Similar Figures 2.5. ACTIVITY: Reducing Photographs. How can you use proportions to help make decisions in art, design, and magazine layouts?
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1 .5 Similar Figures How can you use proportions to help make decisions in art, design, and magazine layouts? In a computer art program, when you click and drag on a side of a photograph, you distort it. But when you click and drag on a corner of the photograph, the dimensions remain proportional to the original. Original photograph Distorted Distorted Proportional 1 ACTIVITY: Reducing Photographs Work with a partner. You are trying to reduce the photograph to the indicated size for a nature magazine. Can you reduce the photograph to the indicated size without distorting or cropping? Eplain your reasoning. a. b. Geometry In this lesson, you will name corresponding angles and corresponding sides of similar figures. identify similar figures. find unknown measures of similar figures. 5 in. in. in. in. 4 in. 3 in. 5 in. 4 in. 70 Chapter Transformations
2 ACTIVITY: Creating Designs Math Practice Analyze Relationships How can you use mathematics to determine whether the dimensions are proportional? Work with a partner. a. Tell whether the dimensions of the new designs are proportional to the dimensions of the original design. Eplain your reasoning. Original Design 1 Design b. Draw two designs whose dimensions are proportional to the given design. Make one bigger and one smaller. Label the sides of the designs with their lengths IN YOUR OWN WORDS How can you use proportions to help make decisions in art, design, and magazine layouts? Give two eamples. 4. a. Use a computer art program to draw two rectangles whose dimensions are proportional to each other. b. Print the two rectangles on the same piece of paper. c. Use a centimeter ruler to measure the length and the width of each rectangle. d. Find the following ratios. What can you conclude? Length of larger Length of smaller Width of larger Width of smaller I love this statue. It seems similar to a big statue I saw in New York. Use what you learned about similar figures to complete Eercises 4 and 5 on page 74. Section.5 Similar Figures 71
3 .5 Lesson Lesson Tutorials Key Vocabulary similar figures, p. 7 Reading The symbol means is similar to. Common Error When writing a similarity statement, make sure to list the vertices of the figures in the correct order. Similar Figures Figures that have the same shape but not necessarily the same size are called similar figures. E B Words A C Triangle ABC is similar to Triangle DEF. Two figures are similar when D corresponding side lengths are proportional and corresponding angles are congruent. Symbols Side Lengths Angles Figures AB DE = BC EF = AC A D ABC DEF DF B E C F F EXAMPLE 1 Identifying Similar Figures Which rectangle is similar to Rectangle A? Rectangle A Rectangle B Rectangle C 3 Each figure is a rectangle. So, corresponding angles are congruent. Check to see if corresponding side lengths are proportional. Rectangle A and Rectangle B Length of A Length of B = = 1 Width of A Width of B = 3 Rectangle A and Rectangle C Length of A Length of C = 4 = 3 Width of A Width of C = 3 So, Rectangle C is similar to Rectangle A. 4 Not proportional Proportional Eercises Rectangle D is 3 units long and 1 unit wide. Which rectangle is similar to Rectangle D? 7 Chapter Transformations
4 EXAMPLE Finding an Unknown Measure in Similar Figures The triangles are similar. Find. Because the triangles are similar, corresponding side lengths are proportional. So, write and solve a proportion to find. m m m = = 7 Write a proportion. Cross Products Property = 1 Divide each side by. So, is 1 meters. The figures are similar. Find. Eercises cm ft ft ft 7 cm 1 cm EXAMPLE 3 Real-Life Application An artist draws a replica of a painting that is on the Berlin Wall. The painting includes a red trapezoid. The shorter base of the similar trapezoid in the replica is 3.75 inches. What is the height h of the trapezoid in the replica? 15 in. 1 in. Because the trapezoids are similar, corresponding side lengths are proportional. So, write and solve a proportion to find h = h Write a proportion = 1 h 1 3 = h Simplify. Multiplication Property of Equality Painting h 3.75 in. Replica So, the height of the trapezoid in the replica is 3 inches. 4. WHAT IF? The longer base in the replica is 4.5 inches. What is the length of the longer base in the painting? Section.5 Similar Figures 73
5 .5 Eercises Help with Homework 1. VOCABULARY How are corresponding angles of two similar figures related?. VOCABULARY How are corresponding side lengths of two similar figures related? 3. CRITICAL THINKING Are two figures that have the same size and shape similar? Eplain. +(-)=3 3+(-3)= 4+(-)= +(-1)= 1 Tell whether the two figures are similar. Eplain your reasoning In a coordinate plane, draw the figures with the given vertices. Which figures are similar? Eplain your reasoning.. Rectangle A: (0, 0), (4, 0), (4, ), (0, ) 7. Figure A: ( 4, ), (, ), (, 0), ( 4, 0) Rectangle B: (0, 0), (, 0), (, 3), (0, 3) Figure B: (1, 4), (4, 4), (4, 1), (1, 1) Rectangle C: (0, 0), (4, 0), (4, ), (0, ) Figure C: (, 1), (5, 1), (5, 3), (, 3) 15 The figures are similar. Find MEXICO A Meican flag is 3 inches long and 3 inches wide. Is the drawing at the right similar to the Meican flag? 13. DESKS A student s rectangular desk is 30 inches long and 1 inches wide. The teacher s desk is similar to the student s desk and has a length of 50 inches. What is the width of the teacher s desk? 11 in..5 in. 74 Chapter Transformations
6 14. LOGIC Are the following figures always, sometimes, or never similar? Eplain. a. two triangles b. two squares c. two rectangles d. a square and a triangle 15. CRITICAL THINKING Can you draw two quadrilaterals each having two 130 angles and two 50 angles that are not similar? Justify your answer. 1. SIGN All the angle measures in the sign are 0. a. You increase each side length by 0%. Is the new sign similar to the original? b. You increase each side length by inches. Is the new sign similar to the original? ft 10 ft 17. STREETLIGHT A person standing 0 feet from a streetlight casts a shadow as shown. How many times taller is the streetlight than the person? Assume the triangles are similar. 1. REASONING Is an object similar to a scale drawing of the object? Eplain. 1. GEOMETRY Use a ruler to draw two different isosceles triangles similar to the one shown. Measure the heights of each triangle to the nearest centimeter. a. Is the ratio of the corresponding heights proportional to the ratio of the corresponding side lengths? b. Do you think this is true for all similar triangles? Eplain. 5 5 height 0. Given ABC DEF and DEF JKL, is ABC JKL? Give an eample or a non-eample. Simplify. 1. ( 4 (Skills Review Handbook) ). ( 3 ) 3. ( 7 4) 4. (.5 5. MULTIPLE CHOICE You solve the equation S = w + wh for w. Which equation is correct? (Section 1.4) A w = S h B w = S h S C w = + h D ) w = S h Section.5 Similar Figures 75
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