text stringlengths 235 3.08k |
|---|
A and point C on AB. Hide AB and construct AC. 578 CHAPTER 11 Similarity Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 D C B D C B Rotate AC, point B, and point C by an angle of 45°. You now have AC, point B, and point C. Construct CD and DC, where D is any point in the region between the circles and between BC and BC. Select, in order, AB and AC. Choose Mark Segment Ratio from the Transform menu. This marks a ratio of a shorter segment to a longer segment. Because this ratio is less than 1, dilating by this scale factor will shrink objects. Select CD, DC, and point D. Choose Dilate from the Transform menu and dilate by the marked ratio. The dilated images are BD, DB, and D. B A Construct three polygon interiors—two triangles and a quadrilateral. Select the three polygon interiors and dilate them by the marked ratio. Repeat this process two or three times Step 9 Step 10 Step 10 Select all the polygon interiors in the sector and rotate them by an angle of 45°. Repeat the rotation by 45° until you’ve gone all the way around the circle. Now you have a design that has the same basic mathematical properties as Escher’s Path of Life I. EXPLORATION Constructing a Dilation Design 579 Step 11 Experiment with changing the design by moving different points. Answer these questions. a. What locations of point D result in both rotational and reflectional symmetry? b. Drag point C away from point A. What does this do to the numerical dilation ratio? What effect does that have on the geometric figure? c. Drag point C toward point A. What happens when the circle defined by point C becomes smaller than the circle defined by point B? Why? Experiment with other dilation-rotation designs of your own. Try different angles of rotation or different polygons. Here are some examples. Mathematician Doris Schattschneider of Moravian College is an expert on M. C. Escher and the mathematics he explored. She made this sketch based on Path of Life I. X This design uses a different angle of rotation and polygons that go outside the sector. This design uses a two-step transformation, a dilation followed by a rotation, called a spiral similarity. 580 CHAPTER 11 Similarity L E S S O N |
11.3 Never be afraid to sit awhile and think. LORRAINE HANSBERRY You will need ● metersticks ● masking tape or a soluble pen ● a mirror Indirect Measurement with Similar Triangles You can use similar triangles to calculate the height of tall objects that you can’t reach. This is called indirect measurement. One method uses mirrors. Try it in the next investigation. Investigation Mirror, Mirror Choose a tall object with a height that would be difficult to measure directly, such as a football goalpost, a basketball hoop, a flagpole, or the height of your classroom. F E P X B Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Mark crosshairs on your mirror. Use tape or a soluble pen. Call the intersection point X. Place the mirror on the ground several meters from your object. An observer should move to a point P in line with the object and the mirror in order to see the reflection of an identifiable point F at the top of the object at point X on the mirror. Make a sketch of your setup, like this one. Measure the distance PX and the distance from X to a point B at the base of the object directly below F. Measure the distance from P to the observer’s eye level, E. Think of FX as a light ray that bounces back to the observer’s eye along XE. Why is B P? Name two similar triangles. Tell why they are similar. Set up a proportion using corresponding sides of similar triangles. Use it to calculate FB, the approximate height of the tall object. Write a summary of what you and your group did in this investigation. Discuss possible causes for error. Another method of indirect measurement uses shadows. LESSON 11.3 Indirect Measurement with Similar Triangles 581 EXAMPLE A person 5 feet 3 inches tall casts a 6-foot shadow. At the same time of day, a lamppost casts an 18-foot shadow. What is the height of the lamppost? 5.25 ft x ft 6 ft 18 ft Solution The light rays that create the shadows hit the ground at congruent angles. Assuming both the person and the lamppost are perpendicular to the ground, you have similar triangles by the AA Similarity Conjecture. Solve a proportion that relates corresponding lengths. 25 5. x 1 8 6 25 x 18 5. 6 15.75 x The height of the lamppost is 15 feet 9 inches. EX |
ERCISES 1. A flagpole 4 meters tall casts a 6-meter shadow. At the same time of day, a nearby building casts a 24-meter shadow. How tall is the building? 2. Five-foot-tall Melody casts an 84-inch shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot shorter than hers? 3. A 10 m rope from the top of a flagpole reaches to the end of the flagpole’s 6 m shadow. How tall is the nearby football goalpost if, at the same moment, it has a shadow of 4 m? You will need Construction tools for Exercise 14 Geometry software for Exercises 17 and 18 4. Private eye Samantha Diamond places a mirror on the ground between herself and an apartment building and stands so that when she looks into the mirror, she sees into a window. The mirror’s crosshairs are 1.22 meters from her feet and 7.32 meters from the base of the building. Sam’s eye is 1.82 meters above the ground. How high is the window? 582 CHAPTER 11 Similarity 5. APPLICATION Juanita, who is 1.82 meters tall, wants to find the height of a tree in her backyard. From the tree’s base, she walks 12.20 meters along the tree’s shadow to a position where the end of her shadow exactly overlaps the end of the tree’s shadow. She is now 6.10 meters from the end of the shadows. How tall is the tree? 6. While vacationing in Egypt, the Greek mathematician Thales calculated the height of the Great Pyramid. According to legend, Thales placed a pole at the tip of the pyramid’s shadow and used similar triangles to calculate its height. This involved some estimating since he was unable to measure the distance from directly beneath the height of the pyramid to the tip of the shadow. From the diagram, explain his method. Calculate the height of the pyramid from the information given in the diagram. 7. Calculate the distance across this river, PR, by sighting a pole, at point P, on the opposite bank. Points R and O are collinear with point P. Point C is chosen so that OC PO. Lastly, point E is chosen so that P, E, and C are collinear and that RE PO. Also explain why PRE POC. 8. A pinhole camera |
is a simple device. Place unexposed film at one end of a shoe box, and make a pinhole at the opposite end. When light comes through the pinhole, an inverted image is produced on the film. Suppose you take a picture of a painting that is 30 cm wide by 45 cm high with a pinhole box camera that is 20 cm deep. How far from the painting should the pinhole be to make an image that is 2 cm wide by 3 cm high? Sketch a diagram of this situation. 1.82 m 6.10 m 12.20 m H 240 m 6.2 m 10 m P R 45 m O E 60 m 90 m C LESSON 11.3 Indirect Measurement with Similar Triangles 583 9. APPLICATION A guy wire attached to a high tower needs to be replaced. The contractor does not know the height of the tower or the length of the wire. Find a method to measure the length of the wire indirectly. Guy wire Ground 10. Kristin has developed a new method for indirectly measuring the height of her classroom. Her method uses string and a ruler. She tacks a piece of string to the base of the wall and walks back from the wall holding the other end of the string to her eye with her right hand. She holds a 12-inch ruler parallel to the wall in her left hand and adjusts her distance to the wall until the bottom of the ruler is in line with the bottom edge of the wall and the top of the ruler is in line with the top edge of the wall. Now with two measurements, she is able to calculate the height of the room. Explain her method. If the distance from her eye to the bottom of the ruler is 23 inches and the distance from her eye to the bottom of the wall is 276 inches, calculate the height of the room. String Ruler Ceiling Floor Wall Review For Exercises 11–13, first identify similar triangles and explain why they are similar. Then find the missing lengths. 11. Find x. 12. Find y. 13. Find x, y, and h. M 9 15 U 10 S N x A B A 54 78 C D y 91 E F 20 h x H G 48 y 52 K 14. Construction Draw an obtuse triangle. a. Use a compass and straightedge to construct two altitudes. b. Use a ruler to measure both altitudes and their corresponding bases. c. Calculate the area using both altitude-base pairs. Compare your results. 15 |
. Find the radius of the circle. 5 3 584 CHAPTER 11 Similarity 16. Give the vertex arrangement of each tessellation. a. b. X B A A, where 17. Technology On a segment AB, point X is called the golden cut if X B A X X B and A A AX XB. The golden ratio is the value of when they are equal. Use X B A X geometry software to explore the location of the golden cut on any segment AB. What is the value of the golden ratio? Find a way to construct the golden cut. A X B AB___ AX AX___ = =? XB 18. Technology Imagine that a rod of a given length is attached at one end to a circular track and passes through a fixed pivot point. As one endpoint moves around the circular track, the other endpoint traces a curve. a. Predict what type of curve will be traced. b. Model this situation with geometry software. Describe the curve that is traced. c. Experiment with changing the size of the circular track, the length of the rod, or the location of the pivot point. Describe your results. Trace of the other endpoint Fixed point Circular track IMPROVING YOUR VISUAL THINKING SKILLS TIC-TAC-NO! Is it possible to shade six of the nine squares of a 3-by-3 grid so that no three of the shaded squares are in a straight line (row, column, or diagonal)? LESSON 11.3 Indirect Measurement with Similar Triangles 585 Corresponding Parts of Similar Triangles Is there more to similar triangles than just proportional sides and congruent angles? For example, are there relationships between corresponding altitudes, corresponding medians, or corresponding angle bisectors in similar triangles? Let’s investigate. Investigation 1 Corresponding Parts Use unlined paper for this investigation. Draw any triangle and construct a triangle of a different size similar to it. State the scale factor you used. Construct a pair of corresponding altitudes and use your compass to compare their lengths. How do they compare? How does the comparison relate to the scale factor you used? Construct a pair of corresponding medians. How do their lengths compare? Construct a pair of corresponding angle bisectors. How do their lengths compare? Compare your results with the results of others near you. You should be ready to make a conjecture. Proportional Parts Conjecture If two triangles are similar, then the corresponding?, |
?, and? are? to the corresponding sides. C-96 The discovery you made in the investigation probably seems very intuitive. Let’s see how you can prove one part of your conjecture. You will prove the other two parts in the exercises. L E S S O N 11.4 Big doesn’t necessarily mean better. Sunflowers aren’t better than violets. EDNA FERBER You will need ● a compass ● a straightedge Step 1 Step 2 Step 3 Step 4 Step 5 586 CHAPTER 11 Similarity EXAMPLE Solution Prove that corresponding medians of similar triangles are proportional to corresponding sides. E O L V M H A T Consider similar triangles LVE and MTH with corresponding medians EO and HA. You need to show that the corresponding medians are proportional to L. If you show that LOE MAH E EO corresponding sides, for example M H H A L. E EO then you can show that M H H A If you accept the SAS Similarity Conjecture as true, then you can show that LOE MAH. You already know that L M. Use algebra to show that LO L E. M A M H LV LO OV MA AT O L L LO E A M A M M H 2 LO L E 2 M A M H LO L E A M M H Corresponding sides of similar triangles LVE and MTH are proportional. LV LO OV and MT MA AT. Substitute. Since EO and HA are medians, O and A are midpoints. Since O and A are midpoints, OV LO and AT MA. Substitute. Add. Reduce. So, LOE MAH by the SAS Similarity Conjecture. Therefore you can L, which shows that the corresponding E EO also set up the proportion M H H A medians are proportional to corresponding sides. Recall when you first saw an angle bisector in a triangle. You may have thought that the bisector of an angle in a triangle divides the opposite side into two equal parts as well. A counterexample shows that this is not necessarily true. In ROE, RT bisects R, but point T does not bisect OE. R An angle bisector is not necessarily a median. O T OT TE E The angle bisector does, however, divide the opposite side in a particular way. LESSON 11.4 Corresponding Parts of Similar Triangles 587 |
(4, 1) x LESSON 11.4 Corresponding Parts of Similar Triangles 589 15. Aunt Florence has willed to her two nephews a plot of land in the shape of an isosceles right triangle. The land is to be divided into two unequal parts by bisecting one of the two congruent angles. What is the ratio of the greater area to the lesser area? 16. Construction How would you divide a segment into lengths with a ratio of 2? The Angle Bisector/Opposite Side 3 Conjecture gives you a way to do this. To get you started, here are the first four steps. Step 1 Construct any segment AB. A B Step 2 Construct a second segment. Call its length x. x Step 3 Construct two more segments with lengths 2x and 3x. 2x 3x Step 4 Construct a triangle with lengths 2x, 3x, and AB. A 2x B 3x You’re on your own from here! 17. Prove that corresponding angle bisectors of similar triangles are proportional to corresponding sides. 18. Prove that corresponding altitudes of similar triangles are proportional to corresponding sides. 19. Mini-Investigation This investigation is in two parts. You will need to complete the conjecture in part a before moving on to part b. a. The altitude to the hypotenuse has been constructed in each right triangle below. This construction creates two smaller right triangles within each original right triangle. Calculate the measures of the acute triangles in each diagram. i. a?, b?, c? ii. a?, b?, c? iii 50° c 30° c b a 68° c How do the smaller right triangles compare in each diagram? How do they compare to the original right triangle? You should be ready to state a conjecture. Conjecture: The altitude to the hypotenuse of a right triangle divides the triangle into two right triangles that are? to each other and to the original?. 590 CHAPTER 11 Similarity b. Complete each proportion for these right triangles. s i. h? r y? ii. h x? iii Review the proportions you wrote. How are they alike? You should be ready to state a conjecture. Conjecture: The altitude (length h) to the hypotenuse of a right triangle divides the hypotenuse into two segments (lengths p and q), such that p?.? q Add these conjectures to your notebook. |
Review b c c, then a d. 20. Use algebra to show that if a d b d b 21. A rectangle is divided into four rectangles, each similar to the original rectangle. What is the ratio of short side to long side in the rectangles? 22. In Chapter 5, you discovered that when you construct the three midsegments in a triangle, they divide the triangle into four congruent triangles. Are the four triangles similar to the original? Explain why. 23. Construction Draw any triangle ABC. Select any point X on AB. Construct a line through X parallel to AC that intersects BC in point Y. Find a proportion that relates AX, XB, BY, and YC. 24. A rectangle has sides a and b. For what values of a and b is another rectangle with sides 2a and b 2 a. Congruent to the original? c. Equal in area to the original? 25. Find the volume of this truncated cone. 20 m 10 m 24 m 36 m b. Equal in perimeter to the original? d. Similar but not congruent to the original? 26. The large circles are tangent to the square and tangent to each other. The smaller circle is tangent to each larger circle. Find the radius of the smaller circle in terms of s, the length of each side of the square. s s LESSON 11.4 Corresponding Parts of Similar Triangles 591 Proportions with Area and Volume You can use similarity to find the surface areas and volumes of objects that are geometrically similar. Suppose an artist wishes to gold-plate a sculpture. If it costs $250 to gold-plate a model that is half as long in each dimension, how much will it cost for the full-size sculpture? Not $500, but $1000! If the model weighs 40 pounds, how much will the full-size sculpture weigh? Not 80 pounds, but 320 pounds! In this lesson you will discover why these answers may not be what you expected. L E S S O N 11.5 It is easy to show that a hare could not be as large as a hippopotamus, or a whale as small as a herring. For every type of animal there is a most convenient size, and a large change in size inevitably carries with it a change of form. J. B. S. HALDANE You might recognize this giant golden man as the Academy Awards statuette. Also called the |
Oscar, smaller versions of the figure are handed out annually for excellence in the motion picture industry. If this statuette were real gold, it would be very expensive and incredibly heavy. You will need ● graph paper Investigation 1 Area Ratios In this investigation you will find the relationship between areas of similar figures. Draw a rectangle on graph paper. Calculate its area. Draw a rectangle similar to your first rectangle by multiplying its sides by a scale factor. Calculate this area. What is the ratio of side lengths (larger to smaller) for your two rectangles? What is the ratio of their areas (larger to smaller)? Step 1 Step 2 Step 3 592 CHAPTER 11 Similarity Step 4 How many copies of the smaller rectangle would you need to fill the larger rectangle? Draw lines in your larger rectangle to show how you would place the copies to fill the area. Step 5 Compare your results with the results of others near you. Step 6 Step 7 Repeat Steps 1–5 using triangles instead of rectangles. Discuss whether or not your findings would apply to any pair of similar polygons. Would they apply to similar circles or other curved figures? You should be ready to state a conjecture. Proportional Areas Conjecture C-98 If corresponding sides of two similar polygons or the radii of two circles compare in the ratio m, then their areas compare in the ratio?. n Similar solids are solids that have the same shape but not necessarily the same size. All cubes are similar, but not all prisms are similar. All spheres are similar, but not all cylinders are similar. Two polyhedrons are similar if all their corresponding faces are similar and the lengths of their corresponding edges are proportional. Two right cylinders (or right cones) are similar if their radii and heights are proportional. EXAMPLE A Are these right rectangular prisms similar? 2 3 7 2 6 14 Solution The two prisms are not similar because the corresponding edges are not proportional. 3 2 7 6 2 1 4 EXAMPLE B Are these right circular cones similar? 21 12 14 8 Solution The two cones are similar because the radii and heights are proportional. 1 4 8 2 1 1 2 LESSON 11.5 Proportions with Area and Volume 593 Investigation 2 Volume Ratios How does the ratio of lengths of corresponding edges of similar solids compare with the ratio of their volumes? Let’s find out. You will need ● interlocking cubes Fish Snake Step 1 Step 2 Step 3 Step |
4 Use blocks to build the “snake.” Calculate its volume. Build a similar snake by multiplying every dimension by a scale factor of 3. Calculate this volume. What is the ratio of side lengths (larger to smaller) for your two snakes? What is the ratio of volumes (larger to smaller)? As in Steps 1–3, use blocks to build the “fish” and another fish similar to it, this time by a scale factor of 2. Find the ratio of the side lengths and the ratio of the volumes. Step 5 How do your results compare with the results in Investigation 1? Discuss how you would calculate the volume of a snake increased by a scale factor of 5. Discuss how you would calculate the volume of a fish increased by a scale factor of 4. Step 6 You should be ready to state a conjecture. Proportional Volumes Conjecture C-99 If corresponding edges (or radii, or heights) of two similar solids compare in the ratio m, then their volumes compare in the ratio?. n 594 CHAPTER 11 Similarity EXERCISES 1. CAT MSE Area of CAT 72 cm2 Area of MSE? E 6 cm T M S 12 cm C 3. TRAP ZOID f o a e r A 1 6 ID. RECT ANGL 9 f o a e r A T EC TR? T R C E G N L 24 A 4. semicircle R semicircle S 3 r 5 s Area of semicircle S 75 cm2 Area of semicircle R? r R s S 5. The ratio of the lengths of corresponding diagonals of two similar kites is 1. What is 7 the ratio of their areas? 6. The ratio of the areas of two similar trapezoids is 1. What is the ratio of the lengths 9 of their altitudes? 7. The ratio of the lengths of the edges of two cubes is m. What is the ratio of their n surface areas? 8. The celestial sphere shown at right has radius 9 inches. The planet in the sphere’s center has radius 3 inches. What is the ratio of the volume of the planet to the volume of the celestial sphere? What is the ratio of the surface area of the planet to the surface area of the celestial sphere? 9. APPLICATION Annie works in a magazine’s advertising department. A client has requested that his 5 cm-by12 cm ad be enlarged: � |
�Double the length and double the width, then send me the bill.” The original ad cost $1500. How much should Annie charge for the larger ad? Explain your reasoning. LESSON 11.5 Proportions with Area and Volume 595 10. The pentagonal pyramids are similar. h 4 7 H Volume of large pyramid? Volume of small pyramid 320 cm3 h H 11. These right cones are similar. 12. These right trapezoidal prisms are similar. H?, h? Volume of large cone? Volume of small cone? Volume of large cone Volume of small cone? Volume of small prism 324 cm3 Area of base of small prism 9 Area of base of large prism 2 5 h? H Volume of large prism Volume of small prism Volume of large prism?? h H 20 cm 3 cm 12 cm h H 13. These right cylinders are similar. Volume of large cylinder 4608 ft3 Volume of small cylinder? Volume of large cylinder Volume of small cylinder H?? 9 24 H 14. The ratio of the lengths of corresponding edges of two similar triangular prisms is 5. What is the ratio of their volumes? 3 8. What is the ratio 15. The ratio of the volumes of two similar pentagonal prisms is 25 1 of their heights? 8. What is the ratio of their 16. The ratio of the weights of two spherical steel balls is 2 7 diameters? 17. APPLICATION The energy (and cost) needed to operate an air conditioner is proportional to the volume of the space that is being cooled. It costs ZAP Electronics about $125 per day to run an air conditioner in their small rectangular warehouse. The company’s large warehouse, a few blocks away, is 2.5 times as long, wide, and high as the small warehouse. Estimate the daily cost of cooling the large warehouse with the same model of air conditioner. 596 CHAPTER 11 Similarity 18. APPLICATION A sculptor creates a small bronze statue that weighs 38 lb. She plans to make a version that will be four times as large in each dimension. How much will this larger statue weigh if it is also bronze? This bronze sculpture by Camille Claudel (1864–1943) is titled La Petite Chatelaine. Claudel was a notable French artist and student of Auguste Rodin, whose famous sculptures include The Thinker. 19. A tabloid magazine at a supermarket checkout exclaims, “Scientists Breed 4- |
Foot Tall Chicken.” A photo shows a giant chicken that supposedly weighs 74 pounds and will solve the world’s hunger problem. What do you think about this headline? Assuming an average chicken stands 14 inches tall and weighs 7 pounds, would a 4-foot chicken weigh 74 pounds? Is it possible for a chicken to be 4 feet tall? Explain your reasoning. 20. The African goliath frog shown in this photo is the largest known frog—about 0.3 m long and 3.2 kg in weight. The Brazilian gold frog is one of the smallest known frogs—about 9.8 mm long. Approximate the weight of a gold frog. What assumptions do you need to make? Explain your reasoning. Review 21. Make four copies of the trapezoid at right. Arrange them into a similar but larger trapezoid. Sketch the final trapezoid and show how the smaller trapezoids fit inside it. 22. Sara rents her goat, Munchie, as a lawn mower. Munchie is tied to a stake with a 10 m rope. Sara wants to find an efficient pattern for Munchie’s stake positions so that all grass in a field 42 m by 42 m is mowed but overlap is minimized. Make a sketch showing all the stake positions needed. 60° 60° LESSON 11.5 Proportions with Area and Volume 597 23 24. XY? D 20 C E A X Y F 36 B 25. Find the area of a regular decagon with an apothem 5.7 cm and a perimeter 37 cm. 26. Find the area of a triangle whose sides measure 13 feet, 13 feet, and 10 feet. 27. True or false? Every cross section of a pyramid has the same shape as, but a different size from the base. 28. What’s wrong with this picture? 80 135 60 105 You can use Fathom to graph your data and find the line of best fit. Choose different types of graphs to get other insights into what your data mean. IN SEARCH OF THE PERFECT RECTANGLE A square is a perfectly symmetric quadrilateral. Yet, people rarely make books, posters, or magazines that are square. Instead, most people seem to prefer rectangles. In fact, some people believe that a particular type of rectangle is more appealing because its proportions fit the golden ratio. You can learn more about the historical importance of the golden ratio by doing research with the links at www.key |
math.com/DG. Do people have a tendency to choose a particular length/width ratio when they design or build common objects? Find at least ten different rectangular objects in your classroom or home: books, postcards, desks, doors, and other everyday items. Measure the longer side and the shorter side of each one. Predict what a graph of your data will look like. Now graph your data. Is there a pattern? Find the line of best fit. How well does it fit the data? What is the range of length/width ratios? What ratio do points on the line of best fit represent? Your project should include A table and graph of your data. Your predictions and your analysis. A paragraph explaining your opinion about whether or not people have a tendency to choose a particular type of rectangle and why. 598 CHAPTER 11 Similarity Why Elephants Have Big Ears The relationship between surface area and volume is of critical importance to all living things. It explains why elephants have big ears, why hippos and rhinos have short, thick legs and must spend a lot of time in water, and why movie monsters like King Kong and Godzilla can’t exist. Activity Convenient Sizes Body Temperature Every living thing processes food for energy. This energy creates heat that radiates from its surface. Imagine two similar animals, one with dimensions three times as large as those of the other. How would the surface areas of these two animals compare? How much more heat could the larger animal radiate through its surface? How would the volumes of these two animals compare? If the animals’ bodies produce energy in proportion to their volumes, how many times as much heat would the larger animal produce? Review your answers from Steps 2 and 3. How many times as much heat must each square centimeter of the larger animal radiate? Would this be good or bad? Use what you have concluded to answer these questions. Consider size, surface area, and volume. a. Why do large objects cool more slowly than similar small objects? b. Why is a beached whale more likely than a beached dolphin to experience overheating? c. Why are larger mammals found closer to the poles than the equator? d. If a woman and a small child fall into a cold lake, why is the child in greater danger of hypothermia? EXPLORATION Why Elephants Have Big Ears 599 Step 1 Step 2 Step 3 Step 4 Step 5 e. When the weather is cold, iguanas |
hardly move. When it warms up, they become active. If a small iguana and a large iguana are sunning themselves in the morning sun, which one will become active first? Why? Which iguana will remain active longer after sunset? Why? f. Why do elephants have big ears? Bone and Muscle Strength The strength of a bone or a muscle is proportional to its cross-sectional area. Imagine a 7-foot-tall basketball player and a 42-inch-tall child. How would the dimensions of the bones of the basketball player compare to the corresponding bones of the child? How would the cross-sectional areas of their corresponding bones compare? How would their weights compare? How many times as much weight would each cross-sectional square inch of bone have to support in the basketball player? What are some factors that may explain why basketball players’ bones don’t usually break? The Spanish artist Salvador Dalí (1904–1989) designed this cover for a 1948 program of the ballet As You Like It. What is the effect created by the elephants on long spindly legs? Could these animals really exist? Explain. Step 6 Step 7 Step 8 Step 9 600 CHAPTER 11 Similarity Step 10 Use what you have concluded to answer these questions. Consider size, surface area, and volume. a. Why are the largest living mammals, the whales, confined to the sea? b. Why do hippos and rhinos have short, thick legs? c. Why are champion weight lifters seldom able to lift more than twice their weight? d. Thoroughbred racehorses are fast runners but break their legs easily, while draft horses are slow moving and rarely break their legs. Why is this? e. Assume that a male gorilla can weigh as much as 450 pounds and can reach about 6 feet tall. King Kong is 30 feet tall. Could King Kong really exist? f. Professional basketball players are not typically similar in shape to professional football players. Discuss the advantages and disadvantages of each body type in each sport. Similarity is a theme in King Kong (1933), a movie about a giant gorilla similar in shape to a real gorilla. Willis O’Brien (1886–1962), an innovator of stop-motion animation, designed the models of Kong that brought the gorilla to life for moviegoers everywhere. Here, special effects master Ray Harryhausen holds a model that was used to create this captivating scene. Gravity and Air Resistance Objects in a vacuum fall at |
the same rate. However, an object falling through air is slowed by air resistance. Air resistance is proportional to the surface area of the falling object. Step 11 Imagine a rat that is 8 times the length, width, and height of a similar mouse. Both animals fall from a cliff. Step 12 How would the volumes of the two animals compare? EXPLORATION Why Elephants Have Big Ears 601 Step 13 How would the air resistance against the two animals compare? Step 14 The mass (related to weight) of an object is a factor in the force of the impact of the object with the ground. But air resistance on an object slows its fall, counteracting some of the force of impact. Assume that the weights of the rat and mouse are proportional to their volumes. Compare the force of the rat’s impact with the ground to the force of the mouse’s impact. Which is more likely to survive the fall? Step 15 Use what you have concluded to answer this question. Consider size, surface area, and volume. An ant can fall 100 times its height and live. This is not true for a human. Why? IMPROVING YOUR VISUAL THINKING SKILLS Painted Faces II Small cubes are assembled to form a larger cube, and then some of the faces of this larger cube are painted. After the paint dries, the larger cube is taken apart. Exactly 60 of the small cubes have no paint on any of their faces. What were the dimensions of the larger cube? How many of its faces were painted? 602 CHAPTER 11 Similarity L E S S O N 11.6 Mistakes are the portals of discovery. JAMES JOYCE Proportional Segments Between Parallel Lines In the figure below, MT LU. Is LUV similar to MTV? Yes, it is. A short paragraph proof can support this observation. Given: LUV with MT LU Show: LUV MTV V M 1 3 T 2 L 4 U Paragraph Proof First assume that the Corresponding Angles Conjecture and the AA Similarity Conjecture are true. If MT LU, then 1 2 and 3 4 by the Corresponding Angles Conjecture. If 1 2 and 3 4, then LUV MTV by the AA Similarity Conjecture. Let’s see how you can use this observation to solve problems. EXAMPLE A EO LN y? N 36 O 48 L y E 60 M Solution Use the fact that EMO |
LMN to write a proportion with the lengths of corresponding sides 36 60 48 y 0 6 4 60 7 y 240 4y 420 4y 180 y 45 Corresponding sides of similar triangles are proportional. Substitute lengths given in the figure. Reduce the left side of the equation. Multiply both sides by 7(60 y), reduce, and distribute. Subtract 240 from both sides. Divide by 4. E is the same as the L Look back at the figure in Example A. Notice that the ratio M E O. So there are more relationships in the figure than the ones we find in N ratio O M similar triangles. Let’s investigate. LESSON 11.6 Proportional Segments Between Parallel Lines 603 You will need ● a ruler ● a protractor Step 1 Investigation 1 Parallels and Proportionality In this investigation, we’ll look at the ratios of segments that have been cut by parallel lines. In each figure below, find x. Then find numerical values for the ratios. a. EC AB x? C? E?, D D BC AE 12 16 D C E 8 x b. KH FG x? JH JK??, G H F K c. QN LM x? Q P PN??, Q L N M A B J 24 x H 12 G K 8 F M 45 N 25 L x Q 20 P Step 2 What do you notice about the ratios of the lengths of the segments that have been cut by the parallel lines? Step 3 Step 4 Step 5 Step 6 Is the converse true? That is, if a line divides two sides of a triangle proportionally, is it parallel to the third side? Let’s see. Draw an acute angle, P. Beginning at point P, use your ruler to mark off lengths of 8 cm and 10 cm on one ray. Label the points A and B. Mark off lengths of 12 cm and 15 cm on the other ray. Label the points 1 2 8 C and D. Notice that. 1 5 1 0 Draw AC and BD. B 10 cm A 8 cm P 12 cm C 15 cm D 604 CHAPTER 11 Similarity Step 7 Step 8 With a protractor, measure PAC and PBD. Are AC and BD parallel? Repeat Steps 3–7, but this time use your ruler to create your own lengths such PC PA. that D C A B Step 9 Compare your results with the results of others near you. You should be |
ready to combine your observations from Steps 2 and 9 into one conjecture. Parallel/Proportionality Conjecture C-100 If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides?. Conversely, if a line cuts two sides of a triangle proportionally, then it is? to the third side. EXAMPLE B If you assume that the AA Similarity Conjecture is true, you can use algebra to prove the Parallel/Proportionality Conjecture. Here’s the first part. Given: ABC with XY BC Show: a b c d (Assume that the lengths a, b, c, and d are all nonzero.) A a b X c B Y d C Solution First, you know that AXY ABC (see the proof on page 603). Use a proportion of corresponding sides. b a b a d c Lengths of corresponding sides of similar triangles are proportional. d) Multiply both sides by (a c)(b d). d) b(a a(a )(b )(b c c (b c) (a d) a(b d) b(a c) ab ad ba bc ab ad ab bc ad bc c b d a d c cd Apply the distributive property. Subtract ab from both sides. Commute ba to ab. We want c and d in the denominator, so divide both sides by cd. Reduce. b a d c Reduce. You’ll prove the converse of the Parallel/Proportionality Conjecture in Exercise 18. Can the Parallel/Proportionality Conjecture help you divide segments into several proportional parts? Let’s investigate. LESSON 11.6 Proportional Segments Between Parallel Lines 605 Investigation 2 Extended Parallel/Proportionality Step 1 Use the Parallel/Proportionality Conjecture to find each missing length. Are the ratios equal? a. FT LA GR x?, y? TA L F? Is A R G L 21 F 35 42 T E x A b. ZE OP IA DR a?, b?, c? P? A IO A I R Is D? Is PE Z O A P IO L 28 G Z 14 O y R a I b D 21 10 P E 20 25 A R c T Step 2 Compare your results with the results of others near you. Complete the conjecture below. Extended Parallel/Proportionality Conjecture |
segment IJ. Construct a regular hexagon with IJ as the perimeter. 15. You can use a sheet of lined paper to divide a segment into equal parts. Draw a segment on a piece of patty paper, and divide it into five equal parts by placing it over lined paper. What conjecture explains why this works? 16. The drafting tool shown at right is called a sector compass. You position a given segment between the 100-marks. What points on the compass should you connect to construct a segment that is three-fourths (or 75%) of BC? Explain why this works 100 17. This truncated cone was formed by cutting off the top of a cone with a slice parallel to the base of the cone. What is the volume of the truncated cone? 10 cm 16 cm 12 cm 608 CHAPTER 11 Similarity 18. Assume that the Corresponding Angles Conjecture and the AA Similarity Conjecture are true. Write a proof to show that if a line cuts two sides of a triangle proportionally, then it is parallel to the third side. b Given: a d c Show: AB YZ (Assume c 0 and d 0.) 19. Another drafting tool used to construct segments is a pair of proportional dividers, shown at right. Two styluses of equal length are connected by a screw. The tool is adjusted for different proportions by moving the screw. Where should the screw be positioned so that AB is three-fourths of CD? The Extended Parallel/Proportionality Conjecture can be extended even further. That is, you don’t necessarily need a triangle. If three or more parallel lines intercept two other lines (transversals) in the same plane, they do so proportionally. For Exercises 20 and 21 use this extension. 20. Find x and y 25 yd z Lot 3 6 yd y Lot 2 8 yd River Road x Lot 1 9 yd 3 cm 7 cm y 8 cm 4 cm x 21. A real estate developer has parceled land between a river and River Road as shown. The land has been divided by segments perpendicular to the road. What is the “river frontage” (lengths x, y, and z) for each of the three lots? Review 9. What is 4 22. The ratio of the surface areas of two cubes is 8 1 the ratio of their volumes? 23. Romunda’s original recipe for her special “cannonball� |
� cookies makes 36 spheres with 4 cm diameters. She reasons that she can make 36 cannonballs with 8 cm diameters by doubling the amount of dough. Is she correct? If not, how many 8 cm diameter cannonballs can she make by doubling the recipe? 24. Find the surface area of a cube with edge x. Find the surface area of a cube with edge 2x. Find the surface area of a cube with edge 3x. LESSON 11.6 Proportional Segments Between Parallel Lines 609 25. A circle of radius r has a chord of length r. Find the length of the minor arc. 26. Technology In Lesson 11.3, Exercise 17, you learned about the golden cut and the golden ratio. A golden rectangle is a rectangle in which the ratio of the length to the width is the golden ratio. That is, a golden rectangle’s length, l, and width, w, satisfy the proportion w l A golden rectangle Some researchers believe Greek architects used golden rectangles to design the Parthenon. You can learn more about the historical importance of golden rectangles using the links at www.keymath.com/DG. l w w l l a. Use geometry software to construct a golden rectangle. Your construction for Exercise 17 in Lesson 11.3 will help. b. When a square is cut off one end of a golden rectangle, the remaining rectangle is a smaller, similar golden rectangle. If you continue this process over and over again, and then connect opposite vertices of the squares with quarter-circles, you create a curve called the golden spiral. Use geometry software to construct a golden spiral. The first three quarter-circles are shown below. D A F J C ABCD is a golden rectangle. EBCF is a golden rectangle. H I G HGCF is a golden rectangle. IJFH is a golden rectangle. The curve from D to E to G to J is the beginning of a golden spiral. E B 27. A circle is inscribed in a quadrilateral. Write a proof showing that the two sums of the opposite sides of the quadrilateral are equal. 28. Copy the figure at right onto your own paper. Divide it into four figures similar to the original figure. IMPROVING YOUR VISUAL THINKING SKILLS Connecting Cubes The two objects shown at right can be placed together to form each of the shapes below except one. Which one? A. B. |
C. D. 610 CHAPTER 11 Similarity Two More Forms of Valid Reasoning In the Chapter 10 Exploration Sherlock Holmes and Valid Forms of Reasoning, you learned about Modus Ponens and Modus Tollens. A third valid form of reasoning is called the Law of Syllogism. According to the Law of Syllogism (LS), if you accept “If P then Q” as true and if you accept “If Q then R” as true, then you must logically accept “If P then R” as true. Here is an example of the law of syllogism. English statement Symbolic translation If I eat pizza after midnight, then I will have nightmares. If I have nightmares, then I will get very little sleep. Therefore, if I eat pizza after midnight, then I will get very little sleep. P: I eat pizza after midnight. Q: I will have nightmares R: I will get very little sleep To work on the next law, you need some new statement forms. Every conditional statement has three other conditionals associated with it. To get the converse of a statement, you switch the “if ” and “then” parts. To get the inverse, you negate both parts. To get the contrapositive, you reverse and negate the two parts. These new forms may be true or false. Statement Converse Inverse If two angles are vertical angles, then they are congruent. If two angles are congruent, then they are vertical angles. If two angles are not vertical angles, then they are not congruent. Contrapositive If two angles are not congruent, then they are not vertical angles. P → Q Q → P true false P → Q false Q → P true EXPLORATION Two More Forms of Valid Reasoning 611 Notice that the original conditional statement and its contrapositive have the same truth value. This leads to a fourth form of valid reasoning. The Law of Contrapositive (LC) says that if a conditional statement is true, then its contrapositive is also true. Conversely, if the contrapositive is true, then the original conditional statement must also be true. This also means that if a conditional statement is false, so is its contrapositive. Often, a logical argument contains multiple steps, applying the same rule more than once, or applying more than one rule. Here is an example. English statement Symbolic translation If the consecutive sides of a parallelogram P: |
The consecutive sides of a are congruent, then it is a rhombus. If a parallelogram is a rhombus, then its diagonals are perpendicular bisectors of each other. The diagonals are not perpendicular bisectors of each other. Therefore the consecutive sides of the parallelogram are not congruent. parallelogram are congruent. Q: The parallelogram is a rhombus. R: The diagonals are perpendicular bisectors of each other. P → Q Q → R R P You can show that this argument is valid in three logical steps. Step 1 Step 2 Step Literature by the Law of Syllogism by the Law of Contrapositive by Modus Ponens Lewis Carroll was the pseudonym of the English novelist and mathematician Charles Lutwidge Dodgson (1832–1898). He is often associated with his famous children’s book Alice’s Adventures in Wonderland. In 1886 he published The Game of Logic, which used a game board and counters to solve logic problems. In 1896 he published Symbolic Logic, Part I, which was an elementary book intended to teach symbolic logic. Here is one of the silly problems from Symbolic Logic. What conclusion follows from these premises? Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Lewis Carroll enjoyed incorporating mathematics and logic into all of his books. Here is a quote from Through the Looking Glass. Is Tweedledee using valid reasoning? “Contrariwise,” said Tweedledee, “if it was so, it might be; and if it were so, it would be, but as it isn’t, it ain’t. That’s logic.” 612 CHAPTER 11 Similarity So far, you have learned four basic forms of valid reasoning. Now let’s apply them in symbolic proofs. Four Forms of Valid Reasoning P → Q P Q by MP P → Q Q P by MT P → Q Q → R P → R by LS P → Q Q → P by LC Activity Symbolic Proofs Determine whether or not each logical argument is valid. If it is valid, state what reasoning form or forms it follows. If it is not valid, write “no valid conclusion.” a. P → Q c. Q → R b. P → R f Step 1 Step 2 Q P d |
Translate parts a–c into symbols, and give the reasoning form(s) or state that the conclusion is not valid. a. If I study all night, then I will miss my late-night talk show. If Jeannine comes over to study, then I study all night. Jeannine comes over to study. Therefore I will miss my late-night talk show. b. If I don’t earn money, then I can’t buy a computer. If I don’t get a job, then I don’t earn money. I have a job. Therefore I can buy a computer. c. If EF is not parallel to side AB in trapezoid ABCD, then EF is not a midsegment of trapezoid ABCD. If EF is parallel to side AB, then ABFE is a trapezoid. EF is a midsegment of trapezoid ABCD. Therefore ABFE is a trapezoid. Step 3 Show how you can use Modus Ponens and the Law of Contrapositive to make the same logical conclusions as Modus Tollens. EXPLORATION Two More Forms of Valid Reasoning 613 ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAP CHAPTER 11 R E V I E W Similarity, like area, volume, and the Pythagorean Theorem, has many applications. Any scale drawing or model, anything that is reduced or enlarged, is governed by the properties of similar figures. So engineers, visual artists, and film-makers all use similarity. It is also useful in indirect measurement. Do you recall the two indirect measurement methods you learned in this chapter? The ratios of area and volume in similar figures are also related to the ratios of their dimensions. But recall that as the dimensions increase, the area increases by a squared factor and volume increases by a cubed factor. EXERCISES For Exercises 1–4, solve each proportion. 8 x 1. 5 1 5 4 2 4 2. 1 1 x 3. 4 x 9 x In Exercises 5 and 6, measurements are in centimeters. 5. ABCDE FGHIJ. ABC DBA x?, y? You will need Construction tools for Exercise 9 3 4 x 4. APPLICATION David is 5 ft 8 in. tall and wants to find the height of an oak tree in his front yard. He walks along the shadow of |
the tree until his head is in a position where the end of his shadow exactly overlaps the end of the tree’s shadow. He is now 11 ft 3 in. from the foot of the tree and 8 ft 6 in. from the end of the shadows. How tall is the oak tree? 5 ft 8 in. 8 ft 6 in. 11 ft 3 in. 614 CHAPTER 11 Similarity ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAP 8. A certain magnifying glass when held 6 in. from an object creates an image that is 10 times the size of the object being viewed. What is the measure of a 20° angle under this magnifying glass? 9. Construction Construct KL. Then find a point P that divides KL into two segments that have a ratio 3. 4 10. Patsy does a juggling act. She sits on a stool that sits on top of a rotating ball that spins at the top of a 20-meter pole. The diameter of the ball is 4 meters, and Patsy’s eye is approximately 2 meters above the ball. Seats for the show are arranged on the floor in a circle so that each spectator can see Patsy’s eyes. Find the radius of the circle of seats to the nearest meter. 11. Charlie builds a rectangular box home for his pet python and uses 1 gallon of paint to cover its surface. Lucy also builds a box for Charlie’s pet, but with dimensions twice as great. How many gallons of paint will Lucy need to paint her box? How many times as much volume does her box have? 12. Suppose you had a real clothespin similar to the sculpture at right and made of the same material. What measurements would you make to calculate the weight of the sculpture? Explain your reasoning. 13. The ratio of the perimeters of two similar parallelograms is 3. What is the ratio of their areas? 7 5 14. The ratio of the areas of two circles is 2. What is the 1 6 ratio of their radii? 15. APPLICATION The Jones family paid $150 to a painting contractor to stain their 12-by-15-foot deck. The Smiths have a similar deck that measures 16 ft by 20 ft. What price should the Smith family expect to pay to have their deck stained? This sculpture, called Clothespin (1976), was created by Swedish-American sculptor Claes Oldenburg |
(b 1929). His art reflects how everyday objects can be intriguing. CHAPTER 11 REVIEW 615 EW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CH 16. The dimensions of the smaller cylinder are two-thirds of the dimensions of the larger cylinder. The volume of the larger cylinder is 2160 cm3. Find the volume of the smaller cylinder. 17? 36 48 42 24 18 30 z w x y 18. Below is a 58-foot statue of Bahubali, in Sravanabelagola, India. Every 12 years, worshipers of the Jain religion bathe the statue with coconut milk. Suppose the milk of one coconut is just enough to cover the surface of the similar 2-foot statuette shown at right. How many coconuts would be required to cover the surface of the full-size statue? This 58-foot statue is carved from a single stone. 19. Greek mathematician Archimedes liked the design at right so much that he wanted it on his tombstone. a. Calculate the ratio of the area of the square, the area of the circle, and the area of the isosceles triangle. Copy and complete this statement of proportionality. Area of square to Area of circle to Area of triangle is? to? to?. 616 CHAPTER 11 Similarity ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAP b. When each of the figures is revolved about the vertical line of symmetry, it generates a solid of revolution—a cylinder, a sphere, and a cone. Calculate their volumes. Copy and complete this statement of proportionality. Volume of cylinder to Volume of sphere to Volume of cone is? to? to?. c. What is so special about this design? 20. Many fanciful stories are about people who accidentally shrink to a fraction of their original height. If a person shrank to one-twentieth his original height, how would that change the amount of food he’d require, or the amount of material needed to clothe him, or the time he’d need to get to different places? Explain. This scene is from the 1957 science fiction movie The Incredible Shrinking Man. 21. Would 15 pounds of 1-inch ice cubes melt faster than a 15-pound block of ice? Explain. TAKE ANOTHER LOOK |
1. You’ve learned that an ordered pair rule such as (x, y) → (x b, y c) is a translation. You discovered in this chapter that an ordered pair rule such as (x, y) → (kx, ky) is a dilation in the coordinate plane, centered at the origin. What transformation is described by the rule (x, y) → (kx b, ky c)? Investigate. 2. In Lesson 11.1, you dilated figures in the coordinate plane, using the origin as the center of dilation. What happens if a different point in the plane is the center of dilation? Copy the polygon at right onto graph paper. Draw the polygon’s image under a dilation with a scale factor of 2 and with point A as the center of dilation. Draw another image using a scale factor of 2. Explain 3 how you found the image points. How does dilating about point A differ from dilating about the origin? 3. True or false? The angle bisector of one of the nonvertex angles of a kite will divide the diagonal connecting the vertex angles into two segments whose lengths are in the same ratio as two unequal sides of the kite. If true, explain why. If false, show a counterexample that proves it false. y (3, 9) A (–6, 6) (–3, 0) (6, 0) x CHAPTER 11 REVIEW 617 EW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CHAPTER 11 REVIEW ● CH 4. A total eclipse of the Sun can occur because the ratio of the Moon’s diameter to its distance from Earth is about the same as the ratio of the Sun’s diameter to its distance to Earth. Draw a diagram and use similar triangles to explain why it works. 5. It is possible for the three angles and two of the sides of one triangle to be congruent to the three angles and two of the sides of another triangle, and yet the two triangles won’t be congruent. Two such triangles are shown below. Use geometry software or patty paper to find another pair of similar (but not congruent) triangles in which five parts of one are congruent to five parts of the other. 16 20 20 25 25 311_ 4 A solar eclipse Explain why these sets of side lengths work |
. Use algebra to explain your reasoning. 6. Is the converse of the Extended Parallel Proportionality Conjecture true? That is, if two lines intersect two sides of a triangle, dividing the two sides proportionally, must the two lines be parallel to the third side? Prove that it is true or find a counterexample showing that it is not true. 7. If the three sides of one triangle are each parallel to one of the three sides of another triangle, what might be true about the two triangles? Use geometry software to investigate. Make a conjecture and explain why you think your conjecture is true. Assessing What You’ve Learned UPDATE YOUR PORTFOLIO If you did the Project Making a Mural, add your mural to your portfolio. ORGANIZE YOUR NOTEBOOK Review your notebook to be sure it’s complete and well organized. Be sure you have each definition and the conjecture. Write a onepage summary of Chapter 11. GIVE A PRESENTATION Give a presentation about one or more of the similarity conjectures. You could even explain how an overhead projector produces similar figures! 618 CHAPTER 11 Similarity CHAPTER 12 Trigonometry I wish I’d learn to draw a little better! What exertion and determination it takes to try and do it well.... It is really just a question of carrying on doggedly, with continuous and, if possible, pitiless self-criticism. M. C. ESCHER Belvedere, M. C. Escher, 1958 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● learn about the branch of mathematics called trigonometry ● define three important ratios between the sides of a right triangle ● use trigonometry to solve problems involving right triangles ● discover how trigonometry extends beyond right triangles L E S S O N 12.1 Research is what I am doing when I don’t know what I’m doing. WERNHER VON BRAUN Trigonometric Ratios Trigonometry is the study of the relationships between the sides and the angles of triangles. In this lesson you will discover some of these relationships for right triangles. Science Trigonometry has origins in astronomy. The Greek astronomer Claudius Ptolemy (100–170 C.E.) used tables of chord ratios in his book known as Almagest. These chord ratios and their related angles were used to describe the |
motion of planets in what were thought to be circular orbits. This woodcut shows Ptolemy using astronomy tools. When studying right triangles, early mathematicians discovered that whenever the ratio of the shorter leg’s length to the longer leg’s length was close to a specific fraction, the angle opposite the shorter leg was close to a specific measure. They found this (and its converse) to be true for all similar right triangles. For example, in every right triangle in which the ratio of the shorter leg’s length to the longer leg’s length is 3, the angle opposite the shorter leg is approximately 31°. 5 3 31° 5 31° 15 9 6 31° 10 31° 55 x What is a good approximation for x? What early mathematicians discovered is supported by what you know about similar triangles. If two right triangles each have an acute angle of the same measure, then the triangles are similar by the AA Similarity Conjecture. And if the triangles are similar, then corresponding sides are proportional. For example, in the similar right triangles shown below, these proportions are true: I K H EF BC L H G E D B A JK C 20° G I H 20° 20° A D F E B J 20° This leg is called the adjacent side because it is next to the 20° angle. L K This leg is called the opposite side because it is across from the 20° angle. The ratio of the length of the opposite side to the length of the adjacent side in a right triangle came to be called the tangent of the angle. 620 CHAPTER 12 Trigonometry In Chapter 11, you used mirrors and shadows to measure heights indirectly. Trigonometry gives you another indirect measuring method. EXAMPLE A At a distance of 36 meters from a tree, the angle from the ground to the top of the tree is 31°. Find the height of the tree. T H 36 m 31° A Solution As you saw in the right triangles on page 620, the ratio of the length of the side opposite a 31° angle divided by the length of the side adjacent to a 31° angle is approximately 3, or 0.6. You can set up a proportion using this tangent ratio. 5 T tan 31° H HA T 0.6 H HA T 0.6 H 36 HT (36)(0.6) HT 22 The definition of tangent. The tangent of 31° is approximately 0.6. Substitute 36 |
for HA. Multiply both sides by 36 and reduce the left side. Multiply. The height of the tree is approximately 22 meters. In order to solve problems like Example A, early mathematicians made tables that related ratios of side lengths to angle measures. They named six possible ratios. You will work with these three: sine, cosine, and tangent, abbreviated sin, cos, and tan. Sine is the ratio of the length of the opposite side to the length of the hypotenuse. Cosine is the ratio of the length of the adjacent side to the length of the hypotenuse. This excerpt from a trigonometric table shows sine, cosine, and tangent ratios for angles measuring from 12.0° to 14.2°. LESSON 12.1 Trigonometric Ratios 621 Trigonometric Ratios (Hypotenuse) c B a (Leg opposite A) A b (Leg adjacent to A) C For an acute angle A in any right triangle ABC: sine of A cosine of A tangent of A length of leg opposite A length of hypotenuse length of leg adjacent to A length of hypotenuse length of leg opposite A length of leg adjacent to A or or or sin A a c cos A b c tan A a b You will need ● a protractor ● a ruler Investigation Trigonometric Tables Step 1 Step 2 Step 3 Step 4 Step 5 In this investigation you will make a small table of trigonometric ratios for angles measuring 20° and 70°. Use your protractor to make a large right triangle ABC with mA 20°, mB 90°, and mC 70°. Measure AB, AC, and BC to the nearest millimeter. Use your side lengths and the definitions of sine, cosine, and tangent to complete a table like this. Round your calculations to the nearest thousandth. mA sin A cos A tan A mC sin C cos C tan C 20° 70° Share your results with your group. Calculate the average of each ratio within your group. Create a new table with your group’s average values. Discuss your results. What observations can you make about the trigonometric ratios you found? What is the relationship between the values for 20° and the values for 70°? Explain why you think these relationships exist. Go to www.keymath.com/DG to find complete tables of trigonometric ratios. 622 |
CHAPTER 12 Trigonometry Step 6 Step 7 Today, trigonometric tables have been replaced by calculators that have sin, cos, and tan keys. Experiment with your calculator to determine how to find the sine, cosine, and tangent values of angles. Use your calculator to find sin 20°, cos 20°, tan 20°, sin 70°, cos 70°, and tan 70°. Check your group’s table. How do the trigonometric ratios found by measuring sides compare with the trigonometric ratios you found on the calculator? Using a table of trigonometric ratios, or using a calculator, you can find the approximate lengths of the sides of a right triangle given the measures of any acute angle and any side. Find the length of the hypotenuse of a right triangle if an acute angle measures 20° and the side opposite the angle measures 410 feet. EXAMPLE B Solution x 410 ft 20° Sketch a diagram. The trigonometric ratio that relates the lengths of the opposite side and the hypotenuse is the sine ratio. 0 sin 20° 41 x x(sin 20°) 410 1 0 4 x 0° 2 n si Substitute 20° for the measure of A and substitute 410 for the length of the opposite side. The length of the hypotenuse is unknown, so use x. Multiply both sides by x and reduce the right side. Divide both sides by sin 20° and reduce the left side. From your table in the investigation, or from a calculator, you know that sin 20° is approximately 0.342. 0 1 4 x 4 2.3 0 x 1199 Sin 20° is approximately 0.342. Divide. The length of the hypotenuse is approximately 1199 feet. LESSON 12.1 Trigonometric Ratios 623 With the help of a calculator, it is also possible to determine the size of either acute angle in a right triangle if you know the length of any two sides of that triangle. For instance, if you know the ratio of the legs in a right triangle, you can find the measure of one acute angle by using the inverse tangent, or tan1, function. Let’s look at an example. The inverse tangent of x is defined as the measure of the acute angle whose tangent is x. The tangent function and inverse tangent function undo each other. That is, tan1(tan A) A and tan(tan1 |
x) x. EXAMPLE C A right triangle has legs of length 8 inches and 15 inches. Find the measure of the angle opposite the 8-inch leg. Solution C 8 in. B A 15 in. Sketch a diagram. In this sketch the angle opposite the 8-inch side is A. The trigonometric ratio that relates the lengths of the opposite side and the adjacent side is the tangent ratio. 8 tan A 1 5 Substitute 8 for the length of the opposite side and substitute 15 for the length of the adjacent side. 8, you can use a To find the angle that has an approximate tangent value of 1 5., or tan1 8 8 calculator to find the inverse tangent of 1 5 1 5 A tan1 8 1 5 A 28 Take the inverse tangent of both sides.. Use your calculator to evaluate tan1 8 1 5 The measure of the angle opposite the 8-inch side is approximately 28°. You can also use inverse sine, or sin1, and inverse cosine, or cos1, to find angle measures. EXERCISES For Exercises 1–3, use a calculator to find each trigonometric ratio accurate to four decimal places. You will need A calculator for Exercises 1–6 and 10–22 1. sin 37° 2. cos 29° 3. tan 8° For Exercises 4–6, solve for x. Express each answer accurate to two decimal places. x 4. sin 40° 8 1 9 5. cos 52° 1 x x 6. tan 29° 2 11 624 CHAPTER 12 Trigonometry For Exercises 7–9, find each trigonometric ratio. 7. sin A? cos A? tan A? R s T t r A 8. sin? cos? tan? 9. sin A? cos A? tan A? sin B? cos B? tan B? y (0, 10) (6, 8) x (10, 0) A 24 B 7 C For Exercises 10–13, find the measure of each angle accurate to the nearest degree. 10. sin A 0.5 12. tan C 0.5773 11. cos B 0.6 4 8 13. tan x 6 0 1 For Exercises 14–20, find the values of a–g accurate to the nearest whole unit. 14. 17. 20. 20 cm 30° a 107 ft 128 ft d g 21 in. 25° 15. |
18. 17 cm 65° b 48 cm 15° e c 70° 36 yd 16. 19. 36 in. 21. Find the perimeter of this quadrilateral. 22. Find x. 66 in. f 85 m 35° 280 ft 55° x 75° LESSON 12.1 Trigonometric Ratios 625 Review For Exercises 23 and 24, solve for x. 7 x 1 23. 3 8 2 5 24. 5 1 1 x 25. APPLICATION Which is the better buy? A pizza with a 16-inch diameter for $12.50, or a pizza with a 20-inch diameter for $20.00? 26. APPLICATION Which is the better buy? Ice cream in a cylindrical container with a base diameter of 6 inches and a height of 8 inches for $3.98, or ice cream in a box (square prism) with a base edge of 6 inches and a height of 8 inches for $4.98? 27. A diameter of a circle is cut at right angles by a chord into a 12 cm segment and a 4 cm segment. How long is the chord? 28. Find the volume and surface area of this sphere. 6 ft IMPROVING YOUR VISUAL THINKING SKILLS 3-by-3 Inductive Reasoning Puzzle II Sketch the figure missing in the lower right corner of this pattern. 626 CHAPTER 12 Trigonometry L E S S O N 12.2 What science can there be more noble, more excellent, more useful... than mathematics? BENJAMIN FRANKLIN Problem Solving with Right Triangles Right triangle trigonometry is often used indirectly to find the height of a tall object. To solve a problem of this type, measure the angle from the horizontal to your line of sight when you look at the top or bottom of the object. Horizontal Angle of depression B A Angle of elevation Horizontal If you look up, you measure the angle of elevation. If you look down, you measure the angle of depression. Here’s an example. EXAMPLE The angle of elevation from a sailboat to the top of a 121-foot lighthouse on the shore measures 16°. To the nearest foot, how far is the sailboat from shore? 121 ft 16° d Solution The height of the lighthouse is opposite the 16° angle. The unknown distance is the adjacent side. Set up a tangent ratio. tan 16° 1 21 d d(tan 16°) |
121 1 2 1 d ta 1 6° n d 422 The sailboat is approximately 422 feet from shore. LESSON 12.2 Problem Solving with Right Triangles 627 EXERCISES 1. According to a Chinese legend from the Han dynasty (206 B.C.E.–220 C.E.), General Han Xin flew a kite over the palace of his enemy to determine the distance between his troops and the palace. If the general let out 800 meters of string and the kite was flying at a 35° angle of elevation, how far away was the palace from General Han Xin’s position? You will need A calculator for Exercises 1–19 2. Benny is flying a kite directly over his friend, Frank, who is 125 meters away. When he holds the kite string down to the ground, the string makes a 39° angle with the level ground. How high is Benny’s kite? 3. APPLICATION The angle of elevation from a ship to the top of a 42-meter lighthouse on the shore measures 33°. How far is the ship from the shore? (Assume the horizontal line of sight meets the bottom of the lighthouse.) 4. APPLICATION A salvage ship’s sonar locates wreckage at a 12° angle of depression. A diver is lowered 40 meters to the ocean floor. How far does the diver need to walk along the ocean floor to the wreckage? 5. APPLICATION A meteorologist shines a spotlight vertically onto the bottom of a cloud formation. He then places an angle-measuring device 65 meters from the spotlight and measures a 74° angle of elevation from the ground to the spot of light on the clouds. How high are the clouds? 6. APPLICATION Meteorologist Wendy Stevens uses a 42 m 33° y theodolite (an angle-measuring device) on a 1-metertall tripod to find the height of a weather balloon. She views the balloon at a 44° angle of elevation. A radio signal from the balloon tells her that it is 1400 meters from her theodolite. a. How high is the balloon? b. How far is she from the point directly below the balloon? c. If Wendy’s theodolite were on the ground rather than on a tripod, would your answers change? Explain your reasoning. The distance from the ground to a cloud formation is called the cloud ceiling. Science Weather balloons carry into the atmosphere what is called a radiosonde, |
an instrument with sensors that detect information about wind direction, temperature, air pressure, and humidity. Twice a day across the world, this upper-air data is transmitted by radio waves to a receiving station. Meteorologists use the information to forecast the weather. 628 CHAPTER 12 Trigonometry 7. APPLICATION A ship’s officer sees a lighthouse at a 42° angle to the path of the ship. After the ship travels 1800 m, the lighthouse is at a 90° angle to the ship’s path. What is the distance between the ship and the lighthouse at this second sighting? When there are no visible landmarks, sailors at sea depend on the location of stars or the Sun for navigation. For example, in the Northern Hemisphere, Polaris (the North Star), stays approximately at the same angle above the horizon for a given latitude. If Polaris appears higher overhead or closer to the horizon, sailors can tell whether their course is taking them north or south. This painting by Winslow Homer (1836–1910) is titled Breezing Up (1876). For Exercises 8–16, find each length or angle measure accurate to the nearest whole unit. 8. a? 9. x? 10. r? 17 cm a 32° 18 m 20 m x 12 cm r 32° 11. e? 12. d1? 13. f? 2.7 m e 62° d1 14 in. 20 in. 56° 16 cm f 28 cm 14.? 15.? 16. h? 16 m 12 m 10 ft 8 ft 15 ft h 58° 40 cm LESSON 12.2 Problem Solving with Right Triangles 629 Review For Exercises 17–19, find the measure of each angle to the nearest degree. 17. sin D 0.7071 18. tan E 1.7321 19. cos F 0.5 Technology The earliest known navigation tool was used by the Polynesians, yet it didn’t measure angles. Early Polynesians carried several different-length hooks made from split bamboo and shells. A navigator held a hook at arm’s length, positioned the bottom of the hook on the horizon, and sighted the North Star through the top of the hook. The length of the hook indicated the navigator’s approximate latitude. Can you use trigonometry to explain how this method works? 20. Solve for x. x a. 4.7.2 3.4 b. 8 16 x |
x c. 0.3736 1 4.5 d. 0.9455 2 x 21. Find x and y. y 8 4 x 7 8 22. A 3-by-5-by-6 cm block of wood is dropped into a cylindrical container of water with radius 5 cm. The level of the water rises 0.8 cm. Does the block sink or float? Explain how you know. 23. Scalene triangle ABC has altitudes AX, BY, and CZ. If AB BC AC, write an inequality that relates the heights. 24. In the diagram at right, PT and PS are tangent to circle O at points T and S, respectively. As point P moves to the right along AB, describe what happens to each of these measures or ratios. a. mTPS c. mATB P e. A BP b. OD d. Area of ATB D f. A BD T A O D B P S 630 CHAPTER 12 Trigonometry 25. Points S and Q, shown at right, are consecutive vertices of square SQRE. Find coordinates for the other two vertices, R and E. There are two possible answers. Try to find both. y S (1, 4) Q (6, 2) x LIGHT FOR ALL SEASONS You have seen that roof design is a practical application of slope—steep roofs shed snow and rain. But have you thought about the overhang of a roof? In a hot climate, a deep overhang shelters windows from the sun. In a cold climate, a narrow overhang lets in more light and warmth. What roof design is common for homes in your area? What factors would an architect consider in the design of a roof relative to the position, size, and orientation of the windows? Do some research and build a shoebox model of the roof design you select. What design is best for your area will depend on your latitude, because that determines the angle of the sun’s light in different seasons. Research the astronomy of solar angles, then use trigonometry and a movable light source to illustrate the effects on your model. Your project should include Research notes on seasonal solar angles. A narrative explanation, with mathematical support, for your choice of roof design, roof overhang, and window placement. Detailed, labeled drawings showing the range of light admitted from season to season, at a given time of day. A model with a movable light source. LESSON |
12.2 Problem Solving with Right Triangles 631 Indirect Measurement In Chapter 11, you used shadows, mirrors, and similar triangles to measure the height of tall objects that you couldn’t measure directly. Right triangle trigonometry gives you yet another method of indirect measurement. In this exploration, you will use two or three different methods of indirect measurement. Then you will compare your results from each method. Activity Using a Clinometer In this activity, you will use a clinometer—a protractor-like tool used to measure angles. You probably will want to make your clinometer in advance, based on one of the designs below. Practice using it before starting the activity. You will need ● a measuring tape or metersticks ● a clinometer (use the Making a Clinometer worksheet or make one of your own design) ● a mirror Clinometer 1 Clinometer 2 Step 1 Locate a tall object that would be difficult to measure directly. Start a table like this one. Name of object Viewing angle Height of observer’s eye Distance from observer to object Calculated height of object 632 CHAPTER 12 Trigonometry Step 2 Step 3 Use your clinometer to measure the viewing angle from the horizontal to the top of the object. Measure the observer’s eye height. Measure the distance from the observer to the base of the object. Step 4 Calculate the approximate height of the object. U.S. Forest Service Ranger Al Sousi uses a clinometer to measure the angle of a mountain slope. In snowy conditions, a slope steeper than 35° can be a high avalanche hazard. Step 5 Use either the shadow method or the mirror method or both to measure the height of the same object. How do your results compare? If you got different results, explain what part of each process could contribute to the differences. Step 6 Repeat Steps 1–5 for another tall object. If you measure the height of the same object as another group, compare your results when you finish. IMPROVING YOUR VISUAL THINKING SKILLS Puzzle Shapes Make five of these shapes and assemble them to form a square. Does it take three, four, or five of the shapes to make a square? EXPLORATION Indirect Measurement 633 L E S S O N 12.3 To think and to be fully alive are the same. HANNAH ARENDT The Law of Sines So far you have used trigonometry only to solve problems with right triangles |
. But you can use trigonometry with any triangle. For example, if you know the measures of two angles and one side of a triangle, you can find the other two sides with a trigonometric property called the Law of Sines. The Law of Sines is related to the area of a triangle. Let’s first see how trigonometry can help you find area. C 100 m 40° B A D 150 m EXAMPLE A Find the area of ABC. Solution Consider AB as the base and use trigonometry to find the height, CD. D C sin 40° 0 0 1 (100)(sin 40°) CD Now find the area. A 0.5bh A (0.5)(AB)(CD) A (0.5)(150)[(100)(sin 40°)] In BCD, CD is the length of the opposite side and 100 is the length of the hypotenuse. Multiply both sides by 100 and reduce the right side. Area formula for a triangle. Substitute AB for the length of the base and CD for the height. Substitute 150 for AB and substitute the expression (100)(sin 40°) for CD. A 4821 Evaluate. The area is approximately 4821 m2. In the next investigation, you will find a general formula for the area of a triangle given the lengths of two sides and the measure of the included angle. Investigation 1 Area of a Triangle Step 1 Find the area of each triangle. Use Example A as a guide. a. b. c. h 29 cm 21 cm 72° h 55° 33 cm 38.45 cm 43 cm 50° h 52° 33.7 634 CHAPTER 12 Trigonometry Step 2 Generalize Step 1 to find the area of this triangle in terms of a, b, and C. State your general formula as your next conjecture. b h a C SAS Triangle Area Conjecture The area of a triangle is given by the formula A?, where a and b are the lengths of two sides and C is the angle between them. C-102 Now use what you’ve learned about finding the area of a triangle to derive the property called the Law of Sines. Investigation 2 The Law of Sines Consider ABC with height h. Find h in terms of a and the sine of an angle. Find h in terms of b and the sine of an angle. Use algebra to show A sin sin B b a C h b A Now |
consider the same ABC using a different height, k. Find k in terms of c and the sine of an angle. Find k in terms of b and the sine of an angle. Use algebra to show B sin sin C c b C b k A Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Combine Steps 3 and 6. Complete this conjecture. a a c c Law of Sines For a triangle with angles A, B, and C and sides of lengths a, b, and c (a opposite A, b opposite B, and c opposite C),?? sin A?? b B B C-103 LESSON 12.3 The Law of Sines 635 Did you notice that you used deductive reasoning rather than inductive reasoning to discover the Law of Sines? You can use the Law of Sines to find the lengths of a triangle’s sides when you know one side’s length and two angles’ measures. EXAMPLE B Find the length of side AC in ABC. C b 350 cm Solution Start with the Law of Sines, and solve for b. A sin sin B b a 59° A 38° B The Law of Sines. Multiply both sides by ab and reduce. b sin A a sin B in s B b a n si A 38°) Substitute 350 for a, 38° for B, and 59° for A. (s in ) b (350 9° 5 in s b 251 Multiply and divide. Divide both sides by sin A and reduce the left side. The length of side AC is approximately 251 cm. You can also use the Law of Sines to find the measure of a missing angle, but only if you know whether the angle is acute or obtuse. Recall from Chapter 4 that SSA failed as a congruence shortcut. For example, if you know in ABC that BC 160 cm, AC 260 cm, and mA 36°, you would not be able to find mB. There are two possible measures for B, one acute and one obtuse. C 160 B1 260 36° A C 160 260 B 2 36° A Because you’ve defined trigonometric ratios only for acute angles, you’ll be asked to find only acute angle measures. EXAMPLE C Find the measure of acute angle B in ABC. C 150 cm 69° A 250 cm B 636 CHAPTER 12 Trigonometry Solution Start |
with the Law of Sines, and solve for B. The Law of Sines. A sin sin B b a in A sin B b s a n 69°) Substitute known values. si ( sin B (150) 2 5 0 n 69°) B sin1(150) si ( 5 0 2 B 34 Solve for sin B. Use your calculator to evaluate. Take the inverse sine of both sides. The measure of B is approximately 34°. EXERCISES In Exercises 1–4, find the area of each polygon to the nearest square centimeter. 1. 3. 29 cm 65° 25 cm 95 cm 100° 2. 4. 50° 3.1 cm 124 cm 104 cm 12 cm You will need A calculator for Exercises 1–16 Construction tools for Exercise 18 78° 115 cm In Exercises 5–7, find each length to the nearest centimeter. 5. w? 6. x? 7. y? 28 cm w 79° 52° x 37° 12 cm 58° 41 cm 46° 87° y LESSON 12.3 The Law of Sines 637 For Exercises 8–10, each triangle is an acute triangle. Find each angle measure to the nearest degree. 8. mA? C 42° 9. mB? C 10. mC? C 36 cm 325 m 445 m 415 cm 362 cm A 29 cm B 77° A B A 63° B 11. Alphonse (point A) is over a 2500-meter landing strip in a hot-air balloon. At one end of the strip, Beatrice (point B) sees Alphonse with an angle of elevation measuring 39°. At the other end of the strip, Collette (point C) sees Alphonse with an angle of elevation measuring 62°. a. What is the distance between Alphonse and Beatrice? b. What is the distance between Alphonse and Collette? c. How high up is Alphonse? A 39° 62° B 2500 m C History For over 200 years, people believed that the entire site of James Fort was washed into the James River. Archaeologists have recently uncovered over 250 feet of the fort’s wall, as well as hundreds of thousands of artifacts dating to the early 1600s. 12. APPLICATION Archaeologists have recently started uncovering remains of James Fort (also known as Jamestown Fort) in Virginia. The fort was in the shape of an is |
osceles triangle. Unfortunately, one corner has disappeared into the James River. If the remaining complete wall measures 300 feet and the remaining corners measure 46.5° and 87°, how long were the two incomplete walls? What was the approximate area of the original fort? 638 CHAPTER 12 Trigonometry C James River B 87° James Fort 300 ft 46.5° A 13. A tree grows vertically on a hillside. The hill is at a 16° angle to the horizontal. The tree casts an 18-meter shadow up the hill when the angle of elevation of the sun measures 68°. How tall is the tree? 68° 18 m 16° Review 14. Read the History Connection below. Each step of El Castillo is 30 cm deep by 26 cm high. How tall is the pyramid, not counting the platform at the top? What is the angle of ascent? History One of the most impressive Mayan pyramids is El Castillo in Chichén Itzá, Mexico. Built in approximately 800 C.E., it has 91 steps on each of its four sides, or 364 steps in all. The top platform adds a level, so the pyramid has 365 levels to represent the number of days in the Mayan year. 15. According to legend, Galileo (1564–1642, Italy) used the Leaning Tower of Pisa to conduct his experiments in gravity. Assume that when he dropped objects from the top of the 55-meter tower (this is the measured length, not the height, of the tower), they landed 4.8 meters from the tower’s base. What was the angle that the tower was leaning from the vertical? 16. Find the volume of this cone. LESSON 12.3 The Law of Sines 639 3 cm 120° 17. Use the circle diagram at right and write a paragraph proof to show that ABE is isosceles. A B E C D 18. Construction Put two points on patty paper. Assume these points are opposite vertices of a square. Find the two missing vertices. 4 in. 19. Find AC, AE, and AF. All measurements are in centimeters. 20. Both boxes are right rectangular prisms. In which is the diagonal rod longer? 9 in. 5 in. Box 1 6 in. 5 in. 6 in. Box 2 IMPROVING YOUR VISUAL THINKING SKILLS Rope Tricks Each rope will be cut 50 times as shown. |
For each rope, how many pieces will result? 1. 1 2 2. 1 2 640 CHAPTER 12 Trigonometry L E S S O N 12.4 A ship in a port is safe, but that is not what ships are built for. JOHN A. SHEDD The Law of Cosines You’ve solved a variety of problems with the Pythagorean Theorem. It is perhaps your most important geometry conjecture. In Chapter 9, you found that the distance formula was really just the Pythagorean Theorem. You even used the Pythagorean Theorem to derive the equation of a circle. You can also derive trigonometry relationships from the Pythagorean Theorem. These are called Pythagorean identities. Complete the steps below to derive one of the Pythagorean identities. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Investigation A Pythagorean Identity Pick any measure of A and find (sin A)2 (cos A)2? Repeat Step 1 for several different measures of A. When you are ready, make a tentative conjecture. Let’s see if you can derive your conjecture. Use this triangle for Steps 3–6. Find ratios for sin A and cos A. Substitute your results from Step 3 into this equation. 2 (sin A)2 (cos A)2?? 2?? Add the two fractions on the right side of your equation. c Triangle ABC is a right triangle. How can you use the Pythagorean Theorem to further simplify your equation? A b Step 7 Does your result in Step 6 support your conjecture in Step 2? You should now be ready to state the Pythagorean identity. Pythagorean Identity For any angle A,?. B a C C-104 The Pythagorean Theorem is very powerful, but its use is still limited to right triangles. Recall from Chapter 9 that the Pythagorean Theorem does not work for acute triangles or obtuse triangles. You might ask, “What happens to the Pythagorean equation for acute triangles or obtuse triangles?” LESSON 12.4 The Law of Cosines 641 a2 b2 25 c2 25 A a2 b2 25 c2 12 b c a2 b2 25 c2 42 In this right triangle c2 a2 b2 In this acute triangle c2 a2 b2 In this obtuse triangle c2 a2 b2 If the legs of a right triangle are brought closer together |
so that the right angle becomes an acute angle, you’ll find that c2 a2 b2. In order to make this inequality into an equality, you would have to subtract something from a2 b2. c2 a2 b2 something If the legs are widened to form an obtuse angle, you’ll find that c2 a2 b2. Here, you’d have to add something to make an equality. c2 a2 b2 something Mathematicians found that the “something” was 2ab cos C. The Pythagorean Theorem generalizes to all triangles with a trigonometric property called the Law of Cosines. The steps used to derive the Law of Cosines are left for you as a Take Another Look activity. Law of Cosines C-105 For any triangle with sides of lengths a, b, and c, and with C the angle opposite the side with length c, c2 a2 b2 2ab cos C You can use the Law of Cosines when you are given three side lengths or two side lengths and the angle measure between them (SSS or SAS). Again, you’ll be asked to work only with acute angles. EXAMPLE A Find the length of side CT in acute triangle CRT. T r 45 cm c C 52 cm t 36° R Solution To find r, use the Law of Cosines: c2 a2 b2 2ab cos C The Law of Cosines. 642 CHAPTER 12 Trigonometry Using the variables in this problem, the Law of Cosines becomes r2 c2 t2 2ct cos R r2 452 522 2(45)(52)(cos 36°) Substitute 45 for c, 52 for t, and 36° for R. r 452 522 2(45)(52)(cos 36°) Take the positive square root of both sides. r 31 Substitute r for c, c for a, t for b, and R for C. Evaluate. The length of side CT is about 31 cm. EXAMPLE B Find the measure of Q in acute triangle QED. D 175 cm e 250 cm q Q d 225 cm E Solution Use the Law of Cosines and solve for Q. q2 e2 d 2 2ed cos Q q2 e2 d2 2ed 2502 1752 2252 2(175)(225) cos Q cos Q The Law of Cosines with respect to Q. |
Solve for cos Q. Substitute known values. Q cos1 Q 76 2502 1752 2252 2(175)(225) Take the inverse cosine of both sides. Evaluate. The measure of Q is about 76°. EXERCISES In Exercises 1–3, find each length to the nearest centimeter. 1. w? Y 41 cm w 2. y? 3. x? E y 32 cm 235 cm X 82° 282 cm 49° W 36 cm H S 78° Y 42 cm K x E You will need A calculator for Exercises 1–14 Construction tools for Exercises 20 and 21 Geometry software for Exercise 22 LESSON 12.4 The Law of Cosines 643 In Exercises 4–6, each triangle is an acute triangle. Find each angle measure to the nearest degree. 4. mA? K 5. mB? T 6. mC? 34 cm 42 cm 350 cm 390 cm 508 cm D 328 cm A 36 cm R B 380 cm E L 418 cm C 7. Two 24-centimeter radii of a circle form a central angle measuring 126°. What is the length of the chord connecting the two radii? 8. Find the measure of the smallest angle in an acute triangle whose side lengths are 4 m, 7 m, and 8 m. 9. Two sides of a parallelogram measure 15 cm and 20 cm, and one of the diagonals measures 19 cm. What are the measures of the angles of the parallelogram to the nearest degree? 10. APPLICATION Captain Malloy is flying a passenger jet. He is heading east at 720 km/hr when he sees an electrical storm straight ahead. He turns the jet 20° to the north to avoid the storm and continues in this direction for 1 hr. Then he makes a second turn, back toward his original flight path. Eighty minutes after his second turn, he makes a third turn and is back on course. By avoiding the storm, how much time did Captain Malloy lose from his original flight plan? Review 11. APPLICATION A cargo company loads truck trailers into ship cargo containers. The trucks drive up a ramp to a horizontal loading platform 30 ft off the ground, but they have difficulty driving up a ramp at an angle steeper than 20°. What is the minimum length that the ramp needs to be? 12. APPLICATION An archaeologist uncovers the remains of a square-based Egyptian pyramid. The |
base is intact and measures 130 meters on each side. The top of the pyramid has eroded away, but what remains of each face of the pyramid forms a 65° angle with the ground. What was the original height of the pyramid? 644 CHAPTER 12 Trigonometry 13. APPLICATION A lighthouse 55 meters above sea level spots a distress signal from a sailboat. The angle of depression to the sailboat measures 21°. How far away is the sailboat from the base of the lighthouse? 14. A painting company has a general safety rule to place ladders at an angle measuring between 55° and 75° from the level ground. Regina places the foot of her 25 ft ladder 6 ft from the base of a wall. What is the angle of the ladder? Is the ladder placed safely? If not, how far from the base of the wall should she place the ladder? in A tan A. 15. Show that s o A s c 16. TRAP is an isosceles trapezoid. a. Find PR in terms of x. b. Write a paragraph proof to show that mTPR 90°. P x A T 2x R 17. As P moves to the right on line happens to a. mPAB b. mAPB 1, describe what P 18. Which of these figures, the cone or the square pyramid, has the greater a. Base perimeter? b. Volume? c. Surface area? A B 8 cm 10 cm 19. What single transformation is equivalent to the composition of each pair of functions? Write a rule for each. a. A reflection over the line x 2 followed by a reflection over the line x 3 b. A reflection over the x-axis followed by a reflection over the y-axis 20. Construction Construct two rectangles that are not similar. 21. Construction Construct two isosceles trapezoids that are similar. Animate points 22. Technology Use geometry software to construct two circles. Connect the circles with a segment and construct the midpoint of the segment. Animate the endpoints of the segment around the circles and trace the midpoint of the segment. What shape does the midpoint of the segment trace? Try adjusting the relative size of the radii of the circles; try changing the distance between the centers of the circles; try starting the endpoints of the segment in different positions; or try animating the endpoints of the segment in different directions. Explain how these changes affect the shape traced by the mid |
point of the segment. 1 2 8 cm 2.5 cm LESSON 12.4 The Law of Cosines 645 JAPANESE TEMPLE TABLETS For centuries it has been customary in Japan to hang colorful wooden tablets in Shinto shrines to honor the gods of this native religion. During Japan’s historical period of isolation (1639–1854), this tradition continued with a mathematical twist. Merchants, farmers, and others who were dedicated to mathematical learning made tablets containing mathematical problems, called sangaku, to inspire and challenge visitors. See if you can answer this sangaku problem. These circles are tangent to each other and to the line. How are the radii of the three circles related? Research other sangaku problems, then design your own tablet. Your project should include Your solution to the problem above. Some problems you found during your research and your sources. Your own decorated sangaku tablet with its solution on the back. r1 r2 r3 These colorful tablets, some with gold engraving, usually contain geometry problems. Photographs by Hiroshi Umeoka. 646 CHAPTER 12 Trigonometry L E S S O N 12.5 One ship drives east and another drives west With the self-same winds that blow, ‘Tis the set of the sails and not the gales Which tells us the way to go. ELLA WHEELER WILCOX EXAMPLE Solution Problem Solving with Trigonometry There are many practical applications of trigonometry. Some of them involve vectors. In earlier vector activities, you used a ruler or a protractor to measure the size of the resulting vector or the angle between vectors. Now you will be able to calculate the resulting vectors with the Law of Sines or the Law of Cosines. Rowing instructor Calista Thomas is in a stream flowing north to south at 3 km/hr. She is rowing northeast at a rate of 4.5 km/hr. At what speed is she moving? What direction (bearing) is she actually moving? First, sketch and label the vector parallelogram. The resultant vector, r, divides the parallelogram into two congruent triangles. In each triangle you know the lengths of two sides and the measure of the included angle. Use the Law of Cosines to find the length of the resultant vector or the speed that it represents. r2 4.52 32 2(4.5)(3)(cos 45° |
) r2 4.52 32 2(4.5)(3)(cos 45°) r 3.2 N 4.5 km/hr 45°? 45° r 3 km/hr E 3 km/hr 4.5 km/hr 45° Calista is moving at a speed of approximately 3.2 km/hr. To find Calista’s bearing (an angle measured clockwise from north), you need to find, and add its measure to 45°. Use the Law of Sines. 5° 4 sin sin 2 3 3. 45°) n sin 3(si 3.2 45°) 42° sin13(si n.2 3 Add 42° and 45° to find that Calista is moving at a bearing of 87°. LESSON 12.5 Problem Solving with Trigonometry 647 EXERCISES 1. APPLICATION The steps to the front entrance of a public building rise a total of 1 m. A portion of the steps will be replaced by a wheelchair ramp. By a city ordinance, the angle of inclination for a ramp cannot measure greater than 4.5°. What is the minimum distance from the entrance that the ramp must begin? You will need A calculator for Exercises 1–11 Geometry software for Exercise 18 2. APPLICATION Giovanni is flying his Cessna airplane on a heading as shown. His instrument panel shows an air speed of 130 mi/hr. (Air speed is the speed in still air without wind.) However, there is a 20 mi/hr crosswind. What is the resulting speed of the plane? 3. APPLICATION A lighthouse is east of a Coast Guard patrol boat. The Coast Guard station is 20 km north of the lighthouse. The radar officer aboard the boat measures the angle between the lighthouse and the station to be 23°. How far is the boat from the station? 130 mi/hr 56° 20 mi/hr wind 4. APPLICATION The Archimedean screw is a water-raising device that consists of a wooden screw enclosed within a cylinder. When the cylinder is turned, the screw raises water. The screw is very efficient at an angle measuring 25°. If a screw needs to raise water 2.5 meters, how long should its cylinder be? Technology Used for centuries in Egypt to lift water from the Nile River, the Archimedean screw is thought to have been invented by Archimedes in the third century B.C.E., when he sailed to Egypt. |
It is also called an Archimedes Snail because of its spiral channels that resemble a snail shell. Once powered by people or animals, the device is now modernized to shift grain in mills and powders in factories. 5. APPLICATION Annie and Sashi are backpacking in the Sierra Nevada. They walk 8 km from their base camp at a bearing of 42°. After lunch, they change direction to a bearing of 137° and walk another 5 km. a. How far are Annie and Sashi from their base camp? b. At what bearing must Sashi and Annie travel to return to their base camp? 648 CHAPTER 12 Trigonometry 6. A surveyor at point A needs to calculate the distance to an island’s dock, point C. He walks 150 meters up the shoreline to point B such that AB AC. Angle ABC measures 58°. What is the distance between A and C? 7. During a strong wind, the top of a tree cracks and bends over, touching the ground as if the trunk were hinged. The tip of the tree touches the ground 20 feet 6 inches from the base of the tree and forms a 38° angle with the ground. What was the tree’s original height? C A Santa Rosa Island B 8. APPLICATION A pocket of matrix opal is known to be 24 meters beneath point A on Alan Ranch. A mining company has acquired rights to mine beneath Alan Ranch, but not the right to bring equipment onto the property. So the mining company cannot dig straight down. Brian Ranch has given permission to dig on its property at point B, 8 meters from point A. At what angle to the level ground must the mining crew dig to reach the opal? What distance must they dig? 9. Todd’s friend Olivia is flying her stunt plane at an elevation of 6.3 km. From the ground, Todd sees the plane moving directly toward him from the west at a 49° angle of elevation. Three minutes later he turns and sees the plane moving away from him to the east at a 65° angle of elevation. How fast is Olivia flying in kilometers per hour? B A C 10. A water pipe for a farm’s irrigation system must go through a small hill. Farmer Golden attaches a 14.5-meter rope to the pipe’s entry point and an 11.2-meter rope to the exit point. When he pulls the ropes taut, their ends meet at a 58° angle. What |
is the length of pipe needed to go through the hill? At what angle with respect to the first rope should the pipe be laid so that it comes out of the hill at the correct exit point? Review 11. Find the volume of this right regular pentagonal prism. 7 cm 2 s n 12. A formula for the area of a regular polygon is A, a n 4 t where n is the number of sides, s is the length of a side, 0. Explain why this formula is correct. 6 and 3 2 n 3 cm LESSON 12.5 Problem Solving with Trigonometry 649 13. Find the volume of the largest cube that can fit into a sphere with a radius of 12 cm. 14. How does the area of a triangle change if its vertices are transformed by the rule (x, y) → (3x, 3y)? Give an example to support your answer. 15. What’s wrong with this picture? 12 cm 4 cm 5 cm 15 cm 16. As P moves to the right on line 1, describe what happens to a. PA b. Area of APB P A B 1 2 17. What single transformation is equivalent to the composition of each pair of functions? Write a rule for each. a. A reflection over the line y x followed by a counterclockwise 270° rotation about the origin b. A rotation 180° about the origin followed by a reflection over the x-axis 18. Technology Tile floors are often designed by creating simple, symmetric patterns on squares. When the squares are lined up, the patterns combine, often leaving the original squares hardly visible. Use geometry software to create your own tile-floor pattern. IMPROVING YOUR ALGEBRA SKILLS Substitute and Solve 0 1. If 2x 3y, y 5w, and w 2, find x in terms of z. 3 z 9 2. If 7x 13y, y 28w, and w, find x in terms of z. 6z 2 650 CHAPTER 12 Trigonometry Trigonometric Ratios and the Unit Circle In Lesson 12.1, you defined trigonometric ratios in terms of the sides of a right triangle. That limited you to talking about acute angles. But in the coordinate plane, it’s possible to define trigonometric ratios for angles with measures less than 0° and greater than 90°. These definitions use a unit circle—a circle with center (0, 0) and radius |
1 unit. You will need ● The Unit Circle worksheet The height of a seat on a Ferris wheel can be modeled by unit-circle trigonometry. This Ferris wheel, called the London Eye, was built for London’s year 2000 celebration. Activity The Unit Circle In this activity, you will use a Sketchpad construction to explore the unit circle. The first part of this activity will give you some understanding of how a unit circle simplifies the trigonometric ratios for acute angles. Then you will use the unit circle to explore the ratios for all angles, from 0° to 360°, and even negative angle measures. C Step 1 Step 2 Step 3 Follow the steps on the worksheet to construct a unit circle with right triangle ADC. In right triangle ADC, write ratios for the sine, cosine, and tangent of DAC. What is the length of the hypotenuse, AC, in this unit circle? Use this length to simplify your trigonometric definitions in Step 2. A D B EXPLORATION Trigonometric Ratios and the Unit Circle 651 Step 4 Step 5 Step 6 Step 7 Measure the coordinates of point C. Which coordinate corresponds to the sine of DAC? Which coordinate corresponds to the cosine of DAC? What parts of your sketch physically represent the sine of DAC and the cosine of DAC? How can you use the coordinates of point C to calculate the tangent of DAC? Follow the steps on the worksheet to add a physical representation of tangent to your unit circle. Measure the coordinates of point E. Which coordinate corresponds to the tangent of DAC? What part of your sketch physically represents the tangent of DAC? Use similar triangles ADC and ABE to explain your answers. So far, you have looked only at right triangle ADC with acute angle DAC. It may seem that you have not gotten any closer to defining trigonometric ratios for angles with measures less than 0° or greater than 90°. That is, even if you move point C into another quadrant, DAC would still be an acute angle. In order to modify the definition of trigonometric ratios in a unit circle, you need to measure BAC instead. That is, measure the amount of rotation from AB to AC. E C E A D B You may recall from Chapter 6 that a point on a rolling tire traces a curve called a cycloid. Cycloids are defined by trigonometry. Step 8 Step 9 Step 10 Step 11 Select point B, point A |
, and point C, in that order. Measure BAC. Move point C around the circle and watch how the measure of BAC changes. Summarize your observations. When point C is in the first quadrant, how is the measure of DAC related to the measure of BAC? How about when point C is in the second quadrant? The third quadrant? The fourth quadrant? Use Sketchpad’s calculator to calculate the sine, cosine, and tangent of BAC. Compare these trigonometric ratios to the coordinates of point C and point E. Do the values support your answers to Steps 4 and 7? 652 CHAPTER 12 Trigonometry Step 12 Move point C around the circle and watch how the trigonometric ratios change for angle measures between 180° and 180°. Answer these questions. a. How does the sine of BAC change as the measure of the angle goes from 0° to 90° to 180°? From 180° to 90° to 0°? b. How does the cosine of BAC change? c. If the sine of one angle is equal to the cosine of another angle, how are the angles related to each other? d. How does the tangent of BAC change? What happens to the tangent of BAC as its measure approaches 90° or 90°? Based on the definition of tangent and the side lengths in ADC, what value do you think the tangent of 90° equals? You can add an interesting animation that will graph the changing sine and tangent values in your sketch. You can construct points that will trace curves that algebraically represent the functions y sin(x) and y tan(x). Animate points 2 E C Tan Sin –2 A D B G 2 4 F 6 Step 13 Step 14 Follow the steps on the worksheet to add an animation that will trace curves representing the sine and tangent functions. Measure the coordinates of point F, then locate it as close to (6.28, 0) as possible. Move point G to the origin and move point C to (1, 0). Press the Animation button to trace the sine and tangent functions. The high and low points of tides can be modeled with trigonometry. This tide table from the Savannah River in Fort Jackson, Georgia, shows a familiar pattern in its data. 6.8 ft 2:09 am 1.2 ft 8:37 am 6.4 ft 2:35 |
pm 0.6 ft 9:06 pm 7.0 ft 3:03 am 1.2 ft 9:49 am 6.4 ft 3:31 pm 0.4 ft 10:08 pm 4 0 01/22 Fort Jackson, GEORGIA Savannah River Noon 6 am 6 pm 01/23 6 am Noon 6 pm 01/24 EXPLORATION Trigonometric Ratios and the Unit Circle 653 Step 15 What’s special about 6.28 as the x-coordinate of point F? Try other locations for point F to see what happens. Use what you know about circles to explain why 6.28 is a special value for the unit circle. You will probably learn much more about unit-circle trigonometry in a future mathematics course. TRIGONOMETRIC FUNCTIONS You’ve seen many applications where you can use trigonometry to find distances or angle measures. Another important application of trigonometry is to model periodic phenomena, which repeat over time. In this project you’ll discover characteristics of the graphs of trigonometric functions, including their periodic nature. The three functions you’ll look at are y sin(x) y cos(x) y tan(x) These functions are defined not only for acute angles but also for angles with measures less than 0° and greater than 90°. The swinging motion of a pendulum is an example of periodic motion that can be modeled by trigonometry. Set your calculator in degree mode and set a window with an x-range of 360 to 360 and a y-range of 2 to 2. One at a time, graph each trigonometric function. Describe the characteristics of each graph, including maximum and minimum values for y and the period—the horizontal distance after which the graph starts repeating itself. Use what you know about the definitions of sine, cosine, and tangent to explain any unusual occurrences. Try graphing pairs of trigonometric functions. Describe any relationships you see between the graphs. Are there any values in common? Prepare an organized presentation of your results. 654 CHAPTER 12 Trigonometry Three Types of Proofs In previous explorations, you learned four forms of valid reasoning: Modus Ponens (MP), Modus Tollens (MT), the Law of Syllogism (LS), and the Law of Contrapositive (LC). You can use these forms of reasoning to make logical arguments, or proofs. In this exploration you will learn the |
three basic types of proofs: direct proofs, conditional proofs, and indirect proofs. In a direct proof, the given information or premises are stated, then valid forms of reasoning are used to arrive directly at a conclusion. Here is a direct proof given in two-column form. In a two-column proof, each statement in the argument is written in the left column, and the reason for each statement is written directly across in the right column. Direct Proof Premises: P → Q R → P Q Conclusion: R 1. P → Q 2. Q 3. P 4. R → P 5. R 1. Premise 2. Premise 3. From lines 1 and 2, using MT 4. Premise 5. From lines 3 and 4, using MT A conditional proof is used to prove that a P → Q statement follows from a set of premises. In a conditional proof, the first part of the conditional statement, called the antecedent, is assumed to be true. Then logical reasoning is used to demonstrate that the second part, called the consequent, must also be true. If this process is successful, it’s demonstrated that if P is true, then Q must be true. EXPLORATION Three Types of Proofs 655 In other words, a conditional proof shows that the antecedent implies the consequent. Here is an example. Conditional Proof Premises: P → R S → R Conclusion: P → S 1. P 2. P → R 3. R 4. S → R 5. S Assuming P is true, the truth of S is established. P → S 1. Assume the antecedent 2. Premise 3. From lines 1 and 2, using MP 4. Premise 5. From lines 3 and 4, using MT An indirect proof is a clever approach to proving something. To prove indirectly that a statement is true, you begin by assuming it is not true. Then you show that this assumption leads to a contradiction. For example, if you are given a set of premises and are asked to show that some conclusion P is true, begin by assuming that the opposite of P, namely P, is true. Then show that this assumption leads to a contradiction of an earlier statement. If P leads to a contradiction, it must be false and P must be true. Here is an example. Indirect Proof Premises: R → S R → P P Conclusion: S 1. Assume the opposite of the conclusion 2. Premise 3. From |
lines 1 and 2, using MT 4. Premise 5. From lines 3 and 4, using MP 6. Premise 1. S 2. R → S 3. R 4. R → P 5. P 6. P But lines 5 and 6 contradict each other. It’s impossible for both P and P to be true. Therefore, S, the original assumption, is false. If S is false, then S is true. S Many logical arguments can be proved using more than one type of proof. For instance, you can prove the argument in the example above by using a direct proof. (Try it!) With practice you will be able to tell which method will work best for a particular argument. 656 CHAPTER 12 Trigonometry Activity Prove It! Step 1 Copy the direct proof below, including the list of premises and the conclusion. Provide each missing reason. Premises: P → Q Q → R R Conclusion: P 1. Q → R 2. R 3. Q 4. P → Q 5. P 1.? 2.? 3.? 4.? 5.? Step 2 Copy the conditional proof below, including the list of premises and the conclusion. Provide each missing statement or reason. Premises: Conclusion. S 2. S → T 3. T 4. T → R 5. R 6.? 7.? Assuming S is true, the truth of Q is established.? 1.? 2.? 3. From lines 1 and 2, using? 4.? 5.? 6.? 7.? Step 3 Copy the indirect proof below and at the top of page 658, including the list of premises and the conclusion. Provide each missing statement or reason. Premises Conclusion: P 1. P 2. P → (Q → R) 3. Q → R 4. Q 1. Assume the? of the? 2.? 3.? 4.? EXPLORATION Three Types of Proofs 657 Step 3 (continued) 5.? 6.? 7. From lines? and?, using? 5. R 6.? 7.? But lines? and? contradict each other. Therefore, P, the assumption, is false. P Step 4 Provide the steps and reasons to prove each logical argument. You will need to decide whether to use a direct, conditional, or indirect proof. a. Premises Conclusion: T → P c. Premises Conclusion: R b. Premises: (R |
→ S Conclusion: (R → S) d. Premises Conclusion: R Step 5 Translate each argument into symbolic terms, then prove it is valid. a. If all wealthy people are happy, then money can buy happiness. If money can buy happiness, then true love doesn’t exist. But true love exists. Therefore, not all wealthy people are happy. b. If Clark is performing at the theater today, then everyone at the theater has a good time. If everyone at the theater has a good time, then Lois is not sad. Lois is sad. Therefore, Clark is not performing at the theater today. c. If Evette is innocent, then Alfa is telling the truth. If Romeo is telling the truth, then Alfa is not telling the truth. If Romeo is not telling the truth, then he has something to gain. Romeo has nothing to gain. Therefore, if Romeo has nothing to gain, then Evette is not innocent. 658 CHAPTER 12 Trigonometry VIEW ● CHAPTER 11 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● C CHAPTER 12 R E V I E W Trigonometry was first developed by astronomers who wanted to map the stars. Obviously, it is hard to directly measure the distances between stars and planets. That created a need for new methods of indirect measurement. As you’ve seen, you can solve many indirect measurement problems by using triangles. Using sine, cosine, and tangent ratios, you can find unknown lengths and angle measures if you know just a few measures in a right triangle. You can extend these methods to any triangle using the Law of Sines or the Law of Cosines. What’s the least you need to know about a right triangle in order to find all its measures? What parts of a nonright triangle do you need to know in order to find the other parts? Describe a situation in which an angle of elevation or depression can help you find an unknown height. EXERCISES For Exercises 1–3, use a calculator to find each trigonometric ratio accurate to four decimal places. You will need A calculator for Exercises 1–3, 7–28, 51, and 53 1. sin 57° 2. cos 9° 3. tan 88° For Exercises 4–6, find each trigonometric ratio. 4. sin A? cos A? tan A? T a N b |
c 5. sin B? cos B? tan B? Y 16 30 A O B 6. sin? cos? tan? y 1 (t, s) x (1, 0) (0, 0) For Exercises 7–9, find the measure of each angle to the nearest degree. 7. sin A 0.5447 10. Shaded area? 8. cos B 0.0696 9. tan C 2.9043 11. Volume? 41 cm 37° h 112° 18 cm CHAPTER 12 REVIEW 659 EW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CH 12. APPLICATION According to the Americans with Disabilities Act, enacted in 1990, the slope of a 1 wheelchair ramp must be less than and there must 1 2 be a minimum 5-by-5 ft landing for every 2.5 ft of rise. These dimensions were chosen to accommodate handicapped people who face physical barriers in public buildings and at work. An architect has submitted the orthographic plan shown below. Does the plan meet the requirements of the act? What will be the ramp’s angle of ascent? 5 5 25 Top view 20 2.5 3 5 1.5 Front view 20 25 Side view 5 1.5 1.5 2.5 2.5 All dimensions are given in feet. The Axis Dance Company includes performers in wheelchairs. Increased tolerance and accessibility laws have broadened the opportunities available to people with disabilities. 13. APPLICATION A lighthouse is east of a sailboat. The sailboat’s dock is 30 km north of the lighthouse. The captain measures the angle between the lighthouse and the dock and finds it to be 35°. How far is the sailboat from the dock? 14. APPLICATION An air traffic controller must calculate the angle of descent (the angle of depression) for an incoming jet. The jet’s crew reports that their land distance is 44 km from the base of the control tower and that the plane is flying at an altitude of 5.6 km. Find the measure of the angle of descent. 15. APPLICATION A new house is 32 feet wide. The rafters will rise at a 36° angle and meet above the center line of the house. Each rafter also needs to overhang the side of the house by 2 feet. How long should the carpenter make each rafter?? 36° 16. APPLICATION During a flood relief effort |
, a Coast Guard 2 ft patrol boat spots a helicopter dropping a package near the Florida shoreline. Officer Duncan measures the angle of elevation to the helicopter to be 15° and the distance to the helicopter to be 6800 m. How far is the patrol boat from the point where the package will land? 32 ft 17. At an air show, Amelia sees a jet heading south away from her at a 42° angle of elevation. Twenty seconds later the jet is still moving away from her, heading south at a 15° angle of elevation. If the jet’s elevation is constantly 6.3 km, how fast is it flying in kilometers per hour? 660 CHAPTER 12 Trigonometry 2 ft ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAP For Exercises 18–23, find each measure to the nearest unit or to the nearest square unit. 18. Area? 19. w? 20. ABC is acute. mA? B 55° 24 cm 40 cm w 53° 25 cm 76° 37 cm 50° C 29 cm A 21. x? 22. mB? 23. Area? x 34 cm 65 cm 101 cm 75° 27 cm B 27 cm 26 cm 48° 24. Find the length of the apothem of a regular pentagon with a side measuring 36 cm. 25. Find the area of a triangle formed by two 12 cm radii and a 16 cm chord in a circle. 26. A circle is circumscribed about a regular octagon with a perimeter of 48 cm. Find the diameter of the circle. 27. A 16 cm chord is drawn in a circle of diameter 24 cm. Find the area of the segment of the circle. 28. Leslie is paddling his kayak at a bearing of 45°. In still water his speed would be 13 km/hr but there is a 5 km/hr current moving west. What is the resulting speed and direction of Leslie’s kayak? 5 km/hr r 13 km/hr CHAPTER 12 REVIEW 661 EW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CH MIXED REVIEW For Exercises 29–41, identify each statement as true or false. For each false statement, explain why it is false or sketch a counterexample. 29. An octahedron is a prism that has an octagonal base |
. 30. If the four angles of one quadrilateral are congruent to the four corresponding angles of another quadrilateral, then the two quadrilaterals are similar. 31. The three medians of a triangle meet at the centroid. 32. To use the Law of Cosines, you must know three side lengths or two side lengths and the measure of the included angle. 33. If the ratio of corresponding sides of two similar polygons is m, then the ratio of n their areas is m. n 34. The measure of an angle inscribed in a semicircle is always 90°. 35. If T is an acute angle in a right triangle, then tangent of T length of side adjacent to T length of side opposite T 36. If C 2 is the area of the base of a pyramid, and C is the height of the pyramid, then the volume of the pyramid is 1 C 3. 3 37. If two different lines intersect at a point, then the sum of the measures of at least one pair of vertical angles will be equal to or greater than 180°. 38. If a line cuts two sides of a triangle proportionally, then it is parallel to the third side. 39. If two sides of a triangle measure 6 cm and 8 cm and the angle between the two sides measures 60°, then the area of the triangle is 123 cm2. 40. A nonvertical line also m. 1 has slope m and is perpendicular to line 2. The slope of 2 is 41. If two sides of one triangle are proportional to two sides of another triangle, then the two triangles are similar. For Exercises 42–53, select the correct answer. 42. The diagonals of a parallelogram i. Are perpendicular to each other. ii. Bisect each other. iii. Form four congruent triangles. B. ii only A. i only 662 CHAPTER 12 Trigonometry C. iii only D. i and ii ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAP 43. What is the formula for the volume of a sphere? A. V 4r2 B. V r2h C. V 4 r3 3 D. V 1 r2h 3 44. For the triangle at right, what is the value of (cos L)2 (sin L)2? A. About 0.22 C. 1 B. About 1 |
.41 D. Cannot be determined J 13 ft L 11 ft 42° K 45. The diagonals of a rhombus i. Are perpendicular to each other. ii. Bisect each other. iii. Form four congruent triangles. B. iii only A. i only C. i and ii D. All of the above 46. The ratio of the surface areas of two similar solids is 4. What is the ratio of the 9 volumes of the solids? A. 2 3 8 B. 2 7 6 4 C. 9 2 7 6 D. 1 8 1 47. A cylinder has height T and base area K. What is the volume of the cylinder? A. V K 2T B. V KT C. V 2KT D. V 1 KT 3 48. Which of the following is not a similarity shortcut? A. SSA B. SSS C. AA D. SAS 49. If a triangle has sides of lengths a, b, and c, and C is the angle opposite the side of length c, which of these statements must be true? A. a2 b2 c2 2ab cos C C. c2 a2 b2 2ab cos C B. c2 a2 b2 2ab cos C D. a2 b2 c2 A 50. In the drawing at right, WX YZ BC. What is the value of m? B. 8 ft D. 16 ft A. 6 ft C. 12 ft 51. When a rock is added to a container of water, it raises the water level by 4 cm. If the container is a rectangular prism with a base that measures 8 cm by 9 cm, what is the volume of the rock? B. 32 cm3 A. 4 cm3 D. 288 cm3 C. 36 cm3 52. Which law could you use to find the value of v? A. Law of Supply and Demand B. Law of Syllogism C. Law of Cosines D. Law of Sines 9 ft W 6 ft Y 3 ft B T m X 4 ft Z C v 25 cm U 70° V 33 cm CHAPTER 12 REVIEW 663 EW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CH 53. A 32-foot telephone pole casts a 12-foot shadow at the same time a boy nearby casts a 1.75-foot shadow. How tall is the boy? A |
. 4 ft 8 in. B. 4 ft 6 in. C. 5 ft 8 in. D. 6 ft Exercises 54–56 are portions of cones. Find the volume of each solid. 54. 55. 6 cm 3 cm 4 cm 12 cm 240° 5 cm 56. 4 cm 5 cm 7.5 cm 6 cm 57. Each person at a family reunion hugs everyone else exactly once. There were 528 hugs. How many people were at the reunion? 58. Triangle TRI with vertices T(7, 0), R(5, 3), and I (1, 0) is translated by the rule (x, y) → (x 2, y 1). Then its image is translated by the rule (x, y) → (x 1, y 2). What single translation is equivalent to the composition of these two translations? 59. LMN PQR. Find w, x, and y. 60. Find x. L 14 cm P 28 cm x 21 cm y M w N Q 36 cm R x 48° 26° 39 cm 61. The diameter of a circle has endpoints (5, 2) and (5, 4). Find the equation of the circle. 62. Explain why a regular pentagon cannot create a monohedral tessellation. 63. Archaeologist Ertha Diggs uses a clinometer to find the height of an ancient temple. She views the top of the temple with a 37° angle of elevation. She is standing 130 meters from the center of the temple’s base, and her eye is 1.5 meters above the ground. How tall is the temple? 664 CHAPTER 12 Trigonometry ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAP 64. In the diagram below, the length of HK 20 ft. Find the radius of the circle. is 65. The shaded area is 10 cm2. Find r. H 80° S K 40° J r 135° 66. Triangle ABC is isosceles with AB congruent to BC. Point D is on AC such that BD is perpendicular to AC. Make a sketch and answer these questions. a. mABC? mABD b. What can you conclude about BD? TAKE ANOTHER LOOK 1. You learned the Law of Sines as B sin A sin sin C c b a Use algebra to show that c b a sin sin sin C B |
A 2. Recall that SSA does not determine a triangle. For that reason, you’ve been asked to find only acute angles using the Law of Sines. Take another look at a pair of triangles, AB1C and AB2C, determined by SSA. How is CB1A related to CB2A? Find mCB1A and mCB2A. Find the sine of each angle. Find the sines of another pair of angles that are related in the same way, then complete this conjecture: for any angle, sin sin(? ). 3. The Law of Cosines is generally stated using C. c2 a2 b2 2ab cos C State the Law of Cosines in two different ways, using A and B. 4. Derive the Law of Cosines. 3 cm B2 C b 6 cm 3 cm B1 C 25° A A a c B CHAPTER 12 REVIEW 665 EW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CHAPTER 12 REVIEW ● CH 5. Is there a relationship between the measure of the central angle of a sector of a circle and the angle at the vertex of the right cone formed when rolled up? x y Is x related to y? Assessing What You’ve Learned UPDATE YOUR PORTFOLIO Choose a real-world indirect measurement problem that uses trigonometry, and add it to your portfolio. Describe the problem, explain how you solved it, and explain why you chose it for your portfolio. ORGANIZE YOUR NOTEBOOK Make sure your notebook is complete and well organized. Write a one-page chapter summary. Reviewing your notes and solving sample test items are good ways to prepare for chapter tests. GIVE A PRESENTATION Demonstrate how to use an angle-measuring device to make indirect measurements of actual objects. Use appropriate visual aids. WRITE IN YOUR JOURNAL The last five chapters have had a strong problem-solving focus. What do you see as your strengths and weaknesses as a problem solver? In what ways have you improved? In what areas could you improve or use more help? Has your attitude toward problem solving changed since you began this course? 666 CHAPTER 12 Trigonometry CHAPTER 13 Geometry as a Mathematical System This search for new possibilities, this discovery of new jigsaw puzzle pieces, which in the first place surprises and astonishes the designer himself, is a game that throug |
h the years has always fascinated and enthralled me anew. M. C. ESCHER Another World (Other World), M. C. Escher, 1947 ©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● look at geometry as a mathematical system ● see how some conjectures are logically related to each other ● review a number of proof strategies, such as working backward and analyzing diagrams L E S S O N 13.1 Geometry is the art of correct reasoning on incorrect figures. GEORGE POLYA The Premises of Geometry As you learned in previous chapters, for thousands of years Babylonian, Egyptian, Chinese, and other mathematicians discovered many geometry principles and developed procedures for doing practical geometry. By 600 B.C.E., a prosperous new civilization had begun to grow in the trading towns along the coast of Asia Minor (present-day Turkey) and later in Greece, Sicily, and Italy. People had free time to discuss and debate issues of government and law. They began to insist on reasons to support statements made in debate. Mathematicians began to use logical reasoning to deduce mathematical ideas. History This map detail shows Sicily, Italy, Greece, and Asia Minor along the north coast of the Mediterranean Sea. The map was drawn by Italian painter and architect Pietro da Cortona (1596–1669). Greek mathematician Thales of Miletus (ca. 625–547 B.C.E.) made his geometry ideas convincing by supporting his discoveries with logical reasoning. Over the next 300 years, the process of supporting mathematical conjectures with logical arguments became more and more refined. Other Greek mathematicians, including Thales’ most famous student, Pythagoras, began linking chains of logical reasoning. The tradition continued with Plato and his students. Euclid, in his famous work about geometry and number theory, Elements, established a single chain of deductive arguments for most of the geometry known then. Timeline of early Greek mathematics THALES PYTHAGORAS PLATO E UC LID ca. 585 B.C.E. ca. 500 B.C.E. ca. 347 B.C.E. ca. 640 B.C.E. ca. 546 B.C.E. ca. 427 B.C.E. ca. 300 B.C.E. You have learned that Euclid used geometric constructions to study properties of lines and shapes. Eucl |
id also created a deductive system—a set of premises, or accepted facts, and a set of logical rules—to organize geometry properties. He started from a collection of simple and useful statements he called postulates. He then systematically demonstrated how each geometry discovery followed logically from his postulates and his previously proved conjectures, or theorems. 668 CHAPTER 13 Geometry as a Mathematical System Up to now, you have been discovering geometry properties inductively, the way many mathematicians have over the centuries. You have studied geometric figures and have made conjectures about them. Then, to explain your conjectures, you turned to deductive reasoning. You used informal proofs to explain why a conjecture was true. However, you did not prove every conjecture. In fact, you sometimes made critical assumptions or relied on unproved conjectures in your proofs. A conclusion in a proof is true if and only if your premises are true and all your arguments are valid. Faulty assumptions can lead to the wrong conclusion. Have all your assumptions been reliable? Inductive reasoning process Deductive reasoning process In this chapter you will look at geometry as Euclid did. You will start with premises: definitions, properties, and postulates. From these premises you will systematically prove your earlier conjectures. Proved conjectures will become theorems, which you can use to prove other conjectures, turning them into theorems, as well. You will build a logical framework using your most important ideas and conjectures from geometry. Premises for Logical Arguments in Geometry 1. Definitions and undefined terms 2. Properties of arithmetic, equality, and congruence 3. Postulates of geometry 4. Previously proved geometry conjectures (theorems) You are already familiar with the first type of premise on the list: the undefined terms—point, line, and plane. In addition, you have a list of basic definitions in your notebook. You used the second set of premises, properties of arithmetic and equality, in your algebra course. LESSON 13.1 The Premises of Geometry 669 Properties of Arithmetic For any numbers a, b, and c : Commutative property of addition a b b a Commutative property of multiplication ab ba Associative property of addition (a b) c a (b c) Associative property of multiplication (ab)c a(bc) Distributive property a(b c) ab ac Properties of Equality For any numbers a, b, c, and d: These |
Mayan stone carvings, found in Tikal, Guatemala, show the glyphs, or symbols, used in the Mayan number system. Learn more about Mayan numerals at www.keymath.com/DG. Reflexive property (also called the identity property) a a Any number is equal to itself. Transitive property If a b and b c, then a c. (This property often takes the form of the substitution property, which says that if b c, you can substitute c for b.) Symmetric property If a b, then b a. Addition property If a b, then a c b c. (Also, if a b and c d, then a c b d.) Subtraction property If a b, then a c b c. (Also, if a b and c d, then a c b d.) Multiplication property If a b, then ac bc. (Also, if a b and c d, then ac bd.) Division property If a b, then a b provided c 0. (Also, if a b and c d, then a b d c c c provided that c 0 and d 0.) Square root property If a2 b, then a b. Zero product property If ab 0, then a 0 or b 0 or both a and b 0. 670 CHAPTER 13 Geometry as a Mathematical System Whether or not you remember their names, you’ve used these properties to solve algebraic equations. The process of solving an equation is really an algebraic proof that your solution is valid. To arrive at a correct solution, you must support each step by a property. The addition property of equality, for example, permits you to add the same number to both sides of an equation to get an equivalent equation. EXAMPLE Solve for x: 5x 12 3(x 2) Solution 5x 12 3(x 2) 5x 12 3x 6 5x 3x 18 2x 18 x 9 Given. Distributive property. Addition property of equality. Subtraction property of equality. Division property of equality. Why are the properties of arithmetic and equality important in geometry? The lengths of segments and the measures of angles involve numbers, so you will often need to use these properties in geometry proofs. And just as you use equality to express a relationship between numbers, you use congruence to express a relationship between geometric figures. Definition of Congruence If AB CD, then AB CD, and |
conversely, if AB CD, then AB CD. If mA mB, then A B, and conversely, if A B, then mA mB. A page from a Latin translation of Euclid’s Elements. Which of these definitions do you recognize? Congruence is defined by equality, so you can extend the properties of equality to a reflexive property of congruence, a transitive property of congruence, and a symmetric property of congruence. This is left for you to do in the exercises. The third set of premises is specific to geometry. These premises are traditionally called postulates. Postulates should be very basic. Like undefined terms, they should be useful and easy for everyone to agree on, with little debate. As you’ve performed basic geometric constructions in this class, you’ve observed some of these “obvious truths.” Whenever you draw a figure or use an auxiliary line, you are using these postulates. LESSON 13.1 The Premises of Geometry 671 Postulates of Geometry Line Postulate You can construct exactly one line through any two points. Line Intersection Postulate The intersection of two distinct lines is exactly one point. Segment Duplication Postulate You can construct a segment congruent to another segment. Angle Duplication Postulate You can construct an angle congruent to another angle. Midpoint Postulate You can construct exactly one midpoint on any line segment. Angle Bisector Postulate You can construct exactly one angle bisector in any angle. Parallel Postulate Through a point not on a given line, you can construct exactly one line parallel to the given line. Perpendicular Postulate Through a point not on a given line, you can construct exactly one line perpendicular to the given line. Segment Addition Postulate If point B is on AC and between points A and C, then AB BC AC. Angle Addition Postulate If point D lies in the interior of ABC, then mABD mDBC mABC. A B There are certain rules that everyone needs to agree on so we can drive safely! What are the “road rules” of geometry? 672 CHAPTER 13 Geometry as a Mathematical System B C A D C Linear Pair Postulate If two angles are a linear pair, then they are supplementary. (Previously called the Linear Pair Conjecture.) Corresponding Angles Postulate (CA Postulate) If two parallel lines |
are cut by a transversal, then the corresponding angles are congruent. Conversely, if two lines are cut by a transversal forming congruent corresponding angles, then the lines are parallel. (Previously called the CA Conjecture.) SSS Congruence Postulate If the three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. (Previously called the SSS Congruence Conjecture.) SAS Congruence Postulate If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent. ASA Congruence Postulate If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent. Mathematics Euclid wrote 13 books covering, among other topics, plane geometry and solid geometry. He started with definitions, postulates, and “common notions” about the properties of equality. He then wrote hundreds of propositions, which we would call conjectures, and used constructions based on the definitions and postulates to show that they were valid. The statements that we call postulates were actually Euclid’s postulates, plus a few of his propositions. LESSON 13.1 The Premises of Geometry 673 To build a logical framework for the geometry you have learned, you will start with the premises of geometry. In the exercises, you will see how these premises are the foundations for some of your previous assumptions and conjectures. You will also use these postulates and properties to see how some geometry statements are logical consequences of others. EXERCISES 1. What is the difference between a postulate and a theorem? 2. Euclid might have stated the addition property of equality (translated from the Greek) in this way: “If equals are added to equals, the results are equal.” State the subtraction, multiplication, and division properties of equality as Euclid might have stated them. (You may write them in English—extra credit for the original Greek!) 3. Write the reflexive property of congruence, the transitive property of congruence, and the symmetric property of congruence. Add these properties to your notebook. Include a diagram for each property. Illustrate one property with congruent triangles, another property with congruent segments, and another property |
with congruent angles. (These properties may seem ridiculously obvious. This is exactly why they are accepted as premises, which require no proof!) 4. When you state AC AC, what property are you using? When you state AC AC, what property are you using? 5. Name the property that supports this statement: If ACE BDF and BDF HKM, then ACE HKM. 6. Name the property that supports this statement: If x 120 180, then x 60. 7. Name the property that supports this statement: If 2(x 14) 36, then x 14 18. In Exercises 8 and 9, provide the missing property of equality or arithmetic as a reason for each step to solve the algebraic equation or to prove the algebraic argument. 8. Solve for x: Solution: 9. Conjecture: Proof: 7x 22 4(x 2) 7x 22 4(x 2) 7x 22 4x 8 3x 22 8 3x 30 x 10 Given.? property.? property of equality.? property of equality.? property of equality. x c d, then x m(c d), provided that m 0. If (d c) x m(c d)???? 674 CHAPTER 13 Geometry as a Mathematical System In Exercises 10–17, identify each statement as true or false. Then state which definition, property of algebra, property of congruence, or postulate supports your answer. 10. If M is the midpoint of AB, then AM BM. 11. If M is the midpoint of CD and N is the midpoint of CD, then M and N are the same point. 12. If AB bisects CAD, then CAB DAB. 13. If AB bisects CAD and AF bisects CAD, then AB and AF are the same ray. 14. Lines and m can intersect at different points A and B. 15. If line passes through points A and B and line m passes through points A and B, lines and m do not have to be the same line. 16. If point P is in the interior of RAT, then mRAP mPAT mRAT. 17. If point M is on AC and between points A and C, then AM MC AC. 18. The Declaration of Independence states “We hold these truths to be selfevident...,” then goes on to list four post |
ulates of good government. Look up the Declaration of Independence and list the four self-evident truths that were the original premises of the United States government. You can find links to this www.keymath.com/DG topic at. Arthur Szyk (1894–1951), a Polish American whose propaganda art helped aid the Allied war effort during World War II, created this patriotic illustrated version of the Declaration of Independence. 19. Copy and complete this flowchart proof. For each reason, state the definition, the property of algebra, or the property of congruence that supports the statement. Given: AO and BO are radii Show: AOB is isosceles 1? Given 2 AO BO 3? Definition of circle Definition of? A O B LESSON 13.1 The Premises of Geometry 675 For Exercises 20–22, copy and complete each flowchart proof. 20. Given: 1 2 Show? Corresponding Angles Postulate 3 3?? 21. Given: AC BD, AD BC Show: D C D C 1 AC BD??? AB BA Identity property 2 3 4 ABC?? A 5? CPCTC 22. Given: Isosceles triangle ABC with AB BC Show: A C 1 Construct angle bisector BD 2? Given A 3 ABD? 5 BAD?? 4 Definition? of BD? BD 6?? B C B D 23. You have probably noticed that the sum of two odd integers is always an even integer. The rule 2n generates even integers and the rule 2n 1 generates odd integers. Let 2n 1 and 2m 1 represent any two odd integers and prove that the sum of two odd integers is always an even integer. 676 CHAPTER 13 Geometry as a Mathematical System 24. Let 2n 1 and 2m 1 represent any two odd integers and prove that the product of any two odd integers is always an odd integer. 25. Show that the sum of any three consecutive integers is always divisible by 3. Review 26. Shannon and Erin are hiking up a mountain. Of course, they are packing the clinometer they made in geometry class. At point A along a flat portion of the trail, Erin sights the mountain peak straight ahead at an angle of elevation of 22°. The level trail continues 220 m straight to the base of the mountain at point B. At that point, Shannon measures the angle of elevation to be 38°. From B the trail follows a ridge straight |
up the mountain to the peak. At point B, how far are they from the mountain peak? T rail 22° 38° A B Trail 220 m 27. 12 cm 6 cm 5 cm 5.5 cm 5.5 cm Cone Sphere Cylinder Arrange the names of the solids in order, greatest to least. Volume: Surface area: Length of the longest rod that will fit inside:????????? 28. Two communication towers stand 64 ft apart. One is 80 ft high and the other is 48 ft high. Each has a guy wire from its top anchored to the base of the other tower. At what height do the two guy wires cross? 80 ft 48 ft? 64 ft LESSON 13.1 The Premises of Geometry 677 In Exercises 29 and 30, all length measurements are given in meters. 29. What’s wrong with this picture? 30. Find angle measures x and y, and length a. E D 6 2 6 140° C 2 3 G B F 52° A 6.8 x 7.3 a 27° y 31. Each arc is a quarter of a circle with its center at a vertex of the square. Given: Each square has side length 1 unit Find: The shaded area a. b. c. 1 1 1 IMPROVING YOUR REASONING SKILLS Logical Vocabulary Here is a logical vocabulary challenge. It is sometimes possible to change one word to another of equal length by changing one letter at a time. Each change, or move, you make gives you a new word. For example, DOG can be changed to CAT in exactly three moves. DOG ⇒ DOT ⇒ COT ⇒ CAT Change MATH to each of the following words in exactly four moves. 1. MATH ⇒? ⇒? ⇒? ⇒ ROSE 3. MATH ⇒? ⇒? ⇒? ⇒ HOST 5. MATH ⇒? ⇒? ⇒? ⇒ LIVE 2. MATH ⇒? ⇒? ⇒? ⇒ CORE 4. MATH ⇒? ⇒? ⇒? ⇒ LESS Now create one of your own. Change MATH to another word in four moves. 678 CHAPTER 13 Geometry as a Mathematical System L E S S O N 13.2 What is now proved was once only imagined. WILLIAM BLAKE Planning a Ge |
ometry Proof A proof in geometry consists of a sequence of statements, starting with a given set of premises and leading to a valid conclusion. Each statement follows from one or more of the previous statements and is supported by a reason. A reason for a statement must come from the set of premises that you learned about in Lesson 13.1. In earlier chapters you informally proved many conjectures. Now you can formally prove them, using the premises of geometry. In this lesson you will identify for yourself what is given and what you must show, in order to prove a conjecture. You will also create your own labeled diagrams As you have seen, you can state many geometry conjectures as conditional statements. For example, you can write the conjecture “Vertical angles are congruent” as a conditional statement: “If two angles are vertical angles, then they are congruent.” To prove that a conditional statement is true, you assume that the first part of the conditional is true, then logically demonstrate the truth of the conditional’s second part. In other words, you demonstrate that the first part implies the second part. The first part is what you assume to be true in the proof; it is the given information. The second part is the part you logically demonstrate in the proof; it is what you want to show Given Show Two angles are vertical angles They are congruent Next, draw and label a diagram that illustrates the given information. Then, use the labels in the diagram to restate graphically what is given and what you must show. Once you’ve created a diagram to illustrate your conjecture and you know where to start and where to go, make a plan. Use your plan to write the proof. Here’s the complete process. Writing a Proof Task 1 From the conditional statement, identify what is given and what you must show. Task 2 Draw and label a diagram to illustrate the given information. Task 3 Restate what is given and what you must show in terms of your diagram. Task 4 Plan a proof. Organize your reasoning mentally or on paper. Task 5 From your plan, write a proof. LESSON 13.2 Planning a Geometry Proof 679 In Chapter 2, you proved the Vertical Angles Conjecture using conjectures that have now become postulates. Lines m and n intersect to form vertical angles 1 and 2 Given Start a theorem list separate from your conjecture list. 1 2 3 m n 1 and 3 are supplementary 3 and 2 are |
supplementary Linear Pair Postulate m1 m3 180° m3 m2 180° Definition of supplementary So the Vertical Angles Conjecture becomes the Vertical Angles (VA) Theorem. It is important when building your mathematical system that you use only the premises of geometry. These include theorems, but not unproved conjectures. You can use all the theorems on your theorem list as premises for proving other theorems. For instance, in Example A you can use the VA Theorem to prove another theorem. m1 m3 m3 m2 Transitive property of equality m1 m2 Subtraction property of equality 1 2 Definition of congruence You may have noticed that in the previous lesson we stated the CA Conjecture as a postulate, but not the AIA Conjecture or the AEA Conjecture. In this first example you will see how to use the five tasks of the proof process to prove the AIA Conjecture. EXAMPLE A Prove the Alternate Interior Angles Conjecture: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Solution For Task 1 identify what is given and what you must show. Given: Two parallel lines are cut by a transversal Show: Alternate interior angles formed by the lines are congruent For Task 2 draw and label a diagram. 3 1 1 2 2 3 680 CHAPTER 13 Geometry as a Mathematical System For Task 3 restate what is given and what you must show in terms of the diagram. Given: Parallel lines angles 1 and 2 2 cut by transversal 3 to form alternate interior 1 and Show: 1 2 For Task 4 plan a proof. Organize your reasoning mentally or on paper. Plan: I need to show that 1 2. Looking over the postulates and theorems, the ones that look useful are the CA Postulate and the VA Theorem. From the CA Postulate, I know that 2 3 and from the VA Theorem, 1 3. If 2 3 and 1 3, then by substitution 1 2. For Task 5 create a proof from your plan. Flowchart Proof 2, with transversal 1 forming AIA 1 and 2 Given 3, 1 3 Vertical Angles Theorem 3 2 Corresponding Angles Postulate 1 2 Transitive property of congruence So the AIA Conjecture becomes the AIA Theorem. Add this theorem to your theorem list. In Chapter |
4, you informally proved the Triangle Sum Conjecture. The proof is short, but clever, too, because it required the construction of an auxiliary line. All the steps in the proof use properties that we now designate as postulates. Example B shows the flowchart proof. For example, the Parallel Postulate guarantees that it will always be possible to construct an auxiliary line through a vertex, parallel to the opposite side. LESSON 13.2 Planning a Geometry Proof 681 EXAMPLE B Prove the Triangle Sum Conjecture: The sum of the measures of the angles of a triangle is 180°. Solution Given: 1, 2, and 3 are the three angles of ABC Show: m1 m2 m3 180° K 4 C 5 2 A 1 1, 2, and 3 of ABC Given Construct KC AB Parallel Postulate 3 B m1 m2 m3 180° Substitution property of equality 1 4 3 5 AIA Theorem m4 m2 mKCB Angle Addition Postulate KCB and 5 are supplementary Linear Pair Postulate m1 m4 m3 m5 Definition of congruence m4 m2 m5 180° Substitution property of equality mKCB m5 180° Definition of supplementary So the Triangle Sum Conjecture becomes the Triangle Sum Theorem. Add it to your theorem list. Notice that each reason we now use in a proof is a postulate, theorem, definition, or property. To make sure a particular theorem has been properly proved, you can also check the “logical family tree” of the theorem. When you create a family tree for a theorem, you trace it back to all the postulates that the theorem relied on. You don’t need to list all the definitions and properties of equality and congruence: list only the theorems and postulates used in the proof. For the theorems that were used in the proof, what postulates and theorems were used in their proofs, and so on. In Chapter 4, you informally proved the Third Angle Conjecture. Let’s look again at the proof. Third Angle Conjecture: If two angles of one triangle are congruent to two angles of a second triangle, then the third pair of angles are congruent. C F A B D E When you trace back your family tree, you include the names of your parents, the names of their parents, and so on. 682 CHAPTER 13 |
Geometry as a Mathematical System ABC and DEF with A D and B E Given mA mD mB mE Definition of congruence mC mF Subtraction property of equality C F Definition of congruence mA mB mC 180° mD mE mF 180° mA mB mC mD mE mF Triangle Sum Theorem Transitive property of equality What does the logical family tree of the Third Angle Theorem look like? You start by putting the Third Angle Theorem in a box. Find all the postulates and theorems used in the proof. The only postulate or theorem used was the Triangle Sum Theorem. Put that box above it. Triangle Sum Theorem Third Angle Theorem Next, locate all the theorems and postulates used to prove the Triangle Sum Theorem. That proof used the Parallel Postulate, the Linear Pair Postulate, the Angle Addition Postulate, and the AIA Theorem. You place these postulates in boxes above the Triangle Sum Theorem. Connect the boxes with arrows showing the logical connection. Now the family tree looks like this: Parallel Postulate AIA Theorem Linear Pair Postulate Angle Addition Postulate Triangle Sum Theorem Third Angle Theorem To prove the AIA Theorem, we used the CA Postulate and the VA Theorem, and we used the Linear Pair Postulate to prove the VA Theorem. Notice that the Linear Pair Postulate is already in the family tree, but we move it up so it’s above both the Triangle Sum Theorem and the VA Theorem. The completed family tree looks like this: Corresponding Angles Postulate Vertical Angles Theorem Linear Pair Postulate Parallel Postulate AIA Theorem Angle Addition Postulate Triangle Sum Theorem Third Angle Theorem LESSON 13.2 Planning a Geometry Proof 683 The family tree shows that, ultimately, the Third Angle Theorem relies on the Parallel Postulate, the CA Postulate, the Linear Pair Postulate, and the Angle Addition Postulate. You might notice that the family tree of a theorem looks similar to a flowchart proof. The difference is that the family tree focuses on the premises and traces them back to the postulates. EXERCISES 1. What postulate(s) does the VA Theorem rely on? 2. What postulate(s) does the Triangle Sum Theorem rely on? 3. If you need a parallel line in a proof |
, what postulate allows you to construct it? 4. If you need a perpendicular line in a proof, what postulate allows you to construct it? In Exercises 5–14, write a paragraph proof or a flowchart proof of the conjecture. Once you have completed their proofs, add the statements to your theorem list. 5. If two angles are both congruent and supplementary, then each is a right angle. (Congruent and Supplementary Theorem) 6. Supplements of congruent angles are congruent. (Supplements of Congruent Angles Theorem) 7. All right angles are congruent. (Right Angles Are Congruent Theorem) 8. If two lines are cut by a transversal forming congruent alternate interior angles, then the lines are parallel. (Converse of the AIA Theorem) 9. If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. (AEA Theorem) 10. If two lines are cut by a transversal forming congruent alternate exterior angles, then the lines are parallel. (Converse of the AEA Theorem) 11. If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. (Interior Supplements Theorem) 12. If two lines are cut by a transversal forming interior angles on the same side of the transversal that are supplementary, then the lines are parallel. (Converse of the Interior Supplements Theorem) 13. If two lines in the same plane are parallel to a third line, then they are parallel to each other. (Parallel Transitivity Theorem) 684 CHAPTER 13 Geometry as a Mathematical System 14. If two lines in the same plane are perpendicular to a third line, then they are parallel to each other. (Perpendicular to Parallel Theorem) 15. Draw a family tree of the Converse of the Alternate Exterior Angles Theorem. Review 16. Suppose the top of a pyramid with volume 1107 cm3 is sliced off and discarded. The remaining portion is called a truncated pyramid. If the cut was parallel to the base and two-thirds of the distance to the vertex, what is the volume of the truncated pyramid? 17. Abraham is building a dog house for his terrier. His plan is shown at right. He will cut a door and |
a window later. After he builds the frame for the structure, can he complete it using one piece of 4-by-8-foot plywood? If the answer is yes, show how he should cut the plywood. If no, explain why not. 18. Find x. 112° x 1 ft 1_ 1 ft 2 3 ft 2 ft 19. M is the midpoint of AC and BD. For each statement, select D always (A), sometimes (S), or never (N). a. BAD and ADC are supplementary. b. ADM and MAD are complementary. c. AD BC AC d. AD CD AC C M B A IMPROVING YOUR REASONING SKILLS Calculator Cunning Using the calculator shown at right, what is the largest number you can form by pressing the keys labeled 1, 2, and 3 exactly once each and the key labeled yx at most once? You cannot press any other keys. LESSON 13.2 Planning a Geometry Proof 685 L E S S O N 13.3 The most violent element in our society is ignorance. EMMA GOLDMAN Triangle Proofs Now that the theorems from the previous lesson have been proved, you should add them to your theorem list. They will be useful to you in proving future theorems. Triangle congruence is so useful in proving other theorems that we will focus next on triangle proofs. You may have noticed that in Lesson 13.1, three of the four triangle congruence conjectures were stated as postulates (the SSS Congruence Postulate, the SAS Congruence Postulate, and the ASA Congruence Postulate). The SAA Conjecture was not stated as a postulate. In Lesson 4.5, you used the ASA Conjecture (now the ASA Postulate) to explain the SAA Conjecture. The family tree for SAA congruence looks like this: ASA Postulate AIA Postulate Triangle Sum Theorem Third Angle Theorem SAA Congruence Theorem EXAMPLE Solution So the SAA Conjecture becomes the SAA Theorem. Add this theorem to your theorem list. This theorem will be useful in some of the proofs in this lesson. Let’s use the five-task proof process and triangle congruence to prove the Angle Bisector Conjecture. Prove the Angle Bisector Conjecture: Any point on the bisector of an angle is equidistant |
from the sides of the angle. Given: Any point on the bisector of an angle Show: The point is equidistant from the sides of the angle Q P A R Given: AP bisecting QAR Show: P is equally distant from sides AQ and AR 686 CHAPTER 13 Geometry as a Mathematical System Plan: The distance from a point to a line is measured along the perpendicular from the point to the line. So I begin by constructing PB AQ and PC AR (the Perpendicular Postulate permits me to do this). I can show that PB PC if they are corresponding parts of congruent triangles. AP AP by the identity property of congruence, and QAP RAP by the definition of an angle bisector. ABP and ACP are right angles and thus they are congruent. So ABP ACP by the SAA Theorem. If the triangles are congruent, then PB PC by CPCTC. Based on this plan, I can write a flowchart proof. AP bisects QAR Given Construct perpendiculars from P to sides of angle. PC AC and PB AB Perpendicular Postulate QAP RAP AP AP Definition of angle bisector Identity property of congruence ABP ACP Right Angles Congruent Theorem mABP 90° mACP 90° Definition of perpendicular ABP ACP SAA Theorem PB PC CPCTC Thus the Angle Bisector Conjecture becomes the Angle Bisector Theorem. As our own proofs build on each other, flowcharts can become too large and awkward. You can also use a two-column format for writing proofs. A two-column proof is identical to a flowchart or paragraph proof, except that the statements are listed in the first column, each supported by a reason (a postulate, definition, property, or theorem) in the second column. Here is the same proof from the example above, following the same plan, presented as a two-column proof. Arrows link the steps. Statement 1. AP bisects QAR 2. AP AP 3. QAP RAP 4. Construct perpendiculars from P to sides of angle so that PC AC and PB AB 5. mABP 90°, mACP 90° 6. ABP ACP 7. ABP ACP 8. PB PC Reason 1. Given 2. Identity property of congruence 3. Definition of angle bisector 4. Perpendicular Postulate 5. Definition |
of perpendicular 6. Right Angles Congruent Theorem 7. SAA Theorem 8. CPCTC LESSON 13.3 Triangle Proofs 687 Compare the two-column proof you just saw with the flowchart proof in Example A. What similarities do you see? What are the advantages of each format? No matter what format you choose, your proof should be clear and easy for someone to follow. EXERCISES In Exercises 1–13, write a proof of the conjecture. Once you have completed the proofs, add the theorems to your list. You will need Geometry software for Exercise 22 1. If a point is on the perpendicular bisector of a segment, then it is equally distant from the endpoints of the segment. (Perpendicular Bisector Theorem) 2. If a point is equally distant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. (Converse of the Perpendicular Bisector Theorem) 3. If a triangle is isosceles, then the base angles are congruent. (Isosceles Triangle Theorem) 4. If two angles of a triangle are congruent, then the triangle is isosceles. (Converse of the Isosceles Triangle Theorem) 5. If a point is equally distant from the sides of an angle, then it is on the bisector of the angle. (Converse of the Angle Bisector Theorem) 6. The three perpendicular bisectors of the sides of a triangle are concurrent. (Perpendicular Bisector Concurrency Theorem) 7. The three angle bisectors of the sides of a triangle are concurrent. (Angle Bisector Concurrency Theorem) 8. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. (Triangle Exterior Angle Theorem) 9. The sum of the measures of the four angles of a quadrilateral is 360°. (Quadrilateral Sum Theorem) 688 CHAPTER 13 Geometry as a Mathematical System 10. In an isosceles triangle, the medians to the congruent sides are congruent. (Medians to the Congruent Sides Theorem) 11. In an isosceles triangle, the angle bisectors to the congruent sides are congruent. (Angle Bisectors to the Congru |
ent Sides Theorem) 12. In an isosceles triangle, the altitudes to the congruent sides are congruent. (Altitudes to the Congruent Sides Theorem) 13. In Lesson 4.8, you were asked to complete informal proofs of these two conjectures: The bisector of the vertex angle of an isosceles triangle is also the median to the base. The bisector of the vertex angle of an isosceles triangle is also the altitude to the base. To demonstrate that the altitude to the base, the median to the base, and the bisector of the vertex angle are all the same segment in an isosceles triangle, you really need to prove three theorems. One possible sequence is diagrammed at right. Prove the three theorems that confirm the conjecture, then add it as a theorem to your theorem list. (Isosceles Triangle Vertex Angle Theorem) If CD is a median, then CD is an angle bisector. If CD is an altitude, then CD is a median. C: A: B: If CD is an angle bisector, then CD is an altitude. Review 14. Find x and y. y x 3 5 12 15. Two bird nests, 3.6 m and 6.1 m high, are on trees across a pond from each other, at points P and Q. The distance between the nests is too wide to measure directly (and there is a pond between the trees). A birdwatcher at point R can sight each nest along a dry path. RP 16.7 m and RQ 27.4 m. QPR is a right angle. What is the distance d between the nests? 16. Apply the glide reflection rule twice to find the first and second images of the point A(2, 9). Glide reflection rule: A reflection across the line x y 5 and a translation (x, y) → (x 4, y 4). 6.1 m Q m 27.4 d R 3.6 m P 1 6.7 m LESSON 13.3 Triangle Proofs 689 17. Explain why 1 2. Given: A 1 B B, G, F, E are collinear mDFE 90° BC FC AF BC BE CD AB FC ED G 2 F C E D 18. Each arc is a quarter of a circle with its center at a vertex of the square. Given: The square has side |
length 1 unit Find: The shaded area a. Shaded area? b. Shaded area? c. Shaded area? 1 1 1 19. Given an arc of a circle on patty paper but not the whole circle or the center, fold the paper to construct a tangent at the midpoint of the arc. A B 20. Find mBAC in this right rectangular prism. A 3 5 B C 12 21. Choose A if the value of the expression is greater in Figure A. Choose B if the value of the expression is greater in Figure B. Choose C if the values are equal for both figures. Choose D if it cannot be determined which value is greater. Y, O, X are collinear. 10 cm W 40° X Y O Y 20 cm W X 10° O Figure A Figure B a. Perimeter of WXY b. Area of XOW 690 CHAPTER 13 Geometry as a Mathematical System 22. Technology Use geometry software to construct a circle and label any three points on the circle. Construct tangents at those three points to form a circumscribed triangle and connect the points of tangency to form an inscribed triangle. a. Drag the points and observe the angle b. What is the relationship between the angle measures of a circumscribed quadrilateral and the inscribed quadrilateral formed by connecting the points of tangency? measures of each triangle. What relationship do you notice between x, a, and c? Is the same true for y and z IMPROVING YOUR VISUAL THINKING SKILLS Picture Patterns III Sketch the figure that goes in box 12 below 10 11 12 LESSON 13.3 Triangle Proofs 691 L E S S O N 13.4 All geometric reasoning is, in the last result, circular. BERTRAND RUSSELL Quadrilateral Proofs In Chapter 5, you discovered and informally proved several quadrilateral properties. As reasons for the statements in some of these proofs, you used conjectures that are now postulates or that you have proved as theorems. So those steps in the proofs are valid. Occasionally, however, you may have used unproven conjectures as reasons. In this lesson you will write formal proofs of some of these quadrilateral conjectures, using only definitions, postulates, and theorems. After you have proved the theorems, you’ll create a family tree tracing them back to postulates and properties. You can prove many quadr |
ilateral theorems by using triangle theorems. For example, you can prove some parallelogram properties by using the fact that a diagonal divides a parallelogram into two congruent triangles. In the example below, we’ll prove this fact as a lemma. A lemma is an auxiliary theorem used specifically to prove other theorems. EXAMPLE Prove: A diagonal of a parallelogram divides the parallelogram into two congruent triangles. Solution Given: Parallelogram ABCD with diagonal AC Prove: ABC CDA D C Two-column Proof Statement 1. ABCD is a parallelogram 2. AB DC and AD BC 3. CAB ACD and BCA DAC 4. AC AC 5. ABC CDA A B Reason 1. Given 2. Definition of parallelogram 3. AIA Theorem 4. Identity property of congruence 5. ASA Congruence Postulate We’ll call the lemma proved in the example the Parallelogram Diagonal Lemma. You can now use it to prove other parallelogram conjectures in the investigation. Investigation Proving Parallelogram Conjectures This investigation is really a proof activity. You will work with your group to prove three of your previous conjectures about parallelograms. Before you try to prove each conjecture, remember to draw a diagram, restate what is given and what you must show in terms of your diagram, and then make a plan. Step 1 The Opposite Sides Conjecture states that the opposite sides of a parallelogram are congruent. Write a two-column proof of this conjecture. 692 CHAPTER 13 Geometry as a Mathematical System Step 2 Step 3 The Opposite Angles Conjecture states that the opposite angles of a parallelogram are congruent. Write a two-column proof of this conjecture. State the converse of the Opposite Sides Conjecture. Then write a two-column proof of this conjecture. After you have successfully proved the parallelogram conjectures above, you can call them theorems and add them to your theorem list. Step 4 Create a family tree that shows the relationship among these theorems in Steps 1–3 and that traces each theorem back to the postulates of geometry. EXERCISES In Exercises 1–12, write a two-column proof or a flowchart proof of the conjecture. Once you have completed the proofs, add |
the theorems to your list. 1. If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. (Converse of the Opposite Angles Theorem) 2. If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. (Opposite Sides Parallel and Congruent Theorem) 3. Each diagonal of a rhombus bisects two opposite angles. (Rhombus Angles Theorem) 4. The consecutive angles of a parallelogram are supplementary. (Parallelogram Consecutive Angles Theorem) 5. If a quadrilateral has four congruent sides, then it is a rhombus. (Four Congruent Sides Rhombus Theorem) 6. If a quadrilateral has four congruent angles, then it is a rectangle. (Four Congruent Angles Rectangle Theorem) 7. The diagonals of a rectangle are congruent. (Rectangle Diagonals Theorem) 8. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. (Converse of the Rectangle Diagonals Theorem) 9. The base angles of an isosceles trapezoid are congruent. (Isosceles Trapezoid Theorem) 10. The diagonals of an isosceles trapezoid are congruent. (Isosceles Trapezoid Diagonals Theorem) 11. If a diagonal of a parallelogram bisects two opposite angles, then the parallelogram is a rhombus. (Converse of the Rhombus Angles Theorem) LESSON 13.4 Quadrilateral Proofs 693 12. If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus. (Double-Edged Straightedge Theorem) 13. Create a family tree for the Parallelogram Consecutive Angles Theorem. 14. Create a family tree for the Double-Edged Straightedge Theorem. Review 15. Find the length and the bearing of the resultant V 2. 1 vector V V V 1 has length 5 and a bearing of 40°. 2 has length 9 and |
a bearing of 90°. V1 V1 V2 V2 16. A triangle has vertices A(7, 4), B(3, 2), and C(4, 1). Find the coordinates of the vertices after a dilation with center (8, 2) and scale factor 2. Complete the mapping rule for the above dilation: (x, y) → (?,? ). 17. Yan uses a 40 ft rope to tie his horse to the corner of the barn to which a fence is attached. How many square feet of grazing, to the nearest square foot, does the horse have? 10 ft 12 ft 25 ft 120° 4 0 ft of ro p e 18. Complete the following chart with the symmetries and names of each type of special quadrilateral: parallelogram, rhombus, rectangle, square, kite, trapezoid, and isosceles trapezoid. Name Lines of symmetry Rotational symmetry trapezoid none 1 diagonal 2 bisectors of sides rhombus 4-fold none 694 CHAPTER 13 Geometry as a Mathematical System 19. Consider the rectangular prisms in Figure A and Figure A. Choose A if the value of the expression is greater in Figure A. Choose B if the value of the expression is greater in Figure B. Choose C if the values are equal in both rectangular prisms. Choose D if it cannot be determined which value is greater. X 4 9 5 Z Figure A Y X 6 Y 6 5 Z Figure B a. Measure of XYZ b. Shortest path from X to Y along the surface of the prism 20. Given: A, O, D are collinear GF is tangent to circle O at point D mEOD 38° OB EC GF Find: a. mAEO b. mDGO c. mBOC d. mEAB e. mHED A E G B C F O H D IMPROVING YOUR VISUAL THINKING SKILLS Mental Blocks In the top figure at right, every cube is lettered exactly alike. Copy and complete the two-dimensional representation of one of the cubes to show how the letters are arranged on the six faces. LESSON 13.4 Quadrilateral Proofs 695 Proof as Challenge and Discovery So far, you have proved many theorems that are useful in geometry. You can also use proof to explore and possibly discover interesting properties. You might make a conjecture, and then use |
proof to decide whether or not it is always true. These activities have been adapted from the book Rethinking Proof with The Geometer’s Sketchpad, 1999, by Michael deVilliers. Activity Exploring Properties of Special Constructions Use Sketchpad to construct these figures. Drag them and notice their properties. Then prove your conjectures. Parallelogram Angle Bisectors Construct a parallelogram and its angle bisectors. Label your sketch as shown. D C H E G F A B Step 1 Step 2 Step 3 Step 4 Step 5 Is EFGH a special quadrilateral? Make a conjecture. Drag the vertices around. Are there cases when EFGH does not satisfy your conjecture? Drag so that ABCD is a rectangle. What happens? Drag so that ABCD is a rhombus. What happens? Prove your conjectures for Steps 1–3. Construct the angle bisectors of another polygon. Investigate and write your observations. Make a conjecture and prove it. 696 CHAPTER 13 Geometry as a Mathematical System Parallelogram Squares Construct parallelogram ABCD and a square on each side. Construct the center of each square, and label your sketch as shown. G F C B D H A E Step 6 Step 7 Step 8 Step 9 Connect E, F, G, and H with line segments. (Try using a different color.) Drag the vertices of ABCD. What do you observe? Make a conjecture. Drag your sketch around. Are there cases when EFGH does not satisfy your conjecture? Drag so that A, B, C, and D are collinear. What happens to EFGH? Prove your conjectures for Steps 6 and 7. Investigate other special quadrilaterals and the shapes formed by connecting the centers of the squares on their sides. Write your observations. Make a conjecture and prove it. IMPROVING YOUR ALGEBRA SKILLS A Precarious Proof You have all the money you need. Let h the money you have. Let n the money you need. Most people think that the money they have is some amount less than the money they need. Stated mathematically, h n p for some positive p. If h n p, then h(h n) (n p)(h n) h2 hn hn n2 hp np h2 hn hp hn n2 np h(h n p) n(h n p) Therefore h n |
. So the money you have is equal to the money you need! Is there a flaw in this proof? EXPLORATION Proof as Challenge and Discovery 697 Indirect Proof In the proofs you have written so far, you have shown directly, through a sequence of statements and reasons, that a given conjecture is true. In this lesson you will write a different type of proof, called an indirect proof. In an indirect proof, you show something is true by eliminating all the other possibilities. You have probably used this type of reasoning when taking multiple-choice tests. If you are unsure of an answer, you can try to eliminate choices until you are left with only one possibility. This mystery story gives an example of an indirect proof. L E S S O N 13.5 How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? SHERLOCK HOLMES IN THE SIGN OF THE FOUR BY SIR ARTHUR CONAN DOYLE D etective Sheerluck Holmes and three other people are alone on a tropical island. One morning, Sheerluck entertains the others by playing show tunes on his ukulele. Later that day, he discovers that his precious ukulele has been smashed to bits. Who could have committed such an antimusical act? Sheerluck eliminates himself as a suspect because he knows he didn’t do it. He eliminates his girlfriend as a suspect because she has been with him all day. Colonel Moran recently injured both arms and therefore could not have smashed the ukulele with such force. There is only one other person on the island who could have committed the crime. So Sheerluck concludes that the fourth person, Sir Charles Mortimer, is the guilty one. For a given mathematical statement, there are two possibilities: either the statement is true or it is not true. To prove indirectly that a statement is true, you start by assuming it is not true. You then use logical reasoning to show that this assumption leads to a contradiction. If an assumption leads to a contradiction, it must be false. Therefore, you can eliminate the possibility that the statement is not true. This leaves only one possibility—namely, that the statement is true! T EXAMPLE A Conjecture: If mN mO in NOT, then NT OT. Given: NOT with mN mO Show: NT OT N O 698 CHAPTER 13 Geometry as a Mathematical System |
Solution To prove indirectly that the statement NT OT is true, start by assuming that it is not true. That is, assume NT OT. Then show that this assumption leads to a contradiction. Paragraph Proof Assume NT OT. If NT OT, then mN mO by the Isosceles Triangle Theorem. But this contradicts the given fact that mN mO. Therefore, the assumption NT OT is false and so NT OT is true. Here is another example of an indirect proof. EXAMPLE B Conjecture: The diagonals of a trapezoid do not bisect each other. Given: Trapezoid ZOID with parallel bases ZO and ID and diagonals DO and IZ intersecting at point Y Show: The diagonals of trapezoid ZOID do not bisect each other; that is, DY OY and ZY IY D I Y Z O Solution Paragraph Proof Assume that at least one of the diagonals of trapezoid ZOID does bisect the other. Then DY OY. Also, by the AIA Theorem, DIY OZY, and IDY YOZ. Therefore, DYI OYZ by the SAA Theorem. By CPCTC, ZO ID. It is given that ZO ID. In Lesson 13.4, you proved that if one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. So, ZOID is a parallelogram. Thus, ZOID has two pairs of opposite sides parallel. But because it is a trapezoid, it has exactly one pair of parallel sides. This is contradictory. So the assumption that its diagonals bisect each other is false and the conjecture is true. In the investigation you’ll write an indirect proof of the Tangent Conjecture from Chapter 6. Investigation Proving the Tangent Conjecture This investigation is really a group proof activity. Copy the information and diagram below, then work with your group to complete an indirect proof of the Tangent Conjecture. Conjecture: A tangent is perpendicular to the radius drawn to the point of tangency. Given: Circle O with tangent AT and radius AO Show: AO AT O T A LESSON 13.5 Indirect Proof 699 Paragraph Proof Step 1 Step 2 Step 3 Step 4 Step 5 |
Step 6 Step 7 Step 8 Step 9 T C O Assume AO is not perpendicular to AT. Construct a perpendicular from point O to AT and label the intersection point B (OB AT). Which postulate allows you to do this? Select a point C on AT so that B is the midpoint of AC. Which postulate allows you to do this? Next, construct OC. Which postulate allows you to do this? ABO CBO. Why? AB BC. What definition tells you this? OB OB. What property of congruence tells you this? Therefore, ABO CBO. Which congruence shortcut tells you the triangles are congruent? If ABO CBO, then AO CO. Why? A B C must be a point on the circle (because a circle is the set of all points in the plane at a given distance from the center and points A and C are both the same distance from the center). Therefore, AT intersects the circle in two points (A and C) and thus AT is not a tangent. But this leads to a contradiction. Why? Step 10 Discuss the steps with your group. What was the contradiction? What does it prove? Use the steps above to write a two-column indirect proof of the Tangent Conjecture. Rename the Tangent Conjecture as the Tangent Theorem and add it to your list of theorems. EXERCISES For Exercises 1 and 2, the correct answer is one of the choices listed. Determine the correct answer by indirect reasoning, explaining how you eliminated each incorrect choice. 1. Which is the capital of Mali? A. Paris B. Tucson C. London D. Bamako 2. Which Italian scientist used a new invention called the telescope to discover the moons of Jupiter? A. Sir Edmund Halley B. Julius Caesar C. Galileo Galilei D. Madonna 700 CHAPTER 13 Geometry as a Mathematical System 3. Is the proof in Example A claiming that if two angles of a triangle are not congruent, then the triangle is not isosceles? Explain. 4. Is the proof in Example B claiming that if one diagonal of a quadrilateral bisects the other, then the quadrilateral is not a trapezoid? Explain. 5. Fill in the blanks in the indirect proof below. Conjecture: No triangle has two right angles. Given: ABC Show: No two angles are right angles Two-column proof Statement 1. |
Assume ABC has two right angles (Assume mA 90° and mB 90° and 0° mC 180°.) 2. mA mB mC 180° 3. 90° 90° mC 180° 4. mC? Reason 1.? 2.? 3.? 4.? C A B But if mC 0, then the two sides AC and BC coincide, and thus there is no angle at C. This contradicts the given information. So the assumption is false. Therefore, no triangle has two right angles. 6. Write an indirect proof of the conjecture below. D I Conjecture: No trapezoid is equiangular. Given: Trapezoid ZOID with bases ZO and ID Show: ZOID is not equiangular 7. Write an indirect proof of the conjecture below. Conjecture: In a scalene triangle, the median cannot be the altitude. Given: Scalene triangle ABC with median CD Show: CD is not the altitude to AB 8. Write an indirect proof of the conjecture below. Conjecture: The bases of a trapezoid have unequal lengths. Given: Trapezoid ZOID with parallel bases and ZO and ID Show: ZO ID. Write the “given” and the “show,” and then plan and write the proof of the Perpendicular Bisector of a Chord Conjecture: The perpendicular bisector of a chord passes through the center of the circle. LESSON 13.5 Indirect Proof 701 Review 10. Find a, b, and c. 58° a b c 105° 11. A clear plastic container is in the shape of a right cone atop a right cylinder, and their bases coincide. Find the volume of the container. 5 ft 3 ft 3 ft 12. Each arc is a quarter of a circle with its center at a vertex of the square. Given: The square has side length 1 unit Find: The shaded area a. Shaded area? b. Shaded area? 1 1 13. For each statement, select always (A), sometimes (S), or never (N). a. An angle inscribed in a semicircle is a right angle. b. An angle inscribed in a major arc is obtuse. c. An arc measure equals the measure of its central angle. d. The measure of an angle formed by two intersecting chords equals the measure of its intercepted arc. e. The measure of |
the angle formed by two tangents to a circle equals the supplement of the central angle of the minor intercepted arc. IMPROVING YOUR REASONING SKILLS Symbol Juggling If V 1 BH, B 1 h(a b), h 2x, a 2b, b x, and Hx 12, find the value 2 3 of V in terms of x. 702 CHAPTER 13 Geometry as a Mathematical System L E S S O N 13.6 When people say,“It can’t be done,” or “You don’t have what it takes,” it makes the task all the more interesting. LYNN HILL Circle Proofs In Chapter 6, you completed the proof of the three cases of the Inscribed Angle Conjecture: The measure of an inscribed angle in a circle equals half the measure of its intercepted arc. There was a lot of algebra in the proof. You may not have noticed that the Angle Addition Postulate was used, as well as a property that we called arc addition. Arc Addition is a postulate that you need to add to your list. Arc Addition Postulate If point B is on AC and between points A and C, then mAB mBC mAC. Is the proof of the Inscribed Angle Conjecture now completely supported by the premises of geometry? Can you call it a theorem? To answer these questions, trace the family tree. SSS Congruence Postulate Linear Pair Postulate CA Postulate Parallel Postulate Angle Addition Postulate Arc Addition Postulate VA Theorem AIA Theorem Triangle Sum Theorem Isosceles Triangle Theorem Exterior Angle Theorem Inscribed Angle Theorem So the Inscribed Angle Conjecture is completely supported by premises of geometry; therefore you can call it a theorem and add it to your theorem list. A double rainbow creates arcs in the sky over Stonehenge near Wiltshire, England. Built from bluestone and sandstone from 3000 to 1500 B.C.E., Stonehenge itself is laid out in the shape of a major arc. LESSON 13.6 Circle Proofs 703 In the exercises, you will create proofs or family trees for many of your earlier discoveries about circles. EXERCISES In Exercises 1–7, set up and write a proof of each conjecture. Once you have completed the proofs, add the theorems to your list. 1. Inscribed angles that |
intercept the same or congruent arcs are congruent. (Inscribed Angles Intercepting Arcs Theorem) 2. The opposite angles of an inscribed quadrilateral are supplementary. (Cyclic Quadrilateral Theorem) You will need Construction tools for Exercise 16 Geometry software for Exercise 14 3. Parallel lines intercept congruent arcs on a circle. (Parallel Secants Congruent Arcs Theorem) 4. If a parallelogram is inscribed within a circle, then the parallelogram is a rectangle. (Parallelogram Inscribed in a Circle Theorem) 5. Tangent segments from a point to a circle are congruent. (Tangent Segments Theorem) 6. The measure of an angle formed by two intersecting chords is half the sum of the measures of the two intercepted arcs. (Intersecting Chords Theorem) 7. Write and prove a theorem about the arcs intercepted by secants intersecting outside a circle, and the angle formed by the secants. (Intersecting Secants Theorem) 8. Create a family tree for the Tangent Segments Theorem. 9. Create a family tree for the Parallelogram Inscribed in a Circle Theorem. Review 10. Find the coordinates of A and P to the nearest tenth. 11. Given: AX 6, XB 2, BC 4, ZC 3 Find: BY?, YC?, AZ? y 5 36° A (a, b) x 3 22° P (p, q) A X B Z Y C 704 CHAPTER 13 Geometry as a Mathematical System 12. List the five segments in order from shortest to longest. 13. AB is a common external tangent. Find the length of AB (to a tenth of a unit). C 60° D 57° B 59° 61° A A 10 B 34 20 14. Technology P is any point inside an equilateral triangle. Is there a relationship between the height h and the sum a b c? Use geometry software to explore the relationship and make a conjecture. Then write a proof of your conjecture. 15. Find each measure or conclude that it “cannot be determined.” a. mP c. mQRN e. mONF b. mQON d. mQMP f. mMN mNQ 94° mOM 40° NF is a tangent F 16. Construction Use a compass and straightedge to construct the |
two tangents to a circle from a point outside the circle. IMPROVING YOUR REASONING SKILLS Seeing Spots The arrangement of green and yellow spots at right may appear to be random, but there is a pattern. Each row is generated by the row immediately above it. Find the pattern and add several rows to the arrangement. Do you think a row could ever consist of all yellow spots? All green spots? Could there ever be a row with one green spot? Does a row ever repeat itself? LESSON 13.6 Circle Proofs 705 L E S S O N 13.7 Mistakes are part of the dues one pays for a full life. SOPHIA LOREN Similarity Proofs To prove conjectures involving similarity, we need to extend the properties of equality and congruence to similarity. Properties of Similarity Reflexive property of similarity Any figure is similar to itself. Symmetric property of similarity If Figure A is similar to Figure B, then Figure B is similar to Figure A. Transitive property of similarity If Figure A is similar to Figure B and Figure B is similar to Figure C, then Figure A is similar to Figure C. The AA Similarity Conjecture is actually a similarity postulate. AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. In Chapter 11, you also discovered the SAS and SSS shortcuts for showing that two triangles are similar. In the example that follows, you will see how to use the AA Similarity Postulate to prove the SAS Similarity Conjecture, making it the SAS Similarity Theorem. F EXAMPLE Prove the SAS Similarity Conjecture: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the two triangles are similar. C D E D E A B Solution C B AB Given: ABC and DEF so that and B E E F D E Show: ABC DEF Plan: The only shortcut for showing that two triangles are similar is the AA Postulate, so you need to find another pair of congruent angles. One way of getting two congruent angles is to find two congruent triangles. You can draw a triangle within ABC that is congruent to DEF. The Segment Duplication Postulate allows you to locate a point P on AB so that PB DE. The Parallel Postulate allows you to construct a line |
PQ parallel to AC. Then A QPB by the CA Postulate. 706 CHAPTER 13 Geometry as a Mathematical System C Q F A P B D E Now, if you can show that PBQ DEF, then you will have two congruent pairs of angles to prove ABC DEF. So, how do you show that PBQ DEF? If you can get ABC PBQ, then A C. It is given B B Q B PB C B AB, and you constructed PB DE. With some algebra and that E F D E substitution, you can get EF BQ. Then the two triangles will be congruent by the SAS Congruence Postulate. Here is the two-column proof. Statement 1. Locate P so that PB DE 2. Construct PQ AC 3. A QPB 4. B B 5. ABC PBQ B 6. A B C PB B Q B AB C 7. B E D Q C B AB 8. F E Q B 10. BQ EF 11. B E 12. PBQ DEF 13. QPB D 14. A D 15. ABC DEF Reason 1. Segment Duplication Postulate 2. Parallel Postulate 3. CA Postulate 4. Identity property of congruence 5. AA Similarity Postulate 6. Corresponding sides of similar triangles are proportional (CSSTP) 7. Substitution 8. Given 9. Transitive property of equality 10. Algebra operations 11. Given 12. SAS Congruence Postulate 13. CPCTC 14. Substitution 15. AA Similarity Postulate This proves the SAS Similarity Conjecture. The proof in the example above may seem complicated, but it relies on triangle congruence and triangle similarity postulates. Reading the plan again can help you follow the steps in the proof. You can now call the SAS Similarity Conjecture the SAS Similarity Theorem and add it to your theorem list. In this investigation you will use the SAS Similarity Theorem to prove the SSS Similarity Conjecture. LESSON 13.7 Similarity Proofs 707 Investigation Can You Prove the SSS Similarity Conjecture? Similarity proofs can be challenging. Follow the steps and work with your group to prove the SSS Similarity Conjecture: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar. Step 1 Step 2 Identify the given and show. |
Restate what is given and what you must show in terms of this diagram. M Q K P N L Step 3 Plan your proof. (Hint: Use an auxiliary line like the one in the example.) M Q S N L P K R Step 4 Copy the first ten statements and provide the reasons. Then write the remaining steps and reasons necessary to complete the proof. Statement 1. Locate R so that RL NP 2. Construct RS KM 3. SRL K 4. RSL M 5. KLM RLS K M M L L 6. K L RL SR S M L KL 7. L S P N KL M L 8. N P Q P KL K M 9. N P SR KL K M 10. P N QN … Reason 1.? Postulate 2.? Postulate 3.? Postulate 4.? Postulate 5.? 6.? 7.? 8.? 9.? 10.? … Step 5 Draw arrows to show the flow of logic in your two-column proof. 708 CHAPTER 13 Geometry as a Mathematical System When you have completed the proof, you can call the SSS Similarity Conjecture the SSS Similarity Theorem and add it to your theorem list. EXERCISES In Exercises 1 and 2, write a proof and draw the family tree of each theorem. If the family tree is completely supported by theorems and postulates, add the theorem to your list. You will need Geometry software for Exercises 17 and 21 1. If two triangles are similar, then corresponding altitudes are proportional to the corresponding sides. (Corresponding Altitudes Theorem) 2. If two triangles are similar, then corresponding medians are proportional to the corresponding sides. (Corresponding Medians Theorem) In Exercises 3–10, write a proof of the conjecture. Once you have completed the proofs, add the theorems to your list. As always, you may use theorems that have been proved in previous exercises in your proofs. 3. If two triangles are similar, then corresponding angle bisectors are proportional to the corresponding sides. (Corresponding Angle Bisectors Theorem) 4. If a line passes through two sides of a triangle parallel to the third side, then it divides the two sides proportionally. (Parallel/Proportionality Theorem) 5. If a line passes through two sides of a triangle dividing them proportionally, then it |
is parallel to the third side. (Converse of the Parallel/Proportionality Theorem) 6. If you drop an altitude from the vertex of a right angle to its hypotenuse, then it divides the right triangle into two right triangles that are similar to each other and to the original right triangle. (Three Similar Right Triangles Theorem) 7. The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments on the hypotenuse. (Altitude to the Hypotenuse Theorem) 8. The Pythagorean Theorem 9. Converse of the Pythagorean Theorem 10. If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. (Hypotenuse Leg Theorem) 11. Create a family tree for the Parallel/Proportionality Theorem. 12. Create a family tree for the SSS Similarity Theorem. 13. Create a family tree for the Pythagorean Theorem. This monument in Wellington, New Zealand, was designed by Maori architect Rewi Thompson. How would you describe the shape of the monument? How might the artist have used geometry in planning the construction? LESSON 13.7 Similarity Proofs 709 Review 14. A circle with diameter 9.6 cm has two parallel chords with lengths 5.2 cm and 8.2 cm. How far apart are the chords? Find two possible answers. 15. Choose A if the value is greater in the regular hexagon. Choose B if the value is greater in the regular pentagon. Choose C if the values are equal in both figures. Choose D if it cannot be determined which value is greater. 10 cm 12 cm Regular hexagon Regular pentagon a. Perimeter d. Sum of interior angles b. Apothem e. Sum of exterior angles c. Area 16. Mini-Investigation Cut out a small nonsymmetric concave quadrilateral. Label the vertices A, B, C, D. Use your cut-out as a template to create a tessellation. Trace about 10 images that fit together to cover part of the plane. Number the vertices of each image to match the numbers on your cut-out. a. Draw two different translation vectors that map your tessellation onto itself. How do these two vectors relate to your original |
quadrilateral? b. Pick a quadrilateral in your tessellation. What transformation will map the quadrilateral you picked onto an adjacent quadrilateral? With that transformation, what happens to the rest of the tessellation 17. Technology Use geometry software to draw a small nonsymmetric concave quadrilateral. a. Describe the transformations that will make the figure tessellate. b. Describe two different translation vectors that map your tessellation onto itself. 18. Given: Circles P and Q PS and PT are tangent to circle Q mSPQ 57° 118° mGS Find: a. mGLT c. mTSQ e. Explain why PSQT is cyclic. f. Explain why SQ is tangent. b. mSQT d. mSL 710 CHAPTER 13 Geometry as a Mathematical System L Q N 118° S G A 57° P T In the figures for Exercises 19 and 20, each arc is a quarter of a circle with its center at a vertex of the square. Given: The square has side length 1 unit Find: The shaded area 19. Shaded area? 20. Shaded area? 1 1 21. Technology The diagram below shows a scalene triangle with angle bisector CG, and perpendicular bisector GE of side AB. Study the diagram. a. Which triangles are congruent? b. You can use congruent triangles to prove that C ABC is isosceles. How? c. Given a scalene triangle, you proved that it is isosceles. What’s wrong with this proof? d. Use geometry software to re-create the construction. What does the sketch tell you about what’s wrong? A D F G E B 22. Dakota Davis is at an archaeological dig where he has uncovered a stone voussoir that resembles an isosceles trapezoidal prism. Each trapezoidal face has bases that measure 27 cm and 32 cm, and congruent legs that measure 32 cm each. Help Dakota determine the rise and span of the arch when it was standing, and the total number of voussoirs. Explain your method. IMPROVING YOUR ALGEBRA SKILLS The Eye Should Be Quicker Than the Hand How fast can you answer these questions? 1. If 2x y 12 and 3x 2y 17, what is 5x y? 2. If 4 |
x 5y 19 and 6x 7y 31, what is 10x + 2y? 3. If 3x 2y 11 and 2x y 7, what is x y? LESSON 13.7 Similarity Proofs 711 ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING USING YOUR ALGEBRA SKILLS 10 Coordinate Proof You can prove conjectures involving midpoints, slope, and distance using analytic geometry. When you do this, you create a coordinate proof. Coordinate proofs rely on the premises of geometry, and these three properties from algebra. Coordinate Midpoint Property are the coordinates of the endpoints of a segment, then If x1, y1 y2. x2, y1 the coordinates of the midpoint are x1 2 2 and x2, y2 Parallel Slope Property In a coordinate plane, two distinct lines are parallel if and only if their slopes are equal. Perpendicular Slope Property In a coordinate plane, two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. For coordinate proofs, you also use the coordinate version of the Pythagorean Theorem, the distance formula. Distance Formula The distance between points Ax1, y1 AB2 x2 2 y2 x1 y1 and Bx2, y2 2 or AB x2 x1 is given by 2 y2 y12. The process you use in a coordinate proof contains the same five tasks that you learned in Lesson 13.2. However, in Task 2, you draw and label a diagram on a coordinate plane. Locate the vertices and other points of your diagram such that they reflect the given information, yet their coordinates should not restrict the generality of your diagram. In other words, do not assume any extra properties for your figure, besides the ones given in its definition. EXAMPLE A Write a coordinate proof of the Square Diagonals Conjecture: The diagonals of a square are congruent and are perpendicular bisectors of each other. Solution Task 1 Given: A square with both diagonals Show: The diagonals are congruent and are perpendicular bisectors of each other 712 CHAPTER 13 Geometry as a Mathematical System ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● |
USING YOUR ALGEBRA SKILLS 10 ● USING ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YO Task 2 y y S (0, 0) x S (0, 0) Q (a, 0) x 1. Placing one vertex at the origin will simplify later calculations because it is easy to work with zeros. 2. Placing the second vertex on the x-axis also simplifies calculations because the y-coordinate is zero. To remain general, call the x-coordinate a. y y R (a, a) E (0, a) R (a, a) S (0, 0) x Q (a, 0) S (0, 0) x Q (a, 0) 3. RQ needs to be vertical to form a right angle with SQ, which is horizontal. RQ also needs to be the same length. So R is placed a units vertically above Q. 4. The last vertex is placed a units above S. You can check that SQRE fits the definition of a square—an equiangular, equilateral parallelogram. 0 0 0 Slope of SQ a a 0 0 SQ (a 0)2 (0 0)2 a2 a 0 a a Slope of QR (undefined) 0 a a QR (a a)2 (a 0)2 a2 a USING YOUR ALGEBRA SKILLS 10 Coordinate Proof 713 ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING 0 a Slope of RE a 0 0 a a RE (0 a)2 (a a)2 a2 a Slope of ES 0 a (undefined) a 0 0 0 ES (0 0)2 (0 a)2 a2 a Opposite sides have the same slope and are therefore parallel, so SQRE is a parallelogram. Also, from the slopes, SQ and RE are horizontal and QR and ES are vertical, so all angles are right angles and the parallelogram is equiangular. Lastly, all the sides have the same length, so the parallelogram is equilateral. SQRE is an equiangular, equilateral parallelogram and is a square by definition. Task 3 |
Given: Square SQRE with diagonals SR and QE Show: SR QE, SR and QE bisect each other, and SR QE Task 4 To show that SR QE, you must show that both segments have the same length. To show that SR and QE bisect each other, you must show that the segments share the same midpoint. To show that SR QE, you must show that the segments have negative reciprocal slopes. Because you know the coordinates of the endpoints of both SR and QE, you can do the necessary calculations to use the distance formula, the coordinate midpoint property, and the perpendicular slope property. Task 5 Use the distance formula to find SR and QE. SR (a 0)2 (a 0)2 2a2 a2 QE (a 0)2 (0 a)2 2a2 a2 So, by the definition of congruence, SR QE because both segments have the same length. Use the coordinate midpoint property to find the midpoints of SR and QE. a (0.5a, 0.5a) Midpoint of SR 0 a, 0 2 2 0 (0.5a, 0.5a) Midpoint of QE 0 a, a 2 2 So, SR and QE bisect each other because both segments have the same midpoint. 714 CHAPTER 13 Geometry as a Mathematical System ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING USING YOUR ALGEBRA SKILLS 1 Finally, compare the slopes of SR and QE. 0 Slope of SR a 1 0 a 0 1 Slope of QE a 0 a So, SR QE by the perpendicular slope property because the segments have negative reciprocal slopes. Therefore, the diagonals of a square are congruent and are perpendicular bisectors of each other. Add the Square Diagonals Theorem to your list. Here’s another example. See if you can recognize how the five tasks result in this proof. EXAMPLE B Write a coordinate proof of this conditional statement: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Solution Given: Quadrilateral ABCD with diagonals AC and BD that bisect each other (common midpoint M) Show: |
ABCD is a parallelogram C (–p, 0) y B (r, s) M (0, 0) x A (p, 0) Proof s s Slope of AB r p r 0 p D (–r, –s) r ( s) ( 0 s s Slope of BC p r p p r) ( s) 0 s Slope of CD r ( ( r p p) r ( ) s s Slope of DA p r ( ) r s 0 p p). Opposite sides BC and DA Opposite sides AB and CD have equal slopes of s r p s have equal slopes of. So each pair is parallel by the parallel slope property. p Therefore, quadrilateral ABCD is a parallelogram by definition. Add this theorem to your list. r It is clear from these examples that creating a diagram on a coordinate plane is a significant challenge in a coordinate proof. The first seven exercises will give you some more practice creating these diagrams. USING YOUR ALGEBRA SKILLS 10 Coordinate Proof 715 ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING EXERCISES In Exercises 1–3, each diagram shows a convenient general position of a polygon on a coordinate plane. Find the missing coordinates. 1. Triangle ABC is isosceles. 2. Quadrilateral ABCD is a 3. Quadrilateral ABCD is a y C (0, b) parallelogram. y rhombus. y D (b, c) C (?,?) D (b,?) C (?,?) A (–a, 0) B (?,?) x A (0, 0) B (a, 0) x A (0, 0) B (a, 0) x In Exercises 4–7, draw each figure on a coordinate plane. Assign general coordinates to each point of the figure. Then use the coordinate midpoint property, parallel slope property, perpendicular slope property, and/or the distance formula to check that the coordinates you have assigned meet the definition of the figure. 4. Rectangle RECT 5. Triangle TRI with its three midsegments 6. Isosceles trapezoid TRAP 7. Equilateral triangle EQU In Exercises 8–13, write a coordinate proof of each conjecture. If it cannot be proven, write “c |
annot be proven.” 8. The diagonals of a rectangle are congruent. 9. The midsegment of a triangle is parallel to the third side and half the length of the third side. 10. The midsegment of a trapezoid is parallel to the bases. 11. If only one diagonal of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite. 12. The figure formed by connecting the midpoints of the sides of a quadrilateral is a parallelogram. 13. The quadrilateral formed by connecting the D midpoint of the base to the midpoint of each leg in an isosceles triangle is a rhombus. B A E F C BC E is the midpoint of base. D and F are the midpoints of the legs. 716 CHAPTER 13 Geometry as a Mathematical System ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING YOUR ALGEBRA SKILLS 10 ● USING SPECIAL PROOFS OF SPECIAL CONJECTURES In this project your task is to research and present logical arguments in support of one or more of these special properties. 1. Prove that there are only five regular polyhedra. 2. You discovered Euler’s rule for determining whether a planar network can or cannot be traveled. Write a proof defending Euler’s formula for networks. Yes 3. You discovered that the formula for the sum of the measures of the interior angles of an n-gon is (n 2)180°. Prove that this formula is correct. 4. You discovered that the composition of two reflections over intersecting lines is equivalent to one rotation. Prove that this always works. No M L O 5. The coordinates of the centroid of a triangle are equal to the average of the coordinates of the triangle’s three vertices. Prove that this is always true. 6. Prove that 2 is irrational. C (ea, b) F B (c, d) 7. When you explored all the 1-uniform tilings (Archimedean tilings), you discovered that there are exactly 11 Archimedean tilings of the plane. Prove that there are exactly 11. USING YOUR ALGEBRA SKILLS 10 Coordinate Proof 717 Non-Euclidean Geometries Have you ever changed the |
rules of a game? Sometimes, changing just one simple rule creates a completely different game. You can compare geometry to a game whose rules are postulates. If you change even one postulate, you may create an entirely new geometry. Euclidean geometry—the geometry you learned in this course—is based on several postulates. A postulate, according to the contemporaries of Euclid, is an obvious truth that cannot be derived from other postulates. The list below contains the first five of Euclid’s postulates. Hungarian mathematician János Bolyai, one of the discoverers of hyperbolic geometry, said, “I have discovered such wonderful things that I was amazed... out of nothing I have created a strange new Universe.” Postulate 1: You can draw a straight line through any two points. Postulate 2: You can extend any segment indefinitely. Postulate 3: You can draw a circle with any given point as center and any given radius. Postulate 4: All right angles are equal. Postulate 5: Through a given point not on a given line, you can draw exactly one line that is parallel to the given line. The fifth postulate, known as the Parallel Postulate, does not seem as obvious as the others. In fact, for centuries, many mathematicians did not believe it was a postulate at all and tried to show that it could be proved using the other postulates. Attempting to use indirect proof, mathematicians began by assuming that the fifth postulate was false and then tried to reach a logical contradiction. If the Parallel Postulate is false, then one of these assumptions must be true. Assumption 1: Through a given point not on a given line, you can draw more than one line parallel to the given line. Assumption 2: Through a given point not on a given line, you can draw no line parallel to the given line. 718 CHAPTER 13 Geometry as a Mathematical System Interestingly, neither of these assumptions contradict any of Euclid’s other postulates. Assumption 1 leads to a new deductive system of non-Euclidean geometry, called hyperbolic geometry. Assumption 2 leads to another non-Euclidean system, called elliptic geometry. One model of elliptic geometry applies to lines and angles on a sphere. On Earth, if you walk in a “straight line” indefinitely, what shape will your path take? Theoretically, |
if you walk long enough, you will end up back at the same point, after walking a complete circle around Earth! (Find a globe and check it!) So, on a sphere, a “straight line” is not a line at all, but a circle. Hyperbolic geometry is confined to a circular disk. The edges of the disk represent infinity so lines curve and come to an end at the edge of the circle. This may sound like a strange model, but it fits physicists’ theory that we live in a closed universe. In hyperbolic geometry, many lines can be drawn through a point parallel to another line. Lines p, r, and s all pass through point M, and are parallel to line t. t s M r p In this activity, you will explore elliptic geometry. You will need ● a sphere that you can draw on ● a compass Activity Elliptic Geometry You can use the surface of a sphere as a model to explore elliptic geometry. Of course, you can’t draw a straight line on a sphere. On a plane, the shortest distance between two points is measured along a line. On a sphere, the shortest distance between two points is measured along a great circle. Recall that a great circle is a circle on the surface of a sphere whose diameter passes through the center of the sphere. So, in this elliptic-geometry model, a “segment” is an arc of a great circle. Great circles Segment EXPLORATION Non-Euclidean Geometries 719 In elliptic geometry, “lines” (that is, great circles) never end; however, their length is finite! Because all great circles have the same diameter, all “lines” have the same length. A model for elliptic geometry must satisfy the assumption that, through a given point not on a given line, there are no lines parallel to the given line. Simply put, there are no parallel lines in elliptic geometry. All great circles intersect, so the spherical model of elliptic geometry supports this assumption. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Write a set of postulates for elliptic geometry by rewriting Euclid’s first five postulates. Replace the word line with the words great circle. In Euclidean geometry, two lines that are perpendicular to the same line are parallel to each other. This is not true in elliptic geometry. On your sphere, |
draw an example of two “lines” that are perpendicular to the same “line” but that are not parallel to each other. On your sphere, show that two points do not always determine a unique “line.” Draw an isosceles triangle on your sphere. (Remember, the “segments” that form the sides of a triangle must be arcs of great circles.) Does the Isosceles Triangle Theorem appear to hold in elliptic geometry? In elliptic geometry, the sum of the measures of the three angles of a triangle is always greater than 180°. Draw a triangle on your model and use it to help you explain why this makes sense. In Euclidean geometry, no triangle can have two right angles. But in elliptic geometry, a triangle can have three right angles. Find such a triangle and sketch it. Japanese temari balls, colorful balls made of thread or scrap material, are embroidered with geometric designs derived from nature, like flowers or trees. Also called “princess balls,” they originated in 700 C.E., when young nobility made them from silk and gave them as gifts. Notice that each “line segment” in the design of a temari ball is actually an arc of a great circle. 720 CHAPTER 13 Geometry as a Mathematical System VIEW ● CHAPTER 11 REVIEW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● C CHAPTER 13 R E V I E W In this course you have discovered geometry properties and made conjectures based on inductive reasoning. You have also used deductive reasoning to explain why some of your conjectures were true. In this chapter you have focused on geometry as a deductive system. You learned about the premises of geometry. Starting fresh with these premises, you built a system of theorems. By discovering geometry, and then examining it as a mathematical system, you have been following in the footsteps of mathematicians throughout history. Your discoveries gave you an understanding of how geometry works. Proofs gave you the tools for taking apart your discoveries and understanding why they work. EXERCISES In Exercises 1–7, identify each statement as true or false. For each false statement, sketch a counterexample or explain why it is false. 1. If one pair of sides of a quadrilateral are parallel and the other pair of sides are congruent, then the quadrilateral is a |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.