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C H A P T E R Similarity © 2010 Carnegie Learning, Inc. 5 A scale model is a representation of an object that is either smaller or larger than the actual object. Small-scale models of houses can be used to help builders and buyers visualize the full-sized house before it is actually built. You will use ratios and proportions to calculate the scale and dimensions of models. 5.1 Ace Reporter Review of Ratio and Proportion | p. 257 5.5 Geometric Mean Similar Right Triangles | p. 287 5.2 Picture Picture on the Wall... Similar Polygons | p. 263 5.6 Indirect Measurement Application of Similar Triangles | p. 293 5.3 To Be or Not To Be Similar? Similar Triangle Postulates | p. 273 5.4 Triangle Side Ratios Angle Bisector/Proportional Side Theorem | p. 281 Chapter 5 | Similarity 5 253 Introductory Problem for Chapter 5 Washington Monument Problem The Washington Monument is a tall obelisk built between 1848 and 1884 in honor of the first President of the United States, George Washington. It is the tallest free-standing masonry structure in the world and has 897 steps from bottom to top. The Washington National Monument Society, formed by Congress in 1833, wanted to construct the largest monument in the world with dimensions and magnificence that would be proportionate to the greatness of George Washington. They wanted to show the gratitude that the people of United States felt toward him. Washington Monument The official dedication ceremony for the memorial took place the day before Washington’s birthday in 1885. The final cost of the project was $1,187,710. • Your eyes are 6 feet above ground level. • The reflecting pool is located between the Lincoln Memorial and the Washington Monument. • As you stand facing the Washington Monument, you can see in the reflection pool the very top of the monument. • The distance from the spot where you are standing to the spot where you see the top of the monument in the reflecting pool is 12 feet. • The distance from the location in the reflecting pool where you see the top of the monument to the base of the monument is 1110 feet. 5 254 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. It is possible to determine the height of the Washington Monument using only a simple tape measure and these few known facts: Lincoln Memorial © 2010 Carnegie Learning, Inc. 1. Draw a diagram of the situation and calculate the height of the Washington Monument. The reflecting pool 5 Be prepared to share your solutions and methods. Chapter 5 | Introductory Problem for Chapter 5 255 5 256 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 5.1 Ace Reporter Review of Ratio and Proportion OBJECTIVES KEY TERMS In this lesson you will: ● Write and simplify ratios. ● Compare ratios. ● Write and solve proportions. ● Use survey results to make predictions. PROBLEM 1 ● ● ● ● ● ratio probability proportion means extremes Survey Says © 2010 Carnegie Learning, Inc. You are a reporter for your school’s newspaper. You are writing an article about the order of classes during the school day and are interviewing students to get their opinions. From the information you have gathered, it seems the students have a strong opinion on when the gym class should occur. You have surveyed many students and recorded the results in the table shown. When Do You Think Gym Classes Should Be Held? Beginning of Day End of Day Any Time 8 14 2 1. How many students did you survey? 5 2. What can you conclude from your survey results? Lesson 5.1 | Ace Reporter 257 3. You want to compare the results of the survey in your article. One way you could compare the results is by writing the statement: “Eight out of 24 students prefer to have gym class at the beginning of the day.” Which two responses from the survey results are being compared? 4. Write a comparison sentence using the results of your survey. Include the number of students who prefer to have gym class at the end of the day. 5. Write a comparison sentence using the results of your survey. Include the number of students who have no preference when gym class is held. A ratio is a comparison of two or more numbers that uses division. You can mathematically compare the results in the table by using ratios. You can write a ratio as a fraction, or by using a colon. For instance, you can write “Eight out of 24 students prefer to have gym class at the beginning of the day” in two ways. 8 students As a fraction: ___________ 24 students Using a colon : 8 students : 24 students When you use a colon, you read the colon as the word “to.” So, the ratio “8 students : 24 students” is read as “8 students to 24 students.” 7. Suppose that you have only surveyed students in your own grade. A friend offers to help you out and surveys students from another grade in your school. Your friend’s results are shown in the table. When Do You Think Gym Classes Should Be Held? 5 258 Beginning of Day End of Day Any Time 15 18 3 a. How many students did your friend survey? Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 6. Write each comparison sentence you wrote for Questions 4 and 5 as a ratio. Write each ratio as a fraction. If possible, simplify your fractions. b. Write three different ratios using the table, and then describe what each ratio represents. Simplify fractions if possible. 8. Do a larger portion of the students in your survey or your friend’s survey prefer: a. Gym class at the beginning of the day? b. Gym class at the end of the day? c. No preference when gym class is held? © 2010 Carnegie Learning, Inc. 9. When are two different ratios equivalent? 10. Complete the table to show the results of your survey and your friend’s survey together. Then write two equivalent ratios for each statement. Write your ratios as fractions. When Do You Think Gym Classes Should Be Held? Beginning of Day End of Day Any Time a. Students who prefer gym class at end of day : Students who prefer gym class at beginning of day 5 b. Students who have no preference : Students who prefer gym class at end of day Lesson 5.1 | Ace Reporter 259 PROBLEM 2 Making Predictions 1. Use the combined results of the surveys in Problem 1 to write the following ratios. Write each ratio as a fraction in simplest form. a. Students who prefer gym class at beginning of day : All students surveyed b. Students who prefer gym class at end of day : All students surveyed c. Students with no preference : All students surveyed Ratios are used to describe probability. The probability of an outcome is the ratio of the number of successful outcomes to the number of possible outcomes. 2. Suppose each student was required to submit a questionnaire to participate in the survey in Problem 1. All of the completed questionnaires were put in a box and one questionnaire was selected at random. a. What is the probability that the questionnaire selected was from a student who preferred to take gym class at the end of the day? c. What is the probability that the questionnaire selected was from a student who had no preference when gym class is held? 5 260 3. Suppose that you want to interview students from the other grades in your school. Would you expect the results you would gather from other grades to be very different from the results you already have? Why or why not? Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. b. What is the probability that the questionnaire selected was from a student who preferred to take gym class at the beginning of the day? 4. Suppose that you interviewed 30 students in a different grade. How many students would you expect to respond that they prefer to have gym class at the end of the day? When two ratios that compare the same quantities are equal, you can write them as a proportion. A proportion is an equation that states that two ratios are equivalent, or equal. You write a proportion by placing an equals sign between two equivalent ratios or by using a double colon in place of the equals sign. For instance, you could have used a proportion to answer Question 4: 8 students ___________ ? students ___________ 15 students 30 students 5. What is the value of the unknown quantity in the proportion? Explain. When you calculated the unknown quantity, you were solving the proportion. Another way to solve a proportion is by using the proportion’s means and extremes. a __ b c __ d © 2010 Carnegie Learning, Inc. means extremes 6. What are the means and extremes of the solved proportion in Question 4? 7. Calculate the product of the means and the product of the extremes from Question 4. What do you notice? 8. Solve the proportion for x. 5 6 4 __ __ 3 x Lesson 5.1 | Ace Reporter 261 9. Suppose that there are 480 students in your school. Use the combined survey results from Problem 1, Question 10, to predict how many students in your school would prefer to have gym class at the beginning of the day, how many students would prefer to have gym class at the end of the day, and how many students have no preference. 10. Suppose the 480 students in your school submitted a questionnaire in response to your survey. All questionnaires were put in a box. One questionnaire was selected at random from the box. Use the combined survey results from Problem 1, Question 10, to calculate each probability. a. What is the probability that the questionnaire selected was from a student who preferred to take gym class at the beginning of the day? b. What is the probability that the questionnaire selected was from a student who preferred to take gym class at the end of the day? Be prepared to share your solutions and methods. 5 262 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. c. What is the probability that the questionnaire selected was from student who had no preference when gym class is held? 5.2 Picture Picture on the Wall... Similar Polygons OBJECTIVES KEY TERMS In this lesson you will: ● ● ● ● ● ● ● Identify similar polygons. Identify corresponding angles and corresponding sides in similar polygons. Calculate unknown measures in similar polygons. Calculate unknown measures in a scale model. Compare side lengths, perimeters, and areas of similar polygons. Compare the ratios of side lengths, perimeters, and areas of similar polygons. PROBLEM 1 ● ● similar scale model scale The Perfect Picture © 2010 Carnegie Learning, Inc. When you frame a picture, it is not unusual to put a mat inside the frame. A mat is a piece of paperboard that is used to provide a transition between a picture and the picture frame. 5 Lesson 5.2 | Picture Picture on the Wall... 263 You are creating your own collage of pictures. You bought a large frame and will cut out rectangular holes in the mat as shown. 4 in. 5 in. 3 in. A 6 in. 7 in. 2 in. 3 in. 2 in. D C B 1. What are the interior angle measures of each mat opening? 2. Write a ratio that compares the length of rectangle A to the length of rectangle B. Then, write a ratio that compares the width of rectangle A to the width of rectangle B. What do you notice? 4. Write a ratio that compares the length of rectangle A to the length of rectangle C. Then, write a ratio that compares the width of rectangle A to the width of rectangle C. What do you notice? 5 264 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 3. Write a ratio that compares the length of rectangle A to the length of rectangle D. Then, write a ratio that compares the width of rectangle A to the width of rectangle D. What do you notice? PROBLEM 2 Similarity Two polygons are similar when the corresponding angles are congruent and the ratios of the measures of the corresponding sides are equal. 1. Which rectangles from Problem 1 are similar? 2. The two triangles shown are similar. You can write 䉭UVW ⬃ 䉭XYZ, where the symbol ⬃ means “is similar to.” Y V X Z U W Again, the order in which you write the vertices in a similarity statement indicates the corresponding angles and the corresponding sides. © 2010 Carnegie Learning, Inc. a. List the corresponding angles and the corresponding sides. b. Write a ratio that compares a side length of 䉭UVW to a corresponding side length of 䉭XYZ. c. Write a ratio that compares a side length of 䉭XYZ to a corresponding side length of 䉭UVW. Lesson 5.2 | Picture Picture on the Wall... 5 265 d. Are the two ratios equal from Questions 2(b) and (c)? Why or why not? e. When you write a proportion relating the corresponding side lengths of two similar polygons, what must be true about both of the ratios? 3. In the figure shown, 䉭GHI ⬃ 䉭KLM. H G L I K M b. Suppose that GH 3 feet, KL 9 feet, and HI 5 feet. Write a proportion to calculate LM. Solve the proportion. 5 266 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. a. Write proportions that relate the ratios side lengths using the corresponding sides of 䉭GHI and 䉭KLM. c. Suppose that you also know that KM 12 feet. Calculate GI. d. Calculate the ratio of the height of 䉭GHI to the height of 䉭KLM. e. Calculate the ratio of the length of the base of 䉭GHI to the length of the base of 䉭KLM. © 2010 Carnegie Learning, Inc. f. Compare the ratios of the lengths and heights. g. Calculate the areas of the triangles. Write the ratio of the area of 䉭GHI to the area of 䉭KLM. Write your ratio as a fraction in simplest form. 5 h. How is the ratio of the areas related to the ratio of the heights and to the ratio of the lengths of the bases? Lesson 5.2 | Picture Picture on the Wall... 267 A scale model (or model) is a replication that is similar to an actual object, but it is either larger or smaller. The ratio of a dimension of the actual object to a corresponding dimension in the model is called the scale of the model. 5. A scale model of a framed picture is being created to put inside a dollhouse. The actual rectangular picture is 4 inches wide and 8 inches long. The scale of the model is 4 : 1. Calculate the length and width of the dollhouse picture. 5 268 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 4. A wall mural is being painted from a picture that is 6 inches long and 4 inches wide. The wall mural should be 48 inches long. a. Complete the statement to calculate the scale of the model. Write your answer as a fraction in simplest form. length of picture _______________ length of mural b. Now use the scale to complete the proportion to calculate the width of the mural. width of picture _______________ width of mural c. Calculate the width of the mural. 6. Determine the ratio of the perimeter of the actual picture and the perimeter of the dollhouse picture. 7. Determine the ratio of the area of the actual picture and the area of the dollhouse picture. PROBLEM 3 Ratios of Area and Perimeter of Similar Rectangles © 2010 Carnegie Learning, Inc. 1. Draw a rectangle on the grid. Each square on the grid represents a square that is one foot long and one foot wide. 2. 3. 4. 5. Calculate the perimeter and area of the rectangle you drew. Draw a similar rectangle on the grid by multiplying each side by a scale factor. Calculate the perimeter and the area of the second rectangle. Repeat steps 3 and 4 two more times. Lesson 5.2 | Picture Picture on the Wall... 5 269 6. Record the length, height, perimeter, and area of each rectangle in the table. Length (in feet) Height (in feet) Perimeter (in feet) Area (in square feet) First Rectangle Second Rectangle Third Rectangle Fourth Rectangle 7. Complete the table by calculating the ratios of side lengths, perimeters, and areas for each rectangle comparison. Simplify all fractions. Ratio of Rectangles Ratio of Side Ratio of Lengths Perimeters Ratio of Areas First Rectangle : Second Rectangle Second Rectangle : Third Rectangle Third Rectangle : Fourth Rectangle Compare your table with others in your group or class. Look for patterns. 8. What is the relationship between the ratio of side lengths of two similar rectangles and the ratio of perimeters of the two similar rectangles? a. 10. The ratio of side lengths of two similar rectangles is __ b a. What is the ratio of perimeters of the two similar rectangles? b. What is the ratio of areas of the two similar rectangles? 5 270 4 . What is the ratio of 11. The ratio of side lengths of two similar rectangles is __ 5 areas of the two similar rectangles? Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 9. What is the relationship between the ratio of side lengths of two similar rectangles and the ratio of areas of the two similar rectangles? PROBLEM 4 Area and Perimeter Ratios of Triangles Is the relationship between the ratio of side lengths of two similar rectangles and the ratio of areas of two similar rectangles the same for all polygons? Let’s consider triangles. © 2010 Carnegie Learning, Inc. 1. Draw a right triangle on grid. Each square on the grid represents a square that is one foot long and one foot wide. For easier calculations, use Pythagorean triples. 2. 3. 4. 5. 6. Calculate the perimeter and area of the triangle you drew. Draw a similar triangle on the grid by multiplying each side by a scale factor. Calculate the perimeter and the area of the second triangle. Repeat steps 3 and 4 two more times. Record the length of the base, height, perimeter, and area of each triangle in the table. Base (in feet) Height (in feet) Perimeter (in feet) Area (in square feet) First Triangle 5 Second Triangle Third Triangle Fourth Triangle Lesson 5.2 | Picture Picture on the Wall... 271 7. Complete the table by calculating the ratios of side lengths, perimeters, and areas for each triangle comparison. Simplify all fractions. Ratio of Triangles Ratio of Side Lengths Ratio of Perimeters Ratio of Areas First Triangle : Second Triangle Second Triangle : Third Triangle Third Triangle : Fourth Triangle Compare your table with others in your group or class. Look for patterns. 8. What is the relationship between the ratio of side lengths of two similar triangles and the ratio of perimeters of the two similar triangles? 9. What is the relationship between the ratio of two side lengths of similar triangles and the ratio of areas of the two similar triangles? a. 10. The ratio of side lengths of two similar triangles is __ b a. What is the ratio of perimeters of the two similar triangles? 4 . What is the 11. The ratio of side lengths of two similar triangles is expressed as __ 5 ratio of areas of the two similar triangles? 12. What might you conclude about the relationships of similar polygons with respect to their ratios of side lengths, ratios of perimeters, and ratios of areas? 5 Be prepared to share your solutions and methods. 272 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. b. What is the ratio of areas of the two similar triangles? 5.3 To Be or Not To Be Similar? Similar Triangle Postulates OBJECTIVE In this lesson you will: ● Use constructions and given information to determine whether two triangles are similar. KEY TERMS ● ● ● ● ● Angle-Angle Similarity Postulate Side-Side-Side Similarity Postulate Side-Angle-Side Similarity Postulate included angle included side © 2010 Carnegie Learning, Inc. An art projector is a piece of equipment that artists use to create exact copies of artwork, to enlarge artwork, or to reduce artwork. A basic art projector uses a light bulb and a lens within a box. The light rays from the art being copied are collected onto a lens at a single point. The lens then projects the image of the art onto a screen as shown. If the projector is set up properly, the triangles will be similar polygons. You can show that these triangles are similar without measuring all of the side lengths and all of the interior angles. Lesson 5.3 | To Be or Not To Be Similar? 5 273 PROBLEM 1 Using Two Angles Two polygons are similar if all corresponding angles are congruent and the ratios of the measures of all corresponding sides are equal. 1. State all corresponding congruent angles and all corresponding proportional sides using the similar triangles shown. 䉭RST ⬃ 䉭WXY R W S X Y T Are there any shortcuts that can be taken? Can you use fewer pairs of angles or fewer pairs of sides to show triangles are similar? Constructions can be used to answer these questions. 2. Construct triangle DEF using only ⬔D and ⬔E in triangle DEF as shown. Be sure that the side lengths of triangle DEF are different than the side lengths of triangle DEF. D 5 E F 274 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. You can conclude that two triangles are similar if you are able to prove that three pairs of corresponding angles are congruent and three pairs of corresponding sides are proportional. 3. Did everyone in your group and class construct the same triangle? If not, describe the difference between the triangles that were constructed. 4. Measure the angles and sides of triangle DEF and triangle DEF. Is the third pair of angles congruent? Are the three pairs of corresponding sides proportional? 5. Are the two triangles similar? 6. Would your answers to Questions 3 through 5 change if you had constructed triangle DEF using a different side length for DE? 7. Two pairs of corresponding angles are congruent. Is this sufficient information to conclude that the triangles are similar? The Angle-Angle Similarity Postulate states: “If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.” © 2010 Carnegie Learning, Inc. B E F C D A If m⬔A m⬔D and m⬔C m⬔F, then 䉭ABC ⬃ 䉭DEF. 8. The triangles shown are isosceles triangles. Do you have enough information to show that the triangles are similar? Explain your reasoning. Q M 5 L N R P Lesson 5.3 | To Be or Not To Be Similar? 275 9. The triangles shown are isosceles triangles. Do you have enough information to show that the triangles are similar? Explain your reasoning. T W S U V PROBLEM 2 X Using Two and Three Proportional Sides ___ ___ 1. Construct triangle DEF by doubling the lengths of sides DE and EF . Be sure to first construct the new DE and EF separately and then construct the triangle; this will ensure a ratio of 2 : 1. Do not duplicate angles. D F 2. Did everyone in your group and class construct the same triangle? If not, describe the difference between the triangles that were constructed. 5 276 3. Measure the angles and sides of triangle DEF and triangle DEF. Are the corresponding angles congruent? Are the three pairs of corresponding sides proportional? Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. E 4. Are the two triangles similar? 5. Two pairs of corresponding sides are proportional. Is this sufficient information to conclude that the triangles are similar? 6. Construct triangle DEF by doubling the lengths of sides DE, EF, and FD. Be sure to first construct the new side lengths separately, and then construct the triangle. Do not duplicate angles. D E © 2010 Carnegie Learning, Inc. F 7. Did everyone in your group and class construct the same triangle? If not, describe the difference between the triangles that were constructed. 8. Measure the angles and sides of triangle DEF and triangle DEF. Are the corresponding angles congruent? 5 Lesson 5.3 | To Be or Not To Be Similar? 277 9. Are the two triangles similar? 10. Three pairs of corresponding sides are proportional. Is this sufficient information to conclude that the triangles are similar? The Side-Side-Side Similarity Postulate states: “If the corresponding sides of two triangles are proportional, then the triangles are similar.” E B F C D A BC ___ AC , then 䉭ABC ⬃ 䉭DEF. AB ___ If ___ DE EF DF 11. If the corresponding sides of two triangles are proportional, what makes the triangles similar? 12. Determine whether 䉭UVW is similar to 䉭XYZ. If so, use symbols to write a similarity statement. Z V 16 meters Y 33 meters W 36 meters U 5 278 Chapter 5 | Similarity 24 meters 22 meters X © 2010 Carnegie Learning, Inc. 24 meters The Side-Angle-Side Similarity Postulate states: “If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.” B E C F D A AC and ⬔A 艑 ⬔D, then 䉭ABC ⬃ 䉭DEF. AB ___ If ___ DE DF PROBLEM 3 Using Two Proportional Sides and an Angle An included angle is an angle formed by two consecutive sides of a figure. An included side is a line segment between two consecutive angles of a figure. 1. Construct triangle DEF by doubling the lengths of any two sides and duplicating the included angle. Be sure to first construct the new side lengths separately, and then construct the triangle. © 2010 Carnegie Learning, Inc. D E F 2. Measure the angles and sides of triangle DEF and triangle DEF. Are the corresponding angles congruent? Are the corresponding sides proportional? Lesson 5.3 | To Be or Not To Be Similar? 5 279 3. Are the two triangles similar? 4. Two pairs of corresponding sides are proportional and the corresponding included angles are congruent. Is this sufficient information to conclude that the triangles are similar? PROBLEM 4 Guess My Triangle Be prepared to share your solutions and methods. 5 280 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 1. Gaelin is thinking of a triangle and he wants everyone in his class to draw a similar triangle. Which combinations of sides and angles could he provide? 5.4 Triangle Side Ratios Angle Bisector/Proportional Side Theorem OBJECTIVES KEY TERM In this lesson you will: ● ● ● Angle Bisector/Proportional Side Theorem Prove the Angle Bisector/ Proportional Side Theorem. Apply the Angle Bisector/ Proportional Side Theorem. PROBLEM 1 When an interior angle of a triangle is bisected, proportional relationships involving the sides of the triangle occur. You will be able to prove that these relationships apply to all triangles. To do this, it will be necessary to extend a side of the triangle and add an auxiliary line parallel to one side of the triangle. © 2010 Carnegie Learning, Inc. The Angle Bisector/Proportional Side Theorem states: “A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle.” To prove the Angle Bisector/Proportional Side Theorem, consider the statements and figure shown. ___ Given: AD bisects ⬔BAC AC AB ___ Prove: ___ BD CD A 5 B D C Lesson 5.4 | Triangle Side Ratios 281 ___ ___ 1. Draw an auxiliary line parallel to AB through point C. Extend AD until it intersects the auxiliary line. Label the point of intersection, point E. A B D C E 2. Complete the two-column proof. 5 282 Chapter 5 | Reasons 1. 1. Given 2. 2. Construction 3. 3. Definition of angle bisector 4. ⬔BAE ⬵ ⬔CEA 4. 5. 5. Transitive Property of 艑 6. 6. If two angles of a triangle are congruent, then the sides opposite the angles are congruent. 7. 7. Definition of congruent segments 8. 8. Alternate Interior Angle Theorem 9. 䉭DAB ⬃ 䉭DEC 9. AB ___ BD 10. ___ EC CD 10. 11. 11. Rewrite as an equivalent proportion AC AB ___ 12. ___ BD CD 12. Similarity © 2010 Carnegie Learning, Inc. Statements PROBLEM 2 Applying the Angle Bisector/Proportional Side Theorem Apply the Angle Bisector/Proportional Side Theorem to solve each problem. 1. On the map shown, North Craig Street bisects the angle formed between Bellefield Avenue and Ellsworth Avenue. • The distance from the ATM to the Coffee Shop is 300 feet. • The distance from the Coffee Shop to the Library is 500 feet. • The distance from your apartment to the Library is 1200 feet. Determine the distance from your apartment to the ATM. © 2010 Carnegie Learning, Inc. ___ 2. CD bisects ⬔C. Solve for DB. A D 8 24 B 30 5 C Lesson 5.4 | Triangle Side Ratios 283 ___ 3. CD bisects ⬔C. Solve for AC. A 9 11 D B 22 C ___ 4. AD bisects ⬔A. AC AB 36. Solve for AC and AB. B A 14 D 7 © 2010 Carnegie Learning, Inc. C 5 284 Chapter 5 | Similarity ___ 5. BD bisects ⬔B. Solve for AC. A 14 B 6 D 16 C © 2010 Carnegie Learning, Inc. Be prepared to share your solutions and methods. 5 Lesson 5.4 | Triangle Side Ratios 285 5 286 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 5.5 Geometric Mean Similar Right Triangles OBJECTIVES KEY TERMS In this lesson you will: ● ● Construct an altitude to the hypotenuse in a right triangle. ● Explore the relationships created when an altitude is drawn to the hypotenuse of a right triangle. PROBLEM 1 ● ● ● Right Triangle/Altitude Similarity Theorem geometric mean Right Triangle Altitude Theorem 1 Right Triangle Altitude Theorem 2 Right Triangles © 2010 Carnegie Learning, Inc. A bridge is needed to cross over a canyon. The dotted line segment connecting points S and R represents the bridge. The distance from point P to point S is 45 yards. The distance from point Q to point S is 130 feet. How long is the bridge? 5 To determine the length of the bridge, you must first explore what happens when an altitude is drawn to the hypotenuse of a right triangle. When an altitude is drawn to the hypotenuse of a right triangle, it forms two smaller triangles. All three triangles have a special relationship. Let’s explore this relationship. Lesson 5.5 | Geometric Mean 287 1. Construct an altitude to the hypotenuse in the right triangle ABC. Label the altitude CD. C D A B 2. Name all right triangles in the figure. © 2010 Carnegie Learning, Inc. 3. Trace each of the triangles on separate pieces of paper and label all the vertices on each triangle. Cut out each triangle. Label the vertex of each triangle. 4. Arrange the triangles so that all of the triangles have the same orientation. The hypotenuse, the shortest leg, and the longest leg should all be in corresponding positions. You may have to flip triangles over to do this. 5. Draw each triangle in the same orientation. 5 288 Chapter 5 | Similarity 6. Is 䉭ABC ⬃ 䉭ACD? Explain. 7. Is 䉭ABC ⬃ 䉭CBD? Explain. 8. Is 䉭ACD ⬃ 䉭CBD? Explain. 9. Write the corresponding sides of 䉭ABC and 䉭ACD as proportions. 10. Write the corresponding sides of 䉭ABC and 䉭CBD as proportions. 11. Write the corresponding sides of 䉭ACD and 䉭CDB as proportions. You have just proven a theorem! The Right Triangle/Altitude Similarity Theorem states: “If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.” © 2010 Carnegie Learning, Inc. You can now use this theorem as a valid reason in proofs. 5 Lesson 5.5 | Geometric Mean 289 PROBLEM 2 Geometric Mean When an altitude of a right triangle is constructed from the right angle to the hypotenuse, three similar right triangles are created. This altitude is a geometric mean. The geometric mean of two positive numbers a and b is the positive number x such a __ x. that __ x b Two theorems are associated with the altitude to the hypotenuse of a right triangle. You will prove these theorems in the Assignments for this lesson. The Right Triangle Altitude Theorem 1 states: “The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.” 1. Which proportion(s) in Problem 1 demonstrate(s) this theorem? The Right Triangle Altitude Theorem 2 states: “If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.” 2. Which proportion(s) in Problem 1 demonstrate(s) this theorem? 3. In each triangle, solve for x. a. x 5 290 Chapter 5 | Similarity 9 © 2010 Carnegie Learning, Inc. 4 b. 4 x 8 c. x 4 20 d. © 2010 Carnegie Learning, Inc. x 2 8 5 Lesson 5.5 | Geometric Mean 291 4. In the triangle shown, solve for x, y, and z. 5 10 z x y PROBLEM 3 Bridge Over the Canyon Be prepared to share your solutions and methods. 5 292 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 1. Solve for the length of the bridge in Problem 1 using the geometric mean. 5.6 Indirect Measurement Application of Similar Triangles OBJECTIVES KEY TERM In this lesson you will: ● Identify similar triangles to calculate indirect measurements. ● Use proportions to solve for unknown measurements. PROBLEM 1 ● indirect measurement How Tall is That Flagpole? At times, situations occur when measuring something directly is impossible, or physically undesirable. When these situations arise, indirect measurement, the technique that uses proportions to calculate measurement, can be implemented. Your knowledge of similar triangles can be very helpful in these situations. Use the following steps to measure the height of the school flagpole or any other tall object outside. You will need a partner, a tape measure, a marker, and a flat mirror. © 2010 Carnegie Learning, Inc. Step 1: Use a marker to create a dot near the center of the mirror. Step 2: Face the object you would like to measure and place the mirror between yourself and the object. You, the object, and the mirror should be collinear. Step 3: Focus your eyes on the dot on the mirror and walk backward until you can see the top of the object on the dot, as shown. 5 Step 4: Ask your partner to sketch a picture of you, the mirror, and the object. Lesson 5.6 | Indirect Measurement 293 Step 5: Review the sketch with your partner. Decide where to place right angles, and where to locate the sides of the two triangles. 1. How can similar triangles be used to calculate the height of the object? 2. Use your sketch to write a proportion to calculate the height of the object and solve the proportion. 3. Compare your answer with others measuring the same object. How do the answers compare? 4. What are some possible sources of error that could result when using this method? 5 5. Switch places with your partner and identify a second object to measure. Repeat this method of indirect measurement to solve for the height of the new object. 294 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. Step 6: Determine which segments in your sketch can easily be measured using the tape measure. Describe their locations and record the measurements on your sketch. PROBLEM 2 How Tall is That Oak Tree? © 2010 Carnegie Learning, Inc. 1. You go to the park and use the mirror method to gather enough information to calculate the height of one of the trees. The figure shows your measurements. Calculate the height of the tree. 2. Take Note Remember, whenever you are solving a problem that involves measurements like length (or weight), you may have to rewrite units so they are the same. For instance, if a problem involves weight, all of the weights should be measured in the same unit of measure. Stacey wants to try the mirror method to measure the height of one of her trees. She calculates that the distance between her and the mirror is 3 feet and the distance between the mirror and the tree is 18 feet. Stacey’s eye height is 60 inches. Draw a diagram of this situation. Then calculate the height of this tree. 5 Lesson 5.6 | Indirect Measurement 295 © 2010 Carnegie Learning, Inc. 3. Stacey notices that another tree casts a shadow and suggests that you could also use shadows to calculate the height of the tree. She lines herself up with the tree’s shadow so that the tip of her shadow and the tip of the tree’s shadow meet. She then asks you to measure the distance from the tip of the shadows to her, and then measure the distance from her to the tree. Finally, you draw a diagram of this situation as shown below. Calculate the height of the tree. Explain your reasoning. 5 296 Chapter 5 | Similarity PROBLEM 3 How Wide is That Creek? It is not reasonable for you to directly measure the width of a creek, but you can use indirect measurement to measure the width. You stand on one side of the creek and your friend stands directly across the creek from you on the other side as shown in the figure. 1. Your friend is standing 5 feet from the creek and you are standing 5 feet from the creek. Mark these measurements on the diagram shown. 2. You and your friend walk away from each other in opposite parallel directions. Your friend walks 50 feet and you walk 12 feet. Mark these measurements on the diagram shown. Draw a line segment that connects your starting point and ending point and draw a line segment that connects your friend’s starting point and ending point. © 2010 Carnegie Learning, Inc. 3. Draw a line segment that connects you and your friend’s starting points and draw a line segment that connects you and your friend’s ending points. Label any angle measures and any angle relationships that you know on the diagram. Explain how you know these angle measures. 4. How do you know that the triangles formed by the lines are similar? Lesson 5.6 | Indirect Measurement 5 297 5. Calculate the distance from your friend’s starting point to your side of the creek. Round your answer to the nearest tenth, if necessary. 6. What is the width of the creek? Explain your reasoning. © 2010 Carnegie Learning, Inc. 7. There is also a ravine (a deep hollow in the earth) on another edge of the park. You and your friend take measurements like those in Problem 3 to indirectly calculate the width of the ravine. The figure shows your measurements. Calculate the width of the ravine. 5 298 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 8. There is a large pond in the park. A diagram of the pond is shown below. You ___ want to calculate the distance across the widest part of the pond, labeled as DE . To indirectly calculate this distance, you first place a stake at point A. You chose point A so that you can see the edge of the pond on both sides at points D and E, where you also place stakes. Then you tie string from point A to point D and from point A to point E. At a narrow portion of the pond,___ you place stakes at ___ points B and C along the string so that BC is parallel to DE . The measurements you make are shown on the diagram. Calculate the distance across the widest part of the pond. Be prepared to share your solutions and methods. 5 Lesson 5.6 | Indirect Measurement 299 Chapter 5 Checklist KEY TERMS ● ● ● ● ratio (5.1) probability (5.1) proportion (5.1) means (5.1) ● ● ● ● extremes (5.1) similar (5.2) scale model (5.2) scale (5.2) ● ● ● ● included angle (5.3) included side (5.3) geometric mean (5.5) indirect measurement (5.6) POSTULATES ● ● ● Angle-Angle Similarity Postulate (5.3) Side-Side-Side Similarity Postulate (5.3) Side-Angle-Side Similarity Postulate (5.3) THEOREMS ● ● Angle Bisector/Proportional Side Theorem (5.4) Right Triangle/Altitude Similarity Theorem (5.5) ● ● Right Triangle Altitude Theorem 1 (5.5) Right Triangle Altitude Theorem 2 (5.5) CONSTRUCTIONS ● triangles (5.3) altitudes (5.5) 5.1 Writing Ratios To write a ratio of one quantity to another quantity, write the ratio as a fraction or use a colon. 5 300 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. ● Example: The table shows the results of a survey in which 55 students were asked to name their favorite type of movie. What is Your Favorite Type of Movie? Comedy 21 Drama 16 Horror 7 Action 6 Science Fiction 2 Other 3 The ratios of the number of students who chose comedy as their favorite type of movie to the number of students surveyed is: 21 students As a fraction: ___________ 55 students Using a colon: 21 students : 55 students The ratio of the number of students who chose drama as their favorite type of movie to the number of students who chose horror as their favorite type of movie is: 16 students As a fraction: ___________ 7 students Using a colon: 16 students : 7 students 5.1 Solving Proportions to Make Predictions © 2010 Carnegie Learning, Inc. To solve a proportion, set the product of the means equal to the product of the extremes. Then, solve for the variable in the proportion. Example: Suppose that there are 495 students in your school. You can use the results from the survey in the previous example to predict the number of students that would choose each type of movie. To predict the number of students in your school who would choose comedy as their favorite type of movie, solve the following proportion. 21 students ____________ x students ___________ Write a proportion. 55x 21 495 Set the product of the means equal to the product of the extremes. 55x 10,395 Multiply. 55 students x 189 495 students 5 Divide each side by 55. You can predict that 189 students from your school would choose comedy as their favorite type of movie. Chapter 5 | Checklist 301 5.2 Identifying Similar Polygons Two polygons are similar when the corresponding angles are congruent and the ratios of the measures of the corresponding sides are equal. Example: B 80° S 18 ft 80° 12 ft 12 ft 8 ft 34° 66° R T 34° 14 ft 66° A C 21 ft Corresponding angles are congruent: ⬔R ⬵ ⬔A, ⬔S ⬵ ⬔B, and ⬔T ⬵ ⬔C. RS ___ ST ___ 8 __ 2, ___ 12 __ 2 , and Ratios of corresponding sides are equal: ___ 12 3 BC 18 3 AB 2. RT ___ 14 __ ___ 21 AC 3 So, triangle RST is similar to triangle ABC. You can write 䉭RST ⬃ 䉭ABC. 5.2 Using a Scale Model to Calculate a Measure A scale model is a replication of an actual object that is similar to the object, but is either larger or smaller. To calculate a measure using a scale model, write and solve a proportion. A museum gift shop sells plastic alligators that are scale models of an actual alligator. The scale of the model is 22 : 1. The tail length of the model alligator is 3 inches. To determine the tail length of the actual alligator, write and solve a proportion. x inches _________ 22 inches ________ 3 inches 1 inch x 1 22 3 x 66 5 302 Write a proportion. Set the product of the means equal to the product of the extremes. Multiply. The tail length of the actual alligator is 66 inches, or 5.5 feet. Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. Example: 5.3 Using the Angle-Angle Similarity Postulate The Angle-Angle Similarity Postulate states: “If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.” Example: D Z Y 28° 32° 32° X 28° F E m⬔D m⬔X and m⬔E m⬔Y, so 䉭DEF ⬃ 䉭XYZ 5.3 Using the Side-Side-Side Similarity Postulate The Side-Side-Side Similarity Postulate states: “If three pairs of corresponding sides of two triangles are proportional, then the triangles are similar.” Example: Q B 32 m 20 m 48 m 30 m A 25 m P C 40 m R 5, ____ BC ___ AC ___ 20 __ 30 __ 5 , and ___ 25 __ 5 , so 䉭ABC ⬃ 䉭PQR AB ___ ___ © 2010 Carnegie Learning, Inc. PQ 5.3 32 8 QR 48 8 PR 40 8 Using the Side-Angle-Side Similarity Postulate The Side-Angle-Side Similarity Postulate states: “If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.” Example: G 100° J 16 ft 5 28 ft H 14 ft L 100° 8 ft F K FG ___ GH ___ 28 2, m⬔G m⬔K, and ____ 16 2, so 䉭FGH ⬃ 䉭JKL ___ JK 14 KL 8 Chapter 5 | Checklist 303 5.4 Using the Angle Bisector/Proportional Side Theorem The Angle Bisector/Proportional Side Theorem states: “A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle.” Example: R 18 cm Q 15 cm 12 cm S x T QR ___ SR ____ QT ST 18 ___ 12 ___ x 15 18x 180 x 10 cm 5.5 Using the Right Triangle/Altitude Similarity Theorem The Right Triangle/Altitude Similarity Theorem states: “If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.” Example: B A C 䉭ABC ⬃ 䉭ACD, 䉭ABC ⬃ 䉭CBD, and 䉭ACD ⬃ 䉭CBD. 5 304 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. D 5.5 Using the Right Triangle Altitude Theorem 1 The Right Triangle Altitude Theorem 1 states: “The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.” A geometric mean of two positive numbers a and b is the positive number x a __ x. such that __ x b Example: x 20 ft 8 ft 8 ___ x __ x 20 x 160 2 ____ x √160 x 艐 12.65 ft 5.5 Using the Right Triangle Altitude Theorem 2 The Right Triangle Altitude Theorem 2 states: “If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.” © 2010 Carnegie Learning, Inc. Example: 15 in. 32 in. x 32 ___ x ___ x 15 x 480 2 ____ 5 x √480 x 21.91 in. Chapter 5 | Checklist 305 5.6 Using Indirect Measurement to Solve a Problem Indirect measurement is a technique that uses proportions to determine a measurement when direct measurement is not possible. To calculate a length by using indirect measurement, create two similar triangles and then write and solve a proportion that uses the ratios of the corresponding lengths of the similar triangles. Example: You want to determine the height of a sign in front of a store. You notice that the sign is casting a shadow on the ground. So, you ask your friend to line herself up with the sign’s shadow so that the tip of her shadow and the tip of the sign’s shadow meet. Then, you measure the distance between your friend and the sign, and you also measure the length of your friend’s shadow. You know that your friend is 5.5 feet tall. You draw and label a diagram of this situation as shown. You can use a proportion to solve for the height x of the sign. 11 x 19 ___ ________ 11 5.5 11 11x 165 x 15 The sign is 15 feet tall. 5 306 Chapter 5 | Similarity © 2010 Carnegie Learning, Inc. 5.5 30 x ___ ___