Name Class Date. Finding an Unknown Distance

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1 Name Class Date 7-5 Using Proportional Relationships Going Deeper Essential question: How can you use similar triangles and similar rectangles to solve problems? When you know that two polygons are similar, you can often use the proportionality of corresponding sides to find unknown side lengths. 1 G-SRT.2.5 EXAMPLE Finding an Unknown Distance You want to find the distance across a canyon. In order to find the distance XY, you locate points as described below. Explain how to use this information and the figure to find XY. 1. Identify a landmark, such as a tree, at X. Place a marker (Y) directly across the canyon from X. X 2. At Y, turn 90 away from X and walk 400 feet in a straight line. Place a marker (Z) at this location. W Z 600 ft 400 ft Y 3. Continue walking another 600 feet. Place a marker (W) at this location. 327 ft V 4. Turn 90 away from the canyon and walk until the marker Z aligns with X. Place a marker (V) at this location. Measure WV. A Show that XYZ VWZ. How can you show that two pairs of angles in the triangles are congruent? What can you conclude? Why? B Use the fact that corresponding sides of similar triangles are proportional. Complete the proportion: XY VW = Substitute the known lengths in the proportion: Solve the proportion: XY = REFLECT 1a. Compare this problem to the example Using the ASA Congruence Criterion in the lesson Triangle Congruence: ASA, AAS, and HL. How are the solution methods similar? How are they different? Chapter Lesson 5

2 2 G-SRT.2.5 EXAMPLE Finding an Unknown Height In order to find the height of a palm tree, you measure the tree s shadow and, at the same time of day, you measure the shadow cast by a meter stick that you hold at a right angle to the ground. The measurements are shown in the figure. Find the height of the tree. X 1 m A A Show that ABC XYZ. Z 1.6 m Y C 7.2 m B You can assume that the rays of the sun are parallel. This means that ZX CA. What can you say about Z and C? Why? Explain how to show that ABC XYZ. B Determine the scale factor k for the dilation in the sequence of similarity transformations that maps XYZ to ABC. Find the ratio of corresponding sides. The scale factor is BC YZ = = 4.5. So, AB = k XY =. REFLECT 2a. How could you solve the problem by writing and solving a proportion? 2b. How can you check that your answer is reasonable? 2c. What must be true about the palm tree in order for this method to work? Chapter Lesson 5

3 3 G-MG.1.3 EXAMPLE Solving a Problem About Similar Rectangles A typographic grid system is a set of horizontal and vertical lines that determine the placement of type or images on a page. The lines create an array of identical rectangles. 54 cm A graphic designer wants to lay out a new grid system for a poster that is 54 cm wide by 72 cm tall. The grid must have margins of 2 cm along all edges and 2 cm between each horizontal row of rectangles. There must be 5 rows of rectangles and each rectangle must be similar to the poster itself. 72 cm What are the dimensions of the rectangles? How many rectangles should appear in each row? How much space should be between the columns of rectangles? Concert August 8 A Determine the number of horizontal 2-centimeter bands that are needed, including the top and bottom margins. B Find the remaining amount of vertical space and divide by 5 to find the height of each rectangle. C To find the width of each rectangle, use the fact that the rectangles are similar to the overall poster. Show how to set up a proportion to find the width of each rectangle. D Determine the maximum number of rectangles that can appear in a row. E Find the total amount of horizontal space taken up by the rectangles and the left and right margins. F Assuming the remaining space is distributed evenly, determine the amount of space that should appear between the columns of rectangles. REFLECT 3a. Is there another solution to the problem? Explain. Chapter Lesson 5

4 PRACTICE 1. To find the distance XY across a lake, you locate points as shown in the figure. Explain how to use this information to find XY. U 300 ft 500 ft V 400 ft Z 800 ft 600 ft 2. In order to find the height of a cliff, you stand at the bottom of the cliff, walk 60 ft from the base, and place a mirror on the ground. Then you face the cliff and step back 5 feet so that you can see the top of the cliff in the mirror. Assuming your eyes are 6 feet above ground, explain how to use this information to find the height of the cliff. (Hint : When light strikes a mirror, the angle of incidence is congruent to the angle of reflection, as marked in the figure.) X P 6 ft M J Y Q 5 ft 60 ft K Mirror r 3. Error Analysis A student who is 72 inches tall wants to find the height of a flagpole. He measures the length of the flagpole s shadow and the length of his own shadow at the same time of day, as shown in his sketch below. Explain the error in the student s work. The triangles are similar by the AA Similarity Criterion, so corresponding sides are proportional. x 72 = x = 72 48, so x = 27 in in. 48 in. 128 in. 4. A graphic designer wants to lay out a grid system for a brochure that is 15 cm wide by 20 cm tall. The grid must have margins of 1 cm along all edges and 1 cm between each horizontal row of rectangles. There must be 4 rows of rectangles and each rectangle must be similar to the brochure itself. What are the dimensions of the rectangles? How many rectangles should appear in each row? How much space should be between the columns of rectangles? Give two different solutions. x Chapter Lesson 5

5 Name Class Date 7-5 Additional Practice Chapter Lesson 5

6 Problem Solving = = Chapter Lesson 5