert is creating |
a magic hexagram on the front of a grid of 19 hexagons. When Bert’s magic hexagram (like its cousin the magic square) is completed, it will have the same sum in every straight hexagonal row, column, or diagonal (whether it is three, four, or five hexagons long). For example, B 12 10 is the same sum as B 2 5 6 9, which is the same sum as C 8 6 11. Bert planned to use just the first 19 positive integers (his age in years), but he only had time to place the first 12 integers before he was interrupted. Your job is to complete Bert’s magic hexagram. What are the values for A, B, C, D, E, F, and G? 534 CHAPTER 10 Volume L E S S O N 10.5 Eureka! I have found it! ARCHIMEDES Displacement and Density What happens if you step into a bathtub that is filled to the brim? If you add a scoop of ice cream to a glass filled with root beer? In each case, you’ll have a mess! The volume of the liquid that overflows in each case equals the volume of the solid below the liquid level. This volume is called an object’s displacement. EXAMPLE A Mary Jo wants to find the volume of an irregularly shaped rock. She puts some water into a rectangular prism with a base that measures 10 cm by 15 cm. When the rock is put into the container, Mary Jo notices that the water level rises 2 cm because the rock displaces its volume of water. This new “slice” of water has a volume of (2)(10)(15), or 300 cm3. So the volume of the rock is 300 cm3. Before After 2 cm 10 cm 15 cm 10 cm 15 cm An important property of a material is its density. Density is the mass of matter in a given volume. You can find the mass of an object by weighing it. You calculate density by dividing the mass by the volume, a ss m density e m lu vo EXAMPLE B A clump of metal weighing 351.4 g is dropped into a cylindrical container, causing the water level to rise 1.1 cm. The radius of the base of the container is 3.0 cm. What is the density of the metal? Given the table, and assuming the metal is pure, what is the metal? Aluminum |
2.81 g/cm3 8.89 g/cm3 8.97 g/cm3 21.40 g/cm3 Platinum Density Density Copper Nickel Metal Metal Gold Lead 19.30 g/cm3 Potassium 0.86 g/cm3 11.30 g/cm3 Silver 10.50 g/cm3 Lithium 0.54 g/cm3 Sodium 0.97 g/cm3 LESSON 10.5 Displacement and Density 535 Solution First, find the volume of displaced water. Then, divide the weight by the volume to get the density of the metal. Volume (3.0)2(1.1) ()(9)(1.1) 31.1 Density 351.4 _____ 31.1 11.3 The density is 11.3 g/cm3. Therefore the metal is lead. History Archimedes solved the problem of how to tell if a crown was made of genuine gold by weighing the crown under water. Legend has it that the insight came to him while he was bathing. Thrilled by his discovery, Archimedes ran through the streets shouting “Eureka!” wearing just what he’d been wearing in the bathtub. EXERCISES 1. When you put a rock into a container of water, it raises the water level 3 cm. If the container is a rectangular prism whose base measures 15 cm by 15 cm, what is the volume of the rock? 2. You drop a solid glass ball into a cylinder with a radius of 6 cm, raising the water level 1 cm. What is the volume of the glass ball? 3. A fish tank 10 by 14 by 12 inches high is the home of a large goldfish named Columbia. She is taken out when her owner cleans the tank, and the water level in the tank drops 1 inch. What 3 is Columbia’s volume? For Exercises 4–9, refer to the table on page 535. 4. How much does a solid block of aluminum weigh if its dimensions are 4 cm by 8 cm by 20 cm? 5. Which weighs more: a solid cylinder of gold with a height of 5 cm and a diameter of 6 cm or a solid cone of platinum with a height of 21 cm and a diameter of 8 cm? 6. Chemist Dean Dalton is given a clump of metal and is told that it is sodium. He finds that the metal weighs 145.5 g. He places it |
into a nonreactive liquid in a square prism whose base measures 10 cm on each edge. If the metal is indeed sodium, how high should the liquid level rise? 7. A square-prism container with a base 5 cm by 5 cm is partially filled with water. You drop a clump of metal that weighs 525 g into the container, and the water level rises 2 cm. What is the density of the metal? Assuming the metal is pure, what is the metal? 536 CHAPTER 10 Volume 8. When ice floats in water, one-eighth of its volume floats above the water level and seven-eighths floats beneath the water level. A block of ice placed into an ice chest causes the water in the chest to rise 4 cm. The right rectangular chest measures 35 cm by 50 cm by 30 cm high. What is the volume of the block of ice? 4 cm 30 cm 35 cm 50 cm Science Buoyancy is the tendency of an object to float in either a liquid or a gas. For an object to float on the surface of water, it must sink enough to displace the volume of water equal to its weight. 9. Sherlock Holmes rushes home to his chemistry lab, takes a mysterious medallion from his case, and weighs it. “It weighs 3088 grams. Now, let’s check its volume.” He pours water into a graduated glass container with a 10-by-10 cm square base, and records the water level, which is 53.0 cm. He places the medallion into the container and reads the new water level, 54.6 cm. He enjoys a few minutes of mental calculation, then turns to Dr. Watson. “This confirms my theory. Quick, Watson! Off to the train station.” “Holmes, you amaze me. Is it gold?” questions the good doctor. “If it has a density of 19.3 grams per cubic centimeter, it is gold,” smiles Mr. Holmes. “If it is gold, then Colonel Banderson is who he says he is. If it is a fake, then so is the Colonel.” “Well?” Watson queries. Holmes smiles and says, “It’s elementary, my dear Watson. Elementary geometry, that is.” What is the volume of the medallion? Is it gold? Is Colonel Banderson who he says he is? Review 10. What is |
the volume of the slice removed from this right cylinder? Give your answer to the nearest cm3. 36 cm 6 cm 8 in. 60° 11. APPLICATION Ofelia has brought home a new aquarium shaped like the regular hexagonal prism shown at right. She isn’t sure her desk is strong enough to hold it. The aquarium, without water, weighs 48 pounds. How much will it weigh when it is filled? (Water weighs 63 pounds per cubic foot.) If a small fish needs about 180 cubic inches of water to swim around in, about how many small fish can this aquarium house? 24 in. LESSON 10.5 Displacement and Density 537 12. ABC is equilateral. M is the centroid. AB 6 Find the area of CEA. C 13. Give a paragraph or flowchart proof explaining why M is the midpoint of PQ. 14. The three polygons are regular polygons. How many sides does the red polygon have 15. A circle passes through the three points (4, 7), (6, 3), and (1, 2). a. Find the center. b. Find the equation. 16. A secret rule matches the following numbers: 2 → 4, 3 → 7, 4 → 10, 5 → 13 Find 20 →?, and n →?. MAXIMIZING VOLUME Suppose you have a 10-inch-square sheet of metal and you want to make a small box by cutting out squares from the corners of the sheet and folding up the sides. What size corners should you cut out to get the biggest box possible? To answer this question, consider the length of the corner cut x and write an equation for the volume of the box, y, in terms of x. Graph your equation using reasonable window values. You should see your graph touch the x-axis in at least two places and reach a maximum somewhere in between. Study these points carefully to find their significance. Your project should include x? x The equation you used to calculate volume in terms of x and y. A sketch of the calculator graph and the graphing window you used. An explanation of important points, for example, when the graph touches the x-axis. A solution for what size corners make the biggest volume. Lastly, generalize your findings. For example, what fraction of the side length could you cut from each corner of a 12-inch-square sheet to make a box of maximum volume? 538 CHAPTER 10 Volume Orthographic |
Drawing If you have ever put together a toy from detailed instructions, or built a birdhouse from a kit, or seen blueprints for a building under construction, you have seen isometric drawings. Isometric means “having equal measure,” so the edges of a cube drawn isometrically all have the same length. In contrast, recall that when you drew a cube in two-point perspective, you needed to use edges of different lengths to get a natural look. When you buy a product from a catalog or off the Internet, you want to see it from several angles. The top, front, and right side views are given in an orthographic drawing. Ortho means “straight,” and the views of an orthographic drawing show the faces of a solid as though you are looking at them “head-on.” A B C A Top B Front C Right side Top Front metric An isometric drawing A two-point perspective drawing An orthographic drawing Side Career Architects create blueprints for their designs, as shown at right. Architectural drawing plans use orthographic techniques to describe the proposed design from several angles. These front and side views are called building elevations. Isometric EXPLORATION Orthographic Drawing 539 EXAMPLE A Make an orthographic drawing of the solid shown in the isometric drawing at right. Top Solution Visualize how the building would look from the top, the front, and the right side. Draw an edge wherever there is a change of depth. The top and front views must have the same width, and the front and right side views must have the same height. EXAMPLE B Draw the isometric view of the object shown here as an orthographic drawing. The dashed lines mean that there is an invisible edge. Solution Find the vertices of the front face and make the shape. Use the width of the side and top views to extend parallel lines. Complete the back edges. You can shade parallel planes to show depth. Right side Front Top Front Right side Top Front Right side British pop artist David Hockney (b 1937) titled this photographic collage Sunday Morning Mayflower Hotel, N.Y., Nov-28, 1982. Each photo shows the view from a different angle, just as an orthographic drawing shows multiple angles at once. 540 CHAPTER 10 Volume You will need ● isometric dot paper ● graph paper ● 12 cubes Step 1 Step 2 Activity Isometric and Orthographic Drawings In this investigation you’ll build block |
models and draw their isometric and orthographic views. Practice drawing a cube on isometric dot paper. What is the shape of each visible face? Are they congruent? What should the orthographic views of a cube look like? Stack three cubes to make a two-step “staircase.” Turn the structure so that you look at it the way you would walk up stairs. Call that view the front. Next, identify the top and right sides. How many planes are visible from each view? Make an isometric drawing of the staircase on dot paper and the three orthographic views on graph paper. Step 3 Build solids A–D from their orthographic views, then draw their isometric views. A B C D Top Top Top Top Front Right side Front Right side Front Right side Front Right side Step 4 Make your own original 8- to 12-cube structure and agree on the orthographic views that represent it. Then, trade places with another group and draw the orthographic views of their structure. Step 5 Make orthographic views for solids E and F, and sketch the isometric views of solids G and H. E F G Top H Top Front Right side Front Right side EXPLORATION Orthographic Drawing 541 L E S S O N 10.6 Satisfaction lies in the effort, not in the attainment. Full effort is full victory. MOHANDAS K. GANDHI You will need ● cylinder and hemisphere with the same radius ● sand, rice, birdseed, or water Step 1 Step 2 Step 3 Step 4 Step 5 Volume of a Sphere In this lesson you will develop a formula for the volume of a sphere. In the investigation you’ll compare the volume of a right cylinder to the volume of a hemisphere. Investigation The Formula for the Volume of a Sphere This investigation demonstrates the relationship between the volume of a hemisphere with radius r and the volume of a right cylinder with base radius r and height 2r—that is, the smallest cylinder that encloses a given sphere. Fill the hemisphere. Carefully pour the contents of the hemisphere into the cylinder. What fraction of the cylinder does the hemisphere appear to fill? Fill the hemisphere again and pour the contents into the cylinder. What fraction of the cylinder do two hemispheres (one sphere) appear to fill? r r 2r If the radius of the cylinder is r and its height is 2r, then what is the volume of the cylinder in terms of r? The volume of the sphere is |
the fraction of the cylinder’s volume that was filled by two hemispheres. What is the formula for the volume of a sphere? State it as your conjecture. Sphere Volume Conjecture The volume of a sphere with radius r is given by the formula?. C-90 EXAMPLE A As an exercise for her art class, Mona has cast a plaster cube, 12 cm on each side. Her assignment is to carve the largest possible sphere from the cube. What percentage of the plaster will be carved away? 12 cm 542 CHAPTER 10 Volume Solution The largest possible sphere will have a diameter of 12 cm, so its radius is 6 cm. Applying the formula for volume of a sphere, you get V 4 r3 3 216 288 or about 905 cm3. The volume of the plaster 63 4 4 3 3 cube is 123 or 1728 cm3. You subtract the volume of the sphere from the volume of the cube to get the amount carved away, which is about 823 cm3. Therefore 3 2 8 48%. the percentage carved away is 7 8 2 1 EXAMPLE B Find the volume of plastic (to the nearest cubic inch) needed for this hollow toy component. The outer-hemisphere diameter is 5.0 in. and the inner-hemisphere diameter is 4.0 in. 5.0 in. 4.0 in. Solution Vo The formula for volume of a sphere is V 4 r3, so the volume of a 3 r3. A radius is half a diameter. hemisphere is half of that, V 2 3 Inner Hemisphere r3 2 3 2 (2)3 3 (8) 2 3 17 Outer Hemisphere r3 2 3 2 (2.5)3 3 (15.625) 2 3 33 Vi Subtracting the volume of the inner hemisphere from the volume of the outer one, 16 in.3 of plastic are needed. EXERCISES In Exercises 1–6, find the volume of each solid. All measurements are in centimeters. You will need Construction tools for Exercise 21 1. 4. 12 3 6 6 2. 5. 1 2 3 5 3. 6. 3 4 18 40° LESSON 10.6 Volume of a Sphere 543 7. What is the volume of the largest hemisphere that you could carve out of a wooden block whose edges measure 3 m by 7 m by 7 m? 8. A sphere of ice cream is placed onto your ice cream cone. Both have a diameter of 8 |
cm. The height of your cone is 12 cm. If you push the ice cream into the cone, will all of it fit? 9. APPLICATION Lickety Split ice cream comes in a cylindrical container with an inside diameter of 6 inches and a height of 10 inches. The company claims to give the customer 25 scoops of ice cream per container, each scoop being a sphere with a 3-inch diameter. How many scoops will each container really hold? 10. Find the volume of a spherical shell with an outer diameter of 8 meters and an inner diameter of 6 meters. 11. Which is greater, the volume of a hemisphere with radius 2 cm or the total volume of two cones with radius 2 cm and height 2 cm? 12. A sphere has a volume of 972 in.3. Find its radius. 13. A hemisphere has a volume of 18 cm3. Find its radius. 14. The base of a hemisphere has an area of 256 cm2. Find its volume. 15. If the diameter of a student’s brain is about 6 inches, and you assume its shape is approximately a hemisphere, then what is the volume of the student’s brain? 16. A cylindrical glass 10 cm tall and 8 cm in diameter is filled to 1 cm from the top with water. If a golf ball 4 cm in diameter is placed into the glass, will the water overflow? 17. APPLICATION This underground gasoline storage tank is a right cylinder with a hemisphere at each end. How many gallons of gasoline will the tank hold? (1 gallon 0.13368 cubic foot) If the service station fills twenty 15-gallon tanks from the storage tank per day, how many days will it take to empty the storage tank? 9 feet 36 feet 544 CHAPTER 10 Volume Review 18. Inspector Lestrade has sent a small piece of metal to the crime lab. The lab technician finds that its mass is 54.3 g. It appears to be lithium, sodium, or potassium, all highly reactive with water. Then the technician places the metal into a graduated glass cylinder of radius 4 cm that contains a nonreactive liquid. The metal causes the level of the liquid to rise 2.0 cm. Which metal is it? (Refer to the table on page 535.) 19. City law requires that any one-story commercial building supply a parking area equal in size to the floor area of the building. A-Round Architects has designed a cylindrical building with a |
150-foot diameter. They plan to ring the building with parking. How far from the building should the parking lot extend? Round your answer to the nearest foot. Parking Building 20. Plot A, B, C, and D onto graph paper. A is (3, 5). C is the reflection of A over the x-axis. B is the rotation of C 180° around the origin. D is a transformation of A by the rule (x, y) → (x 6, y 10). What kind of quadrilateral is ABCD? Give reasons for your answer. 21. Construction Use your geometry tools to construct an inscribed and circumscribed circle for an equilateral triangle. 22. Find w, x, and y. w 100° y x 65° IMPROVING YOUR VISUAL THINKING SKILLS Patchwork Cubes The large cube at right is built from 13 double cubes like the one shown plus one single cube. What color must the single cube be, and where must it be positioned? LESSON 10.6 Volume of a Sphere 545 L E S S O N 10.7 Sometimes it’s better to talk about difficult subjects lying down; the change in posture sort of tilts the world so you can get a different angle on things. MARY WILLIS WALKER Step 1 Step 2 546 CHAPTER 10 Volume Surface Area of a Sphere Earth is so large that it is reasonable to use area formulas for plane figures— rectangles, triangles, and circles—to find the areas of most small land regions. But, to find Earth’s entire surface area, you need a formula for the surface area of a sphere. Now that you know how to find the volume of a sphere, you can use that knowledge to arrive at the formula for the surface area of a sphere. Earth rises over the Moon’s horizon. From there, the Moon’s surface seems flat. Investigation The Formula for the Surface Area of a Sphere In this investigation you’ll visualize a sphere’s surface covered by tiny shapes that are nearly flat. So the surface area, S, of the sphere is the sum of the areas of all the “nearly polygons.” If you imagine radii connecting each of the vertices of the “nearly polygons” to the center of the sphere, you are mentally dividing the volume of the sphere into many “nearly pyramids.” Each of the “n |
early polygons” is a base for a pyramid, and the radius, r, of the sphere is the height of the pyramid. So the volume, V, of the sphere is the sum of the volumes of all the pyramids. Now get ready for some algebra. A horsefly’s eyes resemble spheres covered by “nearly polygons.” Divide the surface of the sphere into 1000 “nearly polygons” with areas B1, B2, B3,..., B1000. Then you can write the surface area, S, of the sphere as the sum of the 1000 B’s: B2... B1000 S B1 B3 B1 The volume of the pyramid with base B1 is 1 sphere, V, is the sum of the volumes of the 1000 pyramids: B2(r)... 1 B1(r) 1 V 1 B1000(r) 3 3 3 3 B1(r), so the total volume of the Step 3 Step 4 What common expression can you factor from each of the terms on the right side? Rewrite the last equation showing the results of your factoring. But the volume of the sphere is V 4 r3. Rewrite your equation from Step 2 by 3 substituting 4 r3 for V and substituting for S the sum of the areas of all the 3 “nearly polygons.” Solve the equation from Step 3 for the surface area, S. You now have a formula for finding the surface area of a sphere in terms of its radius. State this as your next conjecture and add it to your conjecture list. Sphere Surface Area Conjecture The surface area, S, of a sphere with radius r is given by the formula?. C-91 EXAMPLE Find the surface area of a sphere whose volume is 12,348 m3. Solution First, use the volume formula for a sphere to find its radius. Then, use the radius to find the surface area. Radius Calculation Surface Area Calculation 4_ 3 r 3 V 3 12,348 4 12,348 r 3 r 3 4_ 3_ 9261 r 3 r 21 S 4r 2 4(21)2 4(441) S 1764 5541.8 The radius is 21 m, and the surface area is 1764 m2, or about 5541.8 m2. EXERCISES For Exercises 1–3, find |
the volume and total surface area of each solid. All measurements are in centimeters. 1. 2. 1.8 3. You will need Geometry software for Exercises 20 and 21 12 9 LESSON 10.7 Surface Area of a Sphere 547 4. The shaded circle at right has area 40 cm2. Find the surface area of the sphere. 5. Find the volume of a sphere whose surface area is 64 cm2. r 6. Find the surface area of a sphere whose volume is 288 cm3. 7. If the radius of the base of a hemisphere (which is bounded by a great circle) is r, what is the area of the great circle? What is the total surface area of the hemisphere, including the base? How do they compare? r 8. If Jose used 4 gallons of wood sealant to cover the hemispherical ceiling of his vacation home, how many gallons of wood sealant are needed to cover the floor? 9. Assume a Kickapoo wigwam is a semicylinder with a half-hemisphere on each end. The diameter of the semicylinder and each of the half-hemispheres is 3.6 meters. The total length is 7.6 meters. What is the volume of the wigwam and the surface area of its roof? Cultural 3.6 m 4.0 m 7.6 m A wigwam was a domed structure that Native American woodland tribes, such as the Kickapoo, Iroquois, and Cherokee, used for shelter and warmth in the winter. They designed each wigwam with an oval floor pattern, set tree saplings vertically into the ground around the oval, bent the tips of the saplings into an arch, and tied all the pieces together to support the framework. They then wove more branches horizontally around the building and added mats over the entire dwelling, except for the doorway and smoke hole. 10. APPLICATION A farmer must periodically resurface the interior (wall, floor, and ceiling) of his silo to protect it from the acid created by the silage. The height of the silo to the top of the hemispherical dome is 50 ft, and the diameter is 18 ft. a. What is the approximate surface area that needs to be treated? b. If 1 gallon of resurfacing compound covers about 250 ft2, how many gallons are needed? c. There is 0.8 bushel per ft3. Calculate the number of bushel |
s of grain this silo will hold. 11. About 70% of Earth’s surface is covered by water. If the diameter of Earth is about 12,750 km, find the area not covered by water to the nearest 100,000 km2. 548 CHAPTER 10 Volume History From the early 13th century to the late 17th century, the Medici family of Florence, Italy, were successful merchants and generous patrons of the arts. The Medici family crest, shown here, features six spheres—five red spheres and one that resembles the earth. The use of three gold spheres to advertise a pawnshop could have been inspired by the Medici crest. 12. A sculptor has designed a statue that features six hemispheres (inspired by the Medici crest) and three spheres (inspired by the pawnshop logo). He wants to use gold electroplating on the six hemispheres (diameter 6 cm) and the three spheres (diameter 8 cm), which will cost about 14¢/cm2. (The bases of the hemispheres will not be electroplated.) Will he be able to stay under his $150 budget? If not, what diameter spheres should he make to stay under budget? 13. Earth has a thin outer layer called the crust, which averages about 24 km thick. Earth’s diameter is about 12,750 km. What percentage of the volume of Earth is the crust? Crust Review 14. A piece of wood placed in a cylindrical container causes the container’s water level to rise 3 cm. This type of wood floats half out of the water, and the radius of the container is 5 cm. What is the volume of the piece of wood? 15. Find the ratio of the area of the circle inscribed in an equilateral triangle to the area of the circumscribed circle. 16. Find the ratio of the area of the circle inscribed in a square to the area of the circumscribed circle. 17. Find the ratio of the area of the circle inscribed in a regular hexagon to the area of the circumscribed circle. 18. Make a conjecture as to what happens to the ratio in Exercises 15–17 as the number of sides of the regular polygon increases. Make sketches to support your conjecture. 19. Use inductive reasoning to complete each table. a. 1 n f(n) 2 b. n f(n............ 200 n.... |
.. n... 200... LESSON 10.7 Surface Area of a Sphere 549 20. Technology Use geometry software to construct a segment AB, and its midpoint C. Trace C and B, and drag B around to sketch a shape. Compare the shapes they trace. 21. Technology Use geometry software to construct a circle. Choose a point A on the circle and a point B not on the circle, and construct the perpendicular bisector of AB. Trace the perpendicular bisector as you animate A around the circle. Describe the locus of points traced. IMPROVING YOUR REASONING SKILLS Reasonable ’rithmetic II Each letter in these problems represents a different digit. 1. What is the value of C. What is the value of D. What is the value of K? 4. What is the value of N? G J 7H LQN M 2 N P M 2 M 2 550 CHAPTER 10 Volume Sherlock Holmes and Forms of Valid Reasoning “That’s logical!” You’ve probably heard that expression many times. What do we mean when we say someone is thinking logically? One dictionary defines logical as “capable of reasoning or using reason in an orderly fashion that brings out fundamental points.” “Prove it!” That’s another expression you’ve probably heard many times. It is an expression that is used by someone concerned with logical thinking. In daily life, proving something often means you can present some facts to support a point. In Chapter 2 you learned that in geometry—as in daily life—a conclusion is valid when you present rules and facts to support it. You have often used given information and previously proven conjectures to prove new conjectures in paragraph proofs and flowchart proofs. When you apply deductive reasoning, you are “being logical” like detective Sherlock Holmes. The statements you take as true are called premises, and the statements that follow from them are conclusions. When you translate a deductive argument into symbolic form, you use capital letters to stand for simple statements. When you write “If P then Q,” you are writing a conditional statement. Here are two examples. English argument Symbolic translation If Watson has chalk between his fingers, then he has been playing billiards. Watson Q: Watson has been playing billiards. has chalk between his fingers. Therefore, Watson has been playing billiards. P: Watson has chalk |
between his fingers. If P then Q. P Q If triangle ABC is isosceles, then the base angles are congruent. Triangle ABC is isosceles. Therefore, its base angles are congruent. P: Triangle ABC is isosceles. Q: Triangle ABC’s base angles are congruent. If P then Q. P Q EXPLORATION Sherlock Holmes and Forms of Valid Reasoning 551 The symbol means “therefore.” So you can read the last two lines “P, Q” as “P, therefore Q” or “P is true, so Q is true.” Both of these examples illustrate one of the well-accepted forms of valid reasoning. According to Modus Ponens (MP), if you accept “If P then Q” as true and you accept P as true, then you must logically accept Q as true. In geometry—as in daily life—we often encounter “not” in a statement. “Not P” is the negation of statement P. If P is the statement “It is raining,” then “not P,” symbolized P, is the statement “It is not raining” or “It is not the case that it is raining.” To remove negation from a statement, you remove the not. The negation of the statement “It is not raining” is “It is raining.” You can also negate a “not” by adding yet another “not.” So you can also negate the statement “It is not raining” by saying “It is not the case that it is not raining.” This property is called double negation. According to Modus Tollens (MT), if you accept “If P then Q” as true and you accept Q as true, then you must logically accept P as true. Here are two examples. English argument Symbolic translation If Watson wished to invest money with Thurston, then he would have had his checkbook with him. Watson did not have his checkbook with him. Therefore Watson did not wish to invest money with Thurston. If AC is the longest side in ABC, then B is the largest angle in ABC. B is not the largest angle in ABC. Therefore AC is not the longest side in ABC. P: Watson wished to invest money |
with Thurston. Q: Watson had his checkbook with him. If P then Q. Q P P: AC is the longest side in ABC. Q: B is the largest angle in ABC. If P then Q. Q P Activity It’s Elementary! In this activity you’ll apply what you have learned about Modus Ponens (MP) and Modus Tollens (MT). You’ll also get practice using the symbols of logic such as P and P as statements and for “so” or “therefore.” To shorten your work even further you can symbolize the conditional “If P then Q” as P → Q. Then Modus Ponens and Modus Tollens written symbolically look like this: Modus Ponens P → Q P Q Modus Tollens R → S S R 552 CHAPTER 10 Volume Step 1 Use logic symbols to translate parts a–e. Tell whether Modus Ponens or Modus Tollens is used to make the reasoning valid. a. If Watson was playing billiards, then he was playing with Thurston. Watson was playing billiards. Therefore Watson was playing with Thurston. b. Every cheerleader at Washington High School is in the 11th grade. Mark is a cheerleader at Washington High School. Therefore, Mark is in the 11th grade. c. If Carolyn studies, then she does well on tests. Carolyn did not do well on her tests, so she must not have studied. d. If ED is a midsegment in ABC, then ED is parallel to a side of ABC. ED is a midsegment in ABC. Therefore ED is parallel to a side of ABC. e. If ED is a midsegment in ABC, then ED is parallel to a side of ABC. ED is not parallel to a side of ABC. Therefore ED is a not a midsegment in ABC. Step 2 Use logic symbols to translate parts a–d. If the two premises fit the valid reasoning pattern of Modus Ponens or Modus Tollens, state the conclusion symbolically and translate it into English. Tell whether Modus Ponens or Modus Tollens is used to make the reasoning valid. Otherwise write “no valid conclusion.” a. If Aurora passes her Spanish test, then she will graduate. Aurora passes the test. b. The diagonals of ABCD are not congruent. If ABCD is a rectangle, |
then its diagonals are congruent. c. If yesterday was Thursday, then there is no school tomorrow. There is no school tomorrow. d. If you don’t use Shining Smile toothpaste, then you won’t be successful. You do not use Shining Smile toothpaste. e. If squiggles are flitz, then ruggles are bodrum. Ruggles are not bodrum. Step 3 Identify each symbolic argument as Modus Ponens or Modus Tollens. If the argument is not valid, write “no valid conclusion.” a. P → S P S b. T → P T P c. R → Q Q R d. Q → S S Q e. Q → P Q P f. R → S S R g. P → (R → Q) P (R → Q) h. (T → P) → Q Q (T → P) i. P → (R → P) (R → P) P EXPLORATION Sherlock Holmes and Forms of Valid Reasoning 553 ● CHAPTER 11 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAP CHAPTER 10 R E V I E W In this chapter you discovered a number of formulas for finding volumes. It’s as important to remember how you discovered these formulas as it is to remember the formulas themselves. For example, if you recall pouring the contents of a cone into a cylinder with the same base and height, you may recall that the volume of the cone is one-third the volume of the cylinder. Making connections will help, too. Recall that prisms and cylinders share the same volume formula because their shapes—two congruent bases connected by lateral faces—are alike. You should also be able to find the surface area of a sphere. The formula for the surface area of a sphere was intentionally not included in Chapter 8, where you first learned about surface area. Look back at the investigations in Lesson 8.7 and explain why the surface area formula requires that you know volume. As you have seen, volume formulas can be applied to many practical problems. Volume also has many extensions such as calculating displacement and density. EXERCISES 1. How are a prism and a cylinder alike? 2. What does a cone have in common with a pyramid? For Exercises 3–8, find the volume of each solid. Each quadrilateral is a rectangle. |
All solids are right (not oblique). All measurements are in centimeters. 26 3. 6. 12 20 6 4 4 554 CHAPTER 10 Volume 21 4. 7. 14 12 10 5. 12 12 10 8. 8 15 6 ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAP For Exercises 9–12, calculate each unknown length given the volume of the solid. All measurements are in centimeters. 9. Find H. V 768 cm3 10. Find h. V 896 cm3 8 H 17 24 20 h 12 11. Find r. V 1728 cm3 12. Find r. V 256 cm3 36 r r 90° 13. Find the volume of a rectangular prism whose dimensions are twice those of another rectangular prism that has a volume of 120 cm3. 14. Find the height of a cone with a volume of 138 cubic meters and a base area of 46 square meters. 15. Find the volume of a regular hexagonal prism that has a cylinder drilled from its center. Each side of the hexagonal base measures 8 cm. The height of the prism is 16 cm. The cylinder has a radius of 6 cm. Express your answer to the nearest cubic centimeter. 16. Two rectangular prisms have equal heights but unequal bases. Each dimension of the smaller solid’s base is half each dimension of the larger solid’s base. The volume of the larger solid is how many times as great as the volume of the smaller solid? 17. The “extra large” popcorn container is a right rectangular prism with dimensions 3 in. by 3 in. by 6 in. The “jumbo” is a cone with height 12 in. and diameter 8 in. The “colossal” is a right cylinder with diameter 10 in. and height 10 in. a. Find the volume of all three containers. b. Approximately how many times as great is the volume of the “colossal” than the “extra large”? 18. Two solid cylinders are made of the same material. Cylinder A is six times as tall as cylinder B, but the diameter of cylinder B is four times the diameter of cylinder A. Which cylinder weighs more? How many times as much CHAPTER 10 REVIEW 555 EW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CH 19. |
APPLICATION Rosa Avila is a plumbing contractor. She needs to deliver 200 lengths of steel pipe to a construction site. Each cylindrical steel pipe is 160 cm long, has an outer diameter of 6 cm, and has an inner diameter of 5 cm. Rosa needs to know if her quarter-tonne truck can handle the weight of the pipes. To the nearest kilogram, what is the weight of these 200 pipes? How many loads will Rosa have to transport to deliver the 200 lengths of steel pipe? (Steel has a density of about 7.7 g/cm3. One tonne equals 1000 kg.) 20. A ball is placed snugly into the smallest possible box that will completely contain the ball. What percentage of the box is filled by the ball? 21. APPLICATION The blueprint for a cement slab floor is shown at right. How many cubic yards of cement are needed for ten identical floors that are each 4 inches thick? 70 ft 35 ft 50 ft 30 ft 22. A prep chef has just made two dozen meatballs. Each meatball has a 2-inch diameter. Right now, before the meatballs are added, the sauce is 2 inches from the top of the 14-inchdiameter pot. Will the sauce spill over when the chef adds the meatballs to the pot? 23. To solve a crime, Betty Holmes, who claims to be Sherlock’s distant cousin, and her friend Professor Hilton Gardens must determine the density of a metal art deco statue that weighs 5560 g. She places it into a graduated glass prism filled with water and finds that the level rises 4 cm. Each edge of the glass prism’s regular hexagonal base measures 5 cm. Professor Gardens calculates the statue’s volume, then its density. Next, Betty Holmes checks the density table (see page 535) to determine if the statue is platinum. If so, it is the missing piece from her client’s collection and Inspector Clouseau is the thief. If not, then the Baron is guilty of fraud. What is the statue made of? 24. Can you pick up a solid steel ball of radius 6 inches? Steel has a density of 0.28 pound per cubic inch. To the nearest pound, what is the weight of the ball? 25. To the nearest pound, what is the weight of a hollow steel ball with an outer diameter of 14 inches and a thickness of 2 inches? 26. A hollow steel ball has a diameter of 14 inches and weighs 327 |
.36 pounds. Find the thickness of the ball. 556 CHAPTER 10 Volume ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAP 27. A water barrel that is 1 m in diameter and 1.5 m long is partially filled. By tapping on its sides, you estimate that the water is 0.25 m deep at the deepest point. What is the volume of the water in cubic meters? 1 m 0.25 m 1.5 m 28. Find the volume of the solid formed by rotating the shaded figure about the y-axis. y 5 3 1 1 3 5 x TAKE ANOTHER LOOK 1. You may be familiar with the area model of the expression (a b)2, shown below. Draw or build a volume model of the expression (a b)3. How many distinct pieces does your model have? What’s the volume of each type of piece? Use your model to write the expression (a b)3 in expanded form. a b a b (a + b) (a + b) a a 2 ab a a b b ab a b b 2 b 2. Use algebra to show that if you double all three dimensions of a prism, a cylinder, a pyramid, or a cone, the volume is increased eightfold but the surface area is increased only four times. 3. Any sector of a circle can be rolled into a cone. Find a way to calculate the volume of a cone given the radius and central angle of the sector. Bring these radii together to form a cone. r r x CHAPTER 10 REVIEW 557 EW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CHAPTER 10 REVIEW ● CH 4. Build a model of three pyramids with equal volumes that you can assemble into a prism. 5. Derive the Sphere Volume Conjecture by using a pair of hollow shapes different from those you used in the Investigation The Formula for the Volume of a Sphere. Or use two solids made of the same material and compare weights. Explain what you did and how it demonstrates the conjecture. 6. Spaceship Earth, located at the Epcot center in Orlando, Florida, is made of polygonal regions arranged in little pyramids. The building appears spherical, but the surface is not smooth. If a perfectly smooth sphere had the same volume as Spaceship Earth, would it have the same surface area? |
If not, which would be greater, the surface area of the smooth sphere or of the bumpy sphere? Explain. Assessing What You’ve Learned Spaceship Earth, the Epcot center’s signature structure, opened in 1982 in Walt Disney World. It features a ride that chronicles the history of communication technology. UPDATE YOUR PORTFOLIO Choose a project, a Take Another Look activity, or one of the more challenging problems or puzzles you did in this chapter to add to your portfolio. WRITE IN YOUR JOURNAL Describe your own problem-solving approach. Are there certain steps you follow when you solve a challenging problem? What are some of your most successful problem-solving strategies? ORGANIZE YOUR NOTEBOOK Review your notebook to be sure it’s complete and well organized. Be sure you have all the conjectures in your conjecture list. Write a one-page summary of Chapter 10. PERFORMANCE ASSESSMENT While a classmate, friend, family member, or teacher observes, demonstrate how to derive one or more of the volume formulas. Explain what you’re doing at each step. WRITE TEST ITEMS Work with classmates to write test items for this chapter. Include simple exercises and complex application problems. Try to demonstrate more than one approach in your solutions. GIVE A PRESENTATION Give a presentation about one or more of the volume conjectures. Use posters, models, or visual aids to support your presentation. 558 CHAPTER 10 Volume CHAPTER 11 Similarity Nobody can draw a line that is not a boundary line, every line separates a unity into a multiplicity. In addition, every closed contour no matter what its shape, pure circle or whimsical splash accidental in form, evokes the sensation of “inside” and “outside,” followed quickly by the suggestion of “nearby” and “far off,” of object and background. M. C. ESCHER Path of Life I, M. C. Escher, 1958 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● review ratio and proportion ● define similar polygons and solids ● discover shortcuts for similar triangles ● learn about area and volume relationships in similar polygons and solids ● use the definition of similarity to solve problems ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGE |
BRA SKILLS 9 ● USING Y USING YOUR ALGEBRA SKILLS 9 Proportion and Reasoning Working with similar geometric figures involves ratios and proportions. You may be a little rusty with these topics, so let’s review. A ratio is an expression that compares two quantities by division. You can write the ratio of quantity a to quantity b in these three ways: a b a to b a:b In this book you will write ratios in fraction form. As with fractions, you can multiply or divide both parts of a ratio by the same number to get an equivalent ratio. 1 6 A proportion is a statement of equality between two ratios. The equality 3 1 8 is an example of a proportion. Proportions are useful for solving problems involving comparisons. EXAMPLE A In a photograph, Dan is 2.5 inches tall and his sister Emma is 1.5 inches tall. Dan’s actual height is 70 inches. What is Emma’s actual height? Solution The ratio of Dan’s height to Emma’s height is the same in real life as it is in the photo. Let x represent Emma’s height and set up a proportion. Dan’s height in photograph Emma’s height in photograph 2.5 1.5 70 x Dan’s actual height Emma’s actual height Find Emma’s height by solving for x 70 1. 5 2.5x 105 x 42 Emma is 42 inches tall. Original proportion. Multiply both sides by x. Multiply both sides by 1.5. Divide both sides by 2.5. There are other proportions you could have used to solve the problem in Example A. For instance, the ratio of Dan’s actual height to his height in the photo is equal to the ratio of Emma’s actual height to her height in the photo. So you 7 0 x could have found Emma’s height by solving. What other correct.5 1 5. 2 proportion could you use? 560 CHAPTER 11 Similarity ALGEBRA SKILLS 9 ● USING YOUR ALGEBRA SKILLS 9 ● USING YOUR ALGEBRA SKILLS 9 ● USING YO Some proportions require more algebra to solve. EXAMPLE B 6 x 50. 0 Solve 3 20 2 4 Solution 6 x 50 0 3 4 2 20 6 x 50 0 20 3 4 2 255 x 50 205 x Original proportion. |
Multiply both sides by 20. Multiply and divide on the left side. Subtract 50 from both sides. EXERCISES 1. Look at the rectangle at right. Find the ratio of the shaded area to the area of the whole figure. Find the ratio of the shaded area to the unshaded area. D D, and B C, C A. 2. Use the figure below to find these ratios: C B BD D C 3 cm 5 cm 8 cm A C D B 3. Consider these triangles. H 15 R 39 S L 5 M 13 F a. Find the ratio of the perimeter of RSH to the perimeter of MFL. b. Find the ratio of the area of RSH to the area of MFL. In Exercises 4–12, solve the proportion. 7 a 4. 2 1 8 1 7. 4 x 7 5 3 5 1 0 10. 5 6 10 z 0 1 5 5. 1 2 4 b y 8. 2 2 3 y 3 d 11. d 2 0 5 13. Solve this proportion for x. Assume c 0 and z 0. b x z c 0 6 0 6. 2 1 3 c 9 4 9 12. y 2 2 1 USING YOUR ALGEBRA SKILLS 9 Proportion and Reasoning 561 ALGEBRA SKILLS 9 ● USING YOUR ALGEBRA SKILLS 9 ● USING YOUR ALGEBRA SKILLS 9 ● USING YO In Exercises 14–17, use a proportion to solve the problem. 14. APPLICATION A car travels 106 miles on 4 gallons of gas. How far can it go on a full tank of 12 gallons? 15. APPLICATION Ernie is a baseball pitcher. He gave up 34 runs in 152 innings last season. What is Ernie’s earned run average—the number of runs he would give up in 9 innings? Give your answer accurate to two decimal places. 16. APPLICATION The floor plan of a house is drawn to the scale of 1 in. 1 ft. The 4 master bedroom measures 3 in. by 33 in. on the blueprints. What is the actual size 4 of the room? 17. Altor and Zenor are ambassadors from Titan, the largest moon of Saturn. The sum of the lengths of any Titan’s antennae is a direct measure of that Titan’s age. Altor has antennae with lengths 8 cm |
, 10 cm, 13 cm, 16 cm, 14 cm, and 12 cm. Zenor is 130 years old, and her seven antennae have an average length of 17 cm. How old is Altor? C B B 18. Assume A. Find AB and BC. Y Z XY 10.5 cm A B C 2 cm Y X 5 cm Z IMPROVING YOUR ALGEBRA SKILLS Algebraic Magic Squares II In this algebraic magic square, the sum of the entries in every row, column, and diagonal is the same. Find the value of x. 8 x 15 14 11 x 12 2x 1 3 2 2x 2 562 CHAPTER 11 Similarity L E S S O N 11.1 He that lets the small things bind him Leaves the great undone behind him. PIET HEIN Similar Polygons You know that figures that have the same shape and size are congruent figures. Figures that have the same shape but not necessarily the same size are similar figures. To say that two figures have the same shape but not necessarily the same size is not, however, a precise definition of similarity. Is your reflection in a fun-house mirror similar to a regular photograph of you? The images have a lot of features in common, but they are not mathematically similar. In mathematics, you can think of similar shapes as enlargements or reductions of each other with no irregular distortions. Are all rectangles similar? They have common characteristics, but they are not all similar. That is, you could not enlarge or reduce a given rectangle to fit perfectly over every other rectangle. What about other geometric figures: squares, circles, triangles? The uneven surface of a fun-house mirror creates a distorted image of you. Your true proportions look different in your reflection. Rectangles A and B are not similar. You could not enlarge or reduce one to fit perfectly over the other. A B Art Movie scenes are scaled down to small images on strips of film. Then they are scaled up to fit a large screen. So the film image and the projected image are similar. If the distance between the projector and the screen is decreased by half, each dimension of the screen image is cut in half. x 2x LESSON 11.1 Similar Polygons 563 You will need ● patty paper ● a ruler C D E Investigation 1 What Makes Polygons Similar? Let’s explore what makes polygons similar. Hexagon PQRSTU is an enlarg |
ement of hexagon ABCDEF—they are similar Step 1 Step 2 Step 3 Use patty paper to compare all corresponding angles. How do the corresponding angles compare? Measure the corresponding segments in both hexagons. Find the ratios of the lengths of corresponding sides. How do the ratios of corresponding sides compare? Similar objects are often used to create unique buildings. The giant basket shown here is actually an office building for a basket manufacturer. The giant donut advertises a donut shop in Los Angeles, California. 564 CHAPTER 11 Similarity From the investigation, you should be able to state a mathematical definition of similar polygons. Two polygons are similar polygons if and only if the corresponding angles are congruent and the corresponding sides are proportional. Similarity is the state of being similar. The statement CORN PEAS says that quadrilateral CORN is similar to quadrilateral PEAS. Just as in statements of congruence, the order of the letters tells you which segments and which angles in the two polygons correspond. N R Corresponding angles are congruent Corresponding segments are proportional: CO___ RN___ AS PE NC___ SP OR___ EA P E Do you need both conditions—congruent angles and proportional sides—to guarantee that the two polygons are similar? For example, if you know only that the corresponding angles of two polygons are congruent, can you conclude that the polygons have to be similar? Or, if corresponding sides of two polygons are proportional, are the polygons necessarily similar? These counterexamples show that both answers are no. In the figures below, corresponding angles of square SQUE and rectangle RCTL are congruent, but their corresponding sides are not proportional. E 12 10 R 12 Q T C 18 In the figures below, corresponding sides of square SQUE and rhombus RHOM are proportional, but their corresponding angles are not congruent. E 12 S U Q 12 M 60° O 120° 1 2 1 2 1 8 8 1 18 120° R 60° H 18 Clearly, neither pair of polygons is similar. You cannot conclude that two polygons are similar given only the fact that their corresponding angles are congruent or given only the fact that their corresponding sides are proportional. LESSON 11.1 Similar Polygons 565 You can use the definition of similar polygons to find missing measures in similar polygons. EXAMPLE SMAL BIGE Find x and y. 21 ft A |
L 18 ft 78° S 83° M E 24 ft B G x y I Solution The quadrilaterals are similar, so you can use a proportion to find x. 1 2 1 8 4 2 x 18x (24)(21) x 28 A proportion of corresponding sides. Multiply both sides by 24x and reduce. Divide both sides by 18. The measure of the side labeled x is 28 ft. In similar polygons, corresponding angles are congruent, so M I. The measure of the angle labeled y is therefore 83°. Earlier in this book you worked with translations, rotations, and reflections. These rigid transformations preserve both size and shape—the images are congruent to the original figures. One type of nonrigid transformation is called a dilation. Let’s look at an image after a dilation transformation. Investigation 2 Dilations on a Coordinate Plane You will need ● graph paper ● a straightedge ● patty paper ● a compass y 7 B –7 A E –7 C x 7 D Step 1 Step 2 To dilate a pentagon on a coordinate plane, first copy this pentagon onto your graph paper. Have each member of your group multiply the coordinates of the vertices by one, 3 of these numbers: 1, 2, or 3. Each of these factors is called a scale factor. 4 2 Step 3 Locate these new coordinates on your graph paper and draw the new pentagon. 566 CHAPTER 11 Similarity Step 4 Step 5 Copy the original pentagon onto patty paper. Compare the corresponding angles of the two pentagons. What do you notice? Compare the corresponding sides with a compass or with patty paper. The length of each side of the new pentagon is how many times as long as the length of the corresponding side of the original pentagon? Step 6 Compare results with your group. You should be ready to state a conjecture. Dilation Similarity Conjecture If one polygon is the image of another polygon under a dilation, then?. C-92 History Similarity plays an important role in human history. For example, accurate maps of regions of China have been found dating back to the second century B.C.E. Neolithic cave paintings 8000–6000 B.C.E. contain small-scale drawings of the animals people hunted. Giant geoglyphs made by the Nazca people of Peru (110 B.C.E.–800 C.E.) are some |
of the largest scale drawings ever made. This cave art is part of a grouping of over 15,000 drawings in Tassili N’Ajjier National Park of the Algerian Sahara. Interestingly, these drawings depict animals and landscapes that are absent from the region today, such as these elephants or vast lakes. This monkey is a geoglyph found in the Pampa region of Peru in 1920. Called the Nazca Lines, the figure measures over 400 feet long and can only be clearly seen from the air. In order for a map to be accurate, cartographers need to use similarity to reduce the earth’s attributes to a smaller scale. This sixteenth-century French map, a plan of Constantinople, included a mariner’s chart of America, Europe, Africa, and Asia. LESSON 11.1 Similar Polygons 567 EXERCISES For Exercises 1 and 2, match the similar figures. You will need Construction tools for Exercises 22 and 25 1. 2. A. A. B. B. C. C. For Exercises 3–5, sketch on graph paper a similar, but not congruent, figure. 3. 4. 5. 6. Complete the statement: If Figure A is similar to Figure B and Figure B is similar to Figure C, then?. Draw and label figures to illustrate the statement. For Exercises 7–14, use the definition of similar polygons. All measurements are in centimeters. 7. THINK LARGE Find AL, RA, RG, and KN 12 L A 9. SPIDER HNYCMB Find NY, YC, CM, and MB. E 36 D 40 R 36 S 88 H H B 66 56 I 28 P R M N C Y 8. Are these polygons similar? Explain why or why not. 150 120 165 128 G 180 192 140 154 10. Are these polygons similar? Explain why or why not. 18 20 14 39 52 26 27 30 21 78 568 CHAPTER 11 Similarity A 7 C 12 8 11. ACE IKS Find x and y. I y K E S x 4 13. DE BC Are the corresponding angles congruent in AED and ABC? Are the corresponding sides proportional? Is AED ABC 12. RAM XAE Find z. M z 14. ABC DBA Find m and n 15. Copy ROY onto your graph paper. Sketch its dilation with a scale factor of 3. What is the |
ratio of the perimeter of the smaller triangle to the perimeter of the larger triangle? 16. Copy this quadrilateral onto your graph paper. Draw a similar quadrilateral with each side half the length of its corresponding side in the original quadrilateral. y Y (2, 5) R (2, 2) O (6, 2) x 17. APPLICATION The photo at right shows the Crazy Horse Memorial and a scale model of the complete monument’s design. The head of the Crazy Horse Memorial, from the chin to the top of the forehead, is 87.5 ft high. When the arms are carved, how long will each be? Use the photo and explain how you got your answer. The Crazy Horse Memorial is located in South Dakota. Started in 1948, it will be the world’s largest sculpture when complete. You can learn more about this monument using the links at www.keymath.com/DG. LESSON 11.1 Similar Polygons 569 Review For Exercises 18–20, use algebra to answer each proportion question. Assume that a, b, c, and d are all nonzero values. 20 5 18. If 1, then a?. a 12 a c, then ad?. 19. If a d b b?. c, then 20. If a a d b 21. APPLICATION Jade and Omar each put in $1000 to buy an old boat to fix up. Later Jade spent $825 on materials, and Omar spent $1650 for parts. They worked an equal number of hours on the boat and eventually sold it for $6800. How might they divide the $6800 fairly? Explain your reasoning. 22. Construction Use a compass and straightedge to construct a. A rhombus with a 60° angle. b. A second rhombus of different size with a 60° angle. 23. A cubic foot of liquid is about 7.5 gallons. How many gallons of 5 ft liquid are in this tank? 2 ft 1 ft 24. Triangle PQR has side lengths 18 cm, 24 cm, and 30 cm. Is PQR a right triangle? Explain why or why not. 25. Construction Use the triangular figure at right and its rotated image. a. Copy the figure and its image onto a piece of patty paper. Locate the center of rotation. Explain your method. b. Copy the figure and its image onto a sheet of paper. Locate the center of |
rotation using a compass and straightedge. Explain your method. Career Similarity plays an important part in the design of cars, trucks, and airplanes, which is done with small-scale drawings and models. This model airplane is about to be tested in a wind tunnel. 570 CHAPTER 11 Similarity MAKING A MURAL A mural is a large work of art that usually fills an entire wall. This project gives you a chance to use similarity and make your own mural. One way to create a mural from a small picture is to draw a grid of squares lightly over the small picture. Then divide the mural surface into a similar but larger grid. Proceeding square by square, draw the lines and curves of the small grid in each corresponding square of the mural grid. Complete the mural by coloring or painting the regions and erasing the grid lines. You project should include An original drawing, a cartoon, or a photograph divided into a grid of squares. A finished mural drawn on a large sheet of paper. You can learn more about the art of mural making through the links at www.keymath.com/DG. Mural artists use similarity to help them create large artwork. This mural, finished in 1990, is in the North Beach neighborhood of San Francisco, California. LESSON 11.1 Similar Polygons 571 L E S S O N 11.2 Life is change. Growth is optional. Choose wisely. KAREN KAISER CLARK Similar Triangles In Lesson 11.1, you concluded that you must know about both the angles and the sides of two quadrilaterals in order to make a valid conclusion about their similarity. However, triangles are unique. Recall from Chapter 4 that you found four shortcuts for triangle congruence: SSS, SAS, ASA, and SAA. Are there shortcuts for triangle similarity as well? Let’s first look for shortcuts using only angles. The figures below illustrate that you cannot conclude that two triangles are similar given that only one set of corresponding angles are congruent. C F 48° A B 48° D E A D, but ABC is not similar to DEF. How about two sets of congruent angles? You will need ● a compass ● a ruler Investigation 1 Is AA a Similarity Shortcut? If two angles of one triangle are congruent to two angles of another triangle, must the two triangles be similar? Step 1 Step 2 Step 3 Step 4 Draw any triangle ABC. Construct a second triangle, DEF |
, with D A and E B. What will be true about C and F? Why? Carefully measure the lengths of the sides of both triangles. Compare the ratios C B C A AB? of the corresponding sides. Is E F F D D E Compare your results with the results of others near you. You should be ready to state a conjecture. AA Similarity Conjecture If? angles of one triangle are congruent to? angles of another triangle, then?. C-93 As you may have guessed from Step 2 of the investigation, there is no need to investigate the AAA Similarity Conjecture. Thanks to the Third Angle Conjecture, the AA Similarity Conjecture is all you need. 572 CHAPTER 11 Similarity Now let’s look for shortcuts for similarity that use only sides. The figures below illustrate that you cannot conclude that two triangles are similar given that two sets of corresponding sides are proportional. G 48 cm 54 cm 54___ 108 48__ 96 1_ 2 1_ 2 108 cm W B J 96 cm K F W B G G, but GWB is not similar to JFK. F J JK How about all three sets of corresponding sides? Investigation 2 Is SSS a Similarity Shortcut? If three sides of one triangle are proportional to the three sides of another triangle, must the two triangles be similar? Draw any triangle ABC. Then construct a second triangle, DEF, whose side lengths are a multiple of the original triangle. (Your second triangle can be larger or smaller.) Compare the corresponding angles of the two triangles. Compare your results with the results of others near you and state a conjecture. SSS Similarity Conjecture C-94 If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are?. You will need ● a compass ● a straightedge ● a protractor Many dollhouses and other toys are scale models of real objects. LESSON 11.2 Similar Triangles 573 You will need ● a compass ● a protractor ● a ruler In Investigations 1 and 2, you discovered two shortcuts for triangle similarity: AA and SSS. But if AA is a shortcut, then so are ASA, SAA, and AAA. That leaves SAS and SSA as possible shortcuts to consider. Investigation 3 Is SAS a Similarity Shortcut? Is SAS a shortcut for similarity? Try to construct two different triangles that are not similar but have two pairs of sides proportional and the pair of included |
, 30) (15, y) (5, 3) x LESSON 11.2 Similar Triangles 575 Review 16. In the figure below right, find the radius, r, of one of the small circles in terms of the radius, R, of the large circle. R r This Tibetan mandala is a complex design with a square inscribed within a circle and tangent circles inscribed within the corners of a larger circumscribed square. 17. APPLICATION Phoung volunteers at an SPCA that always houses 8 dogs. She notices that she uses seven 35-pound bags of dry dog food every two months. A new, larger SPCA facility that houses 20 dogs will open soon. Help Phoung estimate the amount of dry dog food that the facility should order every three months. Explain your reasoning. 18. APPLICATION Ramon and Sabina are oceanography students studying the habitat of a Hawaiian fish called Humuhumunukunukuapua‘a. They are going to use the capture-recapture method to determine the fish population. They first capture and tag 84 fish, which they release back into the ocean. After one week, Ramon and Sabina catch another 64. Only 12 have tags. Can you estimate the population of Humuhumunukunukuapua‘a? 576 CHAPTER 11 Similarity 19. Points A(9, 5), B(4, 13), and C(1, 7) are connected to form a triangle. Find the area of ABC. y, to relocate the 20. Use the ordered pair rule, (x, y) → 1 x, 1 2 2 coordinates of the vertices of parallelogram ABCD. Call the new parallelogram ABCD. Is ABCD similar to ABCD? If they are similar, what is the ratio of the perimeter of ABCD to the perimeter of ABCD? What is the ratio of their areas? 21. The photo below shows a fragment from an ancient statue of the Roman Emperor Constantine. Use this photo to estimate how tall the entire statue was. List the measurements you need to make. List any assumptions you need to make. Explain your reasoning. y 8 D C A B x 8 History The Emperor Constantine the Great (Roman Emperor 306–337 C.E.) adopted Christianity as the official religion of the Roman Empire. The Roman Catholic Church regards him as Saint Constantine, and the city of Constantinople was named for him. The colossal statue of Constantine was built |
between 315 and 330 C.E., and broke when sculptors tried to add the extra weight of a beard to its face. The pieces of the statue remain close to its original location in Rome, Italy. IMPROVING YOUR VISUAL THINKING SKILLS Build a Two-Piece Puzzle Construct two copies of Figure A, shown at right. Here’s how to construct the figure. Construct a regular hexagon. Construct an equilateral triangle on two alternating edges, as shown. Construct a square on the edge between the two equilateral triangles, as shown. Figure A Figure B Cut out each copy and fold them into two identical solids, as shown in Figure B. Tape the edges. Now arrange your two solids to form a regular tetrahedron. LESSON 11.2 Similar Triangles 577 F P R U O F Constructing a Dilation Design In Lesson 11.1, you saw how to dilate a polygon on a coordinate plane. You can also use a simple construction to dilate any polygon. Draw rays from any point P through the vertices of the polygon. Use a compass to measure the distance from point P to one of the vertices. Then mark this distance two more times along the ray; that will give you a scale factor of 3. Repeat this process for each of the other vertices using the same scale factor. When you connect the image of each vertex, you will get a similar polygon. Try it yourself. How would you create a similar polygon with a scale factor of 2? Of 1? 2 O R U Now take a closer look at Path of Life I, the M. C. Escher woodcut that begins this chapter. Notice that dilations transform the black fishlike creatures, shrinking them again and again as they approach the picture’s center. The same is true for the white fish. (The black-and-white fish around the outside border are congruent to one another, but they’re not similar to the other fish.) Also notice that rotations repeat the dilations in eight sectors. With Sketchpad, you can make a similar design. Activity Dilation Creations Step 1 Step 2 Step 3 Construct a circle with center point A and point B on the circle. Construct AB. Use the Transform menu to mark point A as center, then rotate point B by an angle of 45°. Your new point is B. Construct AB. Construct a larger circle with center point |
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