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Contents Lesson 6-1Proportions Lesson 6-2Similar Polygons Lesson 6-3Similar Triangles Lesson 6-4Parallel Lines and Proportional Parts Lesson 6-5Parts of Similar Triangles Lesson 6-6Fractals and Self-Similarity

Lesson 1 Contents Example 1Write a Ratio Example 2Extended Ratios in Triangles Example 3Solve Proportions by Using Cross Products Example 4Solve Problems Using Proportions

Example 1-1a The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is Find the athlete-to-student ratio to the nearest tenth. Answer: The athlete-to-student ratio is 0.3. To find this ratio, divide the number of athletes by the total number of students. 0.3 can be written as

Example 1-1b The country with the longest school year is China with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.) Answer: 0.7

Example 1-2a Multiple- Choice Test Item In a triangle, the ratio of the measures of three sides is 5:12:13, and the perimeter is 90 centimeters. Find the measure of the shortest side of the triangle. A 15 cm B 18 cm C 36 cm D 39 cm Read the Test Item You are asked to apply the ratio to the three sides of the triangle and the perimeter to find the shortest side. Solve the Test Item We can rewrite 5:12:13 as 5x:12x:13x and use those measures for the sides of the triangle. Write an equation to represent the perimeter of the triangle as the sum of the measures of its sides.

Example 1-2b Perimeter Combine like terms. Divide each side by 30.

Example 1-2c Use this value of x to find the measures of the sides of the triangle. Answer: A The shortest side is 15 centimeters. The answer is A. Check Add the lengths of the sides to make sure that the perimeter is 90.

Example 1-2d Multiple- Choice Test Item In a triangle, the ratio of the measures of three sides is 3:4:5, and the perimeter is 42 feet. Find the measure of the longest side of the triangle. A 10.5 ft B 14 ft C 17.5 ft D 37 ft Answer: C

Example 1-3a Original proportion Cross products Multiply. Answer: 27.3 Solve Divide each side by 6.

Example 1-3b Original proportion Cross products Simplify. Answer: –2 Add 30 to each side. Divide each side by 24. Solve

Example 1-3c Answer: 4.5 Answer: 9 Solve each proportion. a. b.

Example 1-4a A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model. Because the scale model of the boxcar and the boxcar are in proportion, you can write a proportion to show the relationship between their measures. Since both ratios compare feet to inches, you need not convert all the lengths to the same unit of measure.

Example 1-4b Substitution Cross products Multiply. Divide each side by 40. Answer: The width of the model is 3.6 inches.

Example 1-4c Two large cylindrical containers are in proportion. The height of the larger container is 25 meters with a diameter of 8 meters. The height of the smaller container is 7 meters. Find the diameter of the smaller container. Answer: 2.24 m

End of Lesson 1

Lesson 2 Contents Example 1Similar Polygons Example 2Scale Factor Example 3Proportional Parts and Scale Factor Example 4Enlargement of a Figure Example 5Scale Factors on Maps

Example 2-1a Determine whether the pair of figures is similar. Justify your answer. Q The vertex angles are marked as 40º and 50º, so they are not congruent.

Example 2-1b Answer: None of the corresponding angles are congruent, so the triangles are not similar. Since both triangles are isosceles, the base angles in each triangle are congruent. In the first triangle, the base angles measure and in the second triangle, the base angles measure

Example 2-1c Determine whether the pair of figures is similar. Justify your answer. T Thus, all the corresponding angles are congruent.

Example 2-1d Now determine whether corresponding sides are proportional. The ratios of the measures of the corresponding sides are equal. Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so

Example 2-1e Determine whether the pair of figures is similar. Justify your answer. a.

Example 2-1e The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, Answer: Both triangles are isosceles with base angles measuring 76º and vertex angles measuring 28º.

Example 2-1f Answer: Only one pair of angles are congruent, so the triangles are not similar. b. Determine whether the pair of figures is similar. Justify your answer.

Example 2-2a An architect prepared a 12-inch model of a skyscraper to look like a real 1100-foot building. What is the scale factor of the model compared to the real building? Before finding the scale factor you must make sure that both measurements use the same unit of measure. 1100(12) 13,200 inches

Example 2-2b Answer: The ratio comparing the two heights is The scale factor is, which means that the model is the height of the real skyscraper.

Example 2-2c A space shuttle is about 122 feet in length. The Science Club plans to make a model of the space shuttle with a length of 24 inches. What is the scale factor of the model compared to the real space shuttle? Answer:

Example 2-3a The two polygons are similar. Write a similarity statement. Then find x, y, and UV. Use the congruent angles to write the corresponding vertices in order.

Example 2-3b Now write proportions to find x and y. To find x: Similarity proportion Cross products Multiply. Divide each side by 4.

Example 2-3c To find y: Similarity proportion Cross products Multiply. Subtract 6 from each side. Divide each side by 6 and simplify.

Example 2-3d Answer:

Example 2-3e The two polygons are similar. Find the scale factor of polygon ABCDE to polygon RSTUV.

Example 2-3f The scale factor is the ratio of the lengths of any two corresponding sides. Answer:

a. Write a similarity statement. Then find a, b, and ZO. b. Find the scale factor of polygon TRAP to polygon. Example 2-3g Answer: The two polygons are similar. Answer: ;

Example 2-4a Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5. If the length and width of rectangle PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ? Write proportions for finding side measures. Let one long side of each WXYZ and PQRS be and one short side of each WXYZ and PQRS be

Example 2-4b Answer:

Example 2-4c Quadrilateral GCDE is similar to quadrilateral JKLM with a scale factor of If two of the sides of GCDE measure 7 inches and 14 inches, what are the lengths of the corresponding sides of JKLM? Answer: 5 in., 10 in.

Example 2-5a The scale on the map of a city is inch equals 2 miles. On the map, the width of the city at its widest point is inches. The city hosts a bicycle race across town at its widest point. Tashawna bikes at 10 miles per hour. How long will it take her to complete the race? Explore Every equals 2 miles. The distance across the city at its widest point is

Example 2-5b Solve Cross products Divide each side by The distance across the city is 30 miles. Plan Create a proportion relating the measurements to the scale to find the distance in miles. Then use the formula to find the time.

Example 2-5c Divide each side by 10. Answer: 3 hours It would take Tashawna 3 hours to bike across town. Examine To determine whether the answer is reasonable, reexamine the scale. If 0.25 inches 2 miles, then 4 inches 32 miles. The distance across the city is approximately 32 miles. At 10 miles per hour, the ride would take about 3 hours. The answer is reasonable.

Example 2-5d An historic train ride is planned between two landmarks on the Lewis and Clark Trail. The scale on a map that includes the two landmarks is 3 centimeters = 125 miles. The distance between the two landmarks on the map is 1.5 centimeters. If the train travels at an average rate of 50 miles per hour, how long will the trip between the landmarks take? Answer: 1.25 hours

End of Lesson 2

Lesson 3 Contents Example 1Determine Whether Triangles Are Similar Example 2Parts of Similar Triangles Example 3Find a Measurement

Example 3-1a In the figure, and Determine which triangles in the figure are similar.

Example 3-1b Vertical angles are congruent, Answer: Therefore, by the AA Similarity Theorem, by the Alternate Interior Angles Theorem.

Example 3-1c In the figure, OW = 7, BW = 9, WT = 17.5, and WI = Determine which triangles in the figure are similar. Answer: I

Example 3-2a ALGEBRA Given QT 2x 10, UT 10, find RQ and QT.

Example 3-2b Since because they are alternate interior angles. By AA Similarity, Using the definition of similar polygons, Substitution Cross products

Example 3-2c Distributive Property Subtract 8x and 30 from each side. Divide each side by 2. Now find RQ and QT. Answer:

Example 3-2d Answer: ALGEBRA Given and CE x + 2, find AC and CE.

Example 3-3a INDIRECT MEASUREMENT Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 P.M. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at that time. What is the height of the Sears Tower?

Example 3-3b Assuming that the sun’s rays form similar triangles, the following proportion can be written. Now substitute the known values and let x be the height of the Sears Tower. Substitution Cross products

Example 3-3c Simplify. Divide each side by 2. Answer: The Sears Tower is 1452 feet tall.

Example 3-3d INDIRECT MEASUREMENT On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 feet 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot? Answer: 196 ft

End of Lesson 3

Lesson 4 Contents Example 1Find the Length of a Side Example 2Determine Parallel Lines Example 3Midsegment of a Triangle Example 4Proportional Segments Example 5Congruent Segments

Example 4-1a From the Triangle Proportionality Theorem, In and Find SU. S

Example 4-1b Substitute the known measures. Cross products Multiply. Divide each side by 8. Simplify. Answer:

Example 4-1c Answer: In and Find BY. B

Example 4-2a In and Determine whether Explain.

Example 4-2b In order to show that we must show that Since the sides have proportional length. Answer: since the segments have proportional lengths,

Example 4-2c In and AZ = 32. Determine whether Explain. Answer: No; the segments are not in proportion since X

Example 4-3a Triangle ABC has vertices A(–2, 2), B(2, 4,) and C(4, –4). is a midsegment of Find the coordinates of D and E. (-2, 2) (2, 4) (4, –4)

Example 4-3b Use the Midpoint Formula to find the midpoints of Answer: D(0, 3), E(1, –1)

Example 4-3c Triangle ABC has vertices A(–2, 2), B(2, 4) and C(4, –4). is a midsegment of Verify that (-2, 2) (2, 4) (4, –4)

Example 4-3d slope of If the slopes of slope of Answer: Because the slopes of

Example 4-3e Triangle ABC has vertices A(–2, 2), B(2, 4) and C(4, –4). is a midsegment of Verify that (-2, 2) (2, 4) (4, –4)

Example 4-3f First, use the Distance Formula to find BC and DE.

Example 4-3g Answer:

Example 4-3h Triangle UXY has vertices U(–3, 1), X(3, 3), and Y(5, –7). is a midsegment of

a. Find the coordinates of W and Z. b. Verify that c. Verify that Example 4-3i Answer: W(0, 2), Z(1, –3) Answer: Since the slope of and the slope of Answer: Therefore,

Example 4-4a In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x.

Example 4-4b Notice that the streets form a triangle that is cut by parallel lines. So you can use the Triangle Proportionality Theorem. Triangle Proportionality Theorem Cross products Multiply. Divide each side by 13. Answer: 32

Example 4-4c In the figure, Davis, Broad, and Main Streets are all parallel. The figure shows the distances in city blocks that the streets are apart. Find x. Answer: 5

Example 4-5a Find x and y. To find x: Given Subtract 2x from each side. Add 4 to each side.

Example 4-5b To find y: The segments with lengths are congruent since parallel lines that cut off congruent segments on one transversal cut off congruent segments on every transversal.

Example 4-5c Equal lengths Multiply each side by 3 to eliminate the denominator. Subtract 8y from each side. Divide each side by 7. Answer: x = 6; y = 3

Example 4-5d Find a and b. Answer: a = 11; b = 1.5

End of Lesson 4

Lesson 5 Contents Example 1Perimeters of Similar Triangles Example 2Write a Proof Example 3Medians of Similar Triangles Example 4Solve Problems with Similar Triangles

Example 5-1a If and find the perimeter of Let x represent the perimeter of The perimeter of C

Example 5-1b Proportional Perimeter Theorem Substitution Cross products Multiply. Divide each side by 16. Answer: The perimeter of

Example 5-1c If and RX = 20, find the perimeter of Answer: R

Example 5-2a are similar with a ratio of According to Theorem 6.8, if two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides. Answer: The ratio of the lengths of the altitudes is and Find the ratio of the length of an altitude of to the length of an altitude of

Example 5-2b Answer: and Find the ratio of the length of a median of to the length of a median of

Example 5-3a In the figure, is an altitude of and is an altitude of Find x if and K

Example 5-3b Write a proportion. Cross products Divide each side by 36. Answer: Thus, JI = 28.

Example 5-3c Answer: 17.5 N In the figure, is an altitude of and is an altitude of Find x if and

Example 5-4a The drawing below illustrates two poles supported by wires.,, and Find the height of the pole.

Example 5-4b are medians of since and If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides. This leads to the proportion measures 40 ft. Also, since both measure 20 ft. Therefore,

Example 5-4c Write a proportion. Cross products Simplify. Divide each side by 80. Answer: The height of the pole is 15 feet.

Example 5-4d The drawing below illustrates the legs, of a table. The top of the legs are fastened so that AC measures 12 inches while the bottom of the legs open such that GE measures 36 inches. If BD measures 7 inches, what is the height h of the table? Answer: 28 in.

End of Lesson 5

Lesson 6 Contents Example 1Self-Similarity Example 2Create a Fractal Example 3Evaluate a Recursive Formula Example 4Find a Recursive Formula Example 5Solve a Problem Using Iteration

Example 6-1a below is found by connecting the midpoints of the sides of Prove that Given: E, D, and F are midpoints of respectively. Prove: C

Example 6-1b Proof: Statements Reasons 1. Given E, D, and F are midpoints of respectively Triangle Midsegment Theorem Alternate Interior Angles Theorem 3.

Example 6-1c 4. Corresponding Angles Postulate Transitive Property AA Similarity 6. Proof: Statements Reasons

Example 6-1d Prove: Given: Z and X are the midpoints of respectively. is formed by connecting the midpoints of the sides and of Prove that w

Example 6-1e Proof: Statements Reasons 1. Given 2. Triangle Midsegment Theorem 3. Corresponding Angles Postulate Z and X are the midpoints of respectively. 4. Reflexive Property 5. AA Similarity

Example 6-2a Draw an equilateral triangle. Create a fractal by drawing another equilateral triangle within it and shading above or beneath the triangle that shares the horizontal side of the equilateral triangle. Stages 1 and 2 are shown.

Example 6-2b Answer:

Example 6-2c Draw a square. Create a fractal by bisecting the top and left side of the square and drawing a smaller square inside the larger square as shown.

Example 6-2d Answer:

Example 6-3a Find the value of where x initially equals 1. Then use that value as the next x in the expression. Repeat the process three more times and describe your observations. The iterative process is to square the value of x, multiply that value by 3, and then subtract 1. Begin with The value of becomes the next value of x. 393, x Answer: 2, 11, 362, 393,131; x values increase with each iteration, approaching infinity.

Example 6-3b Answer: 3, –15, –447, –399,615; x values decrease with each iteration, approaching negative infinity. Find the value of where x initially equals 0. Then use that value as the next x in the expression. Repeat the process three more times and describe your observations.

Example 6-4a The diagram below represents the odd integers 1, 3, 5, and 7. Find a formula in terms of the row number for the sum of the values in the diagram. Notice that each sum is the row number squared. Answer:

Example 6-4b What is the sum of the first 10 odd numbers? The sum of the values in the tenth row will be 10 2 or 100. Answer: 100

Example 6-4c Answer: 28 Answer: S n = S n–1 n a. Find a formula for the sum of the values in the diagram. b. Find the number in row 7. Row      Sum Examine the pattern shown below.

Example 6-5a BANKING Joaquin has $1500 in a savings account that earns 4.1% interest. If the interest is compounded annually, find the balance of his account after 4 years. First, write an equation to find the balance after one year. Answer: After 4 years, Joaquin will have $ in his account.

Example 6-5b BANKING Anna has $3500 in a savings account that earns 3.8% interest. If the interest is compounded annually, find the balance of her account after 5 years. Answer: $

End of Lesson 6

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