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Similar Figures Octahedral fluorite is a crystal found in nature. It grows in the shape of an octahedron, which is a solid figure with eight triangular faces. The triangles in different-sized fluorite crystals are similar figures. Similar figures have the same shape but not necessarily the same size. 5-5 C2: Similar Figures and Proportions Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. Corresponding sides B A C D E F Corresponding angles SIMILAR FIGURES Tickmarks are symbols used to help you determine: • Congruence of sides and angles • Right angles • Parallel lines If two figures are similar, then the measures of the corresponding angles are equal and the ratios of the lengths of the corresponding sides are proportional. To find out if triangles are similar, determine whether the ratios of the lengths of their corresponding sides are proportional. If the ratios are proportional, then the corresponding angles must have equal measures. 1 Similar Figures In order for figures to be similar 2 things must be true: 1. Corresponding angles must be congruent 2. Corresponding sides must be proportional Reading Math A side of a figure can be named by its endpoints, with a bar above. AB Without the bar, the letters indicate the length of the side. When matching corresponding sides be sure to list segments in the correct order. Additional Example 1: Determining Whether Two Triangles Are Similar Try This: Example 1 Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E AB corresponds to DE. 16 in A 10 in C 28 in BC corresponds to EF. D 4 in 7 in 40 in F AC corresponds to DF. B Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar. E AB corresponds to DE. 9 in 9 in C A 21 in BC corresponds to EF. D 3 in 7 in 27 in F AC corresponds to DF. B AB ? BC ? AC = = Write ratios using the corresponding sides. DE EF DF 4 ? 7 ? 10 Substitute the length of the sides. = = 16 28 40 ? 1 ? 1 1 = Simplify each ratio. = 4 4 4 Since the ratios of the corresponding sides are equivalent, the triangles are similar. AB ? BC ? AC = = Write ratios using the corresponding sides. DE EF DF 3 ? 7 ? 9 Substitute the length of the sides. = = 9 21 27 ? 1 ? 1 1 = Simplify each ratio. = 3 3 3 Since the ratios of the corresponding sides are equivalent, the triangles are similar. In figures with four or more sides, it is possible for the corresponding side lengths to be proportional and the figures to have different shapes. To find out if these figures are similar, first check that their corresponding angles have equal measures. 8m 10 m 4m 4m 5m Additional Example 2: Determining Whether Two Four-Sided Figures are Similar Use the properties of similarity to determine whether the figures are similar. The corresponding angles of the figures have equal measure. 5m 8m 10 m 10 m = 5 m 8m 4m Write each set of corresponding sides as a ratio. 2 Additional Example 2 Continued Additional Example 2 Continued Determine whether the ratios of the lengths of the corresponding sides are proportional. MN MN corresponds to QR. QR ? corresponding ? MP ? NO OP = MNratios Write using = = ST QR RS QT sides. NO NO corresponds to RS. RS OP ST 6 ? 8 ? 4 ? 10 Substitute the length of the 9 = 12 = 6 = 15 sides. OP corresponds to ST. 2? 2? 2? 2 = = ratio. = Simplify 3 each 3 3 3 MP MP corresponds to QT. QT Since the ratios of the corresponding sides are equivalent, the figures are similar. Try This: Example 2 Try This: Example 2 Continued Use the properties of similarity to determine whether the figures are similar. M P 100 m 80° 60 m M 65° 47.5 m 90° 125° 80 m N Q O T 65° 240 m 190 m 90° O T 65° 240 m R MN MN corresponds to QR. QR NO NO corresponds to RS. RS 400 m OP ST OP corresponds to ST. 190 m 125° 90° S 320 m 125° 80 m 80° 125° 90° 65° 47.5 m Q Write each set of corresponding sides as a ratio. P 100 m 80° 60 m N 400 m 80° R The corresponding angles of the figures have equal measure. 320 m S MP MP corresponds to QT. QT Try This: Example 2 Continued Determine whether the ratios of the lengths of the corresponding sides are proportional. M P 100 m 80° 60 m 65° 47.5 m 90° 125° 80 m N Q T 65° 240 m 190 m 125° 90° R 320 m 8-4 C1: Similar Figures O 400 m 80° ? corresponding ? MP ? NO OP = MNratios Write using = = sides. ST QR RS QT 60 ? 80 ? 47.5of Substitute the length the ? 100 240 = 320 = 190 = 400 sides. 1? 1? 1? 1 4 4 4 4 = =each=ratio. Simplify S Since the ratios of the corresponding sides are equivalent, the figures are similar. 3 Insert Lesson Title Here Similar Figures Vocabulary similar corresponding sides corresponding angles Learn to use ratios to identify similar figures. Course 1 Two or more figures are similar if they have exactly the same shape. Similar figures may be different sizes. A 2 cm B Similar figures have corresponding sides and corresponding angles. • Corresponding sides have are proportional. lengths that • Corresponding angles are congruent. 3 cm D 2 cm 3 cm W 9 cm 6 cm Z 6 cm C X Corresponding sides: AB corresponds to WX. BC corresponds to XY. CD corresponds to YZ. AD corresponds to WZ. 9 cm Y Corresponding angles: A corresponds to B corresponds to W. X. C corresponds to D corresponds to Y. Z. Similar Figures A 3 cm 2 cm B D 2 cm 3 cm W 9 cm 6 cm Additional Example 1: Finding Missing Measures in Similar Figures Z 6 cm The two triangles are similar. Find the missing length y and the measure of D. C X 9 cm Y In the rectangles above, one proportion is AB AD 2 3 = , or = . WX WZ 6 9 If you cannot use corresponding side lengths to write a proportion, or if corresponding angles are not congruent, then the figures are not similar. 100 111 Write a proportion using ____ = ___ 200 y corresponding side lengths. 200 • 111 = 100 • y The cross products are equal. Course 1 4 Additional Example 1 Continued Try This: Example 1 The two triangles are similar. Find the missing length y and the measure of B. The two triangles are similar. Find the missing length y and the measure of D. 22,200 = 100y 22,200 = ____ 100y ______ 100 100 A 60 m 65° 50 m 45° 52 m y is multiplied by 100. Divide both sides by 100 to undo the multiplication. B 120 m 100 m y 222 mm = y Angle D is congruent to angle C, and m m D = 70° 100 • 52 = 50 • y Try This: Example 1 Continued 5,200 = 50y _____ ___ 50 50 104 m = y Additional Example 2: Problem Solving Application Divide both sides by 50 to undo the multiplication. The answer will be the width of the actual painting. 1 Understand the Problem List the important information: A= B = 65° • The actual painting and the reduction above are similar. • The reduced painting is 2 cm tall and 3 cm wide. • The actual painting is 54 cm tall. Additional Example 2 Continued 2 Make a Plan Additional Example 2 Continued 3 Draw a diagram to represent the situation. Use the corresponding sides to write a proportion. Actual Reduced 2 The cross products are equal. y is multiplied by 50. Angle B is congruent to angle A, and m 65°. m Write a proportion using corresponding side lengths. This reduction is similar to a picture that Katie painted. The height of the actual painting is 54 centimeters. What is the width of the actual painting? The two triangles are similar. Find the missing length y and the measure of B. 5,200 = 50y 50 52 ____ = ___ 100 y C = 70°. Solve 2 cm 3 cm _____ = w cm Write a proportion. 54 cm 54 • 3 = 2 • w The cross products are equal. 162 = 2w 162 2w ____ = ___ 2 2 81 = w 54 3 w w is multiplied by 2. Divide both sides by 2 to undo the multiplication. The width of the actual painting is 81 cm. 5 Try This: Example 2 Additional Example 2 Continued 4 Look Back Estimate to check your answer. The ratio of the heights is about 2:50 or 1:25. The ratio of the widths is about 3:90, or 1:30. Since these ratios are close to each other, 81 cm is a reasonable answer. This reduction is similar to a picture that Marty painted. The height of the actual painting is 3 in. 39 inches. What is the width of the actual painting? 1 4 in. Understand the Problem The answer will be the width of the actual painting. List the important information: • The actual painting and the reduction above are similar. • The reduced painting is 3 in. tall and 4 in. wide. • The actual painting is 39 in. tall. Try This: Example 2 Continued 2 Make a Plan Try This: Example 2 Continued 3 Draw a diagram to represent the situation. Use the corresponding sides to write a proportion. Solve 3 in 4 in _____ = ____ Write a proportion. 39 in w in 39 • 4 = 3 • w The cross products are equal. Actual Reduced 3 156 = 3w 3w 156 = ___ ____ 3 3 52 = w 39 4 w w is multiplied by 3. Divide both sides by 3 to undo the multiplication. The width of the actual painting is 52 inches. Try This: Example 2 Continued 4 Look Back Estimate to check your answer. The ratio of the heights is about 4:40, or 1:10. The ratio of the widths is about 5:50, or 1:10. Since these ratios are the same, 52 inches is a reasonable answer. 6