Similar Shapes and Proportions 4.1 Similar Shapes and Proportions How can you use ratios to determine if two figures are similar?

Texas Essential Knowledge and Skills The student is expected to: Proportionality—7.5.A Generalize the critical attributes of similarity, including ratios within and between similar shapes Mathematical Processes 7.1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Warm Up Find the cross products, and then tell whether the ratios are equal. 16 6 , 40 15 1. 240 = 240; equal 3 8 18 46 , 2. 138 = 144; not equal 8 9 , 24 27 3. 216 = 216; equal 28 12 , 42 18 4. 504 = 504; equal

Octahedral fluorite is a crystal found in nature 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.

When naming similar figures, list the letters of the corresponding vertices in the same order. In the previous table ∆ABC ~ ∆DEF. Writing Math

Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. 82◦ Corresponding angles D E F Corresponding sides A B C 55◦ 43◦

SIMILAR FIGURES Two figures are similar if The measures of their corresponding angles are equal. The ratios of the lengths of the corresponding sides are proportional.

A side of a figure can be named by its endpoints, with a bar above. Without the bar, the letters indicate the length of the side. Reading Math

Additional Example 1: Determining Whether Two Triangles Are Similar 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 10 in A C 28 in BC corresponds to EF. 4 in D 7 in 40 in F AC corresponds to DF. B AB DE = ? BC EF = ? AC DF Write ratios using the corresponding sides. 4 16 = ? 7 28 = ? 10 40 Substitute the length of the sides. 1 4 = ? 1 4 = ? 1 4 Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Check It Out: 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. 9 in 9 in A C 21 in BC corresponds to EF. 3 in D 7 in 27 in F AC corresponds to DF. B AB DE = ? BC EF = ? AC DF Write ratios using the corresponding sides. 3 9 = ? 7 21 = ? 9 27 Substitute the length of the sides. 1 3 = ? 1 3 = ? 1 3 Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Additional Example 2: Determining Whether Two Four-Sided Figures are Similar Tell whether the figures are similar. The corresponding angles of the figures have equal measure. Write each set of corresponding sides as a ratio.

Additional Example 2 Continued MN QR MN corresponds to QR. NO RS NO corresponds to RS. OP ST OP corresponds to ST. MP QT MP corresponds to QT.

Additional Example 2 Continued Determine whether the ratios of the lengths of the corresponding sides are proportional. MN QR = ? NO RS OP ST MP QT Write ratios using corresponding sides. 6 9 = ? 8 12 4 10 15 Substitute the length of the sides. 2 3 = ? Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the figures are similar.

Tell whether the figures are similar. Check It Out: Example 2 Tell whether the figures are similar. 100 m 80 m 60 m 47.5 m 80° 90° 125° 65° M P N O 400 m 320 m 190 m 240 m Q T R S The corresponding angles of the figures have equal measure. Write each set of corresponding sides as a ratio.

Check It Out: Example 2 Continued 80° 90° 125° 65° M P N O 400 m 320 m 190 m 240 m Q T R S MN QR MN corresponds to QR. NO RS NO corresponds to RS. OP ST OP corresponds to ST. MP QT MP corresponds to QT.

Check It Out: Example 2 Continued Determine whether the ratios of the lengths of the corresponding sides are proportional. 100 m 80 m 60 m 47.5 m 80° 90° 125° 65° M P N O 400 m 320 m 190 m 240 m Q T R S MN QR = ? NO RS OP ST MP QT Write ratios using corresponding sides. 60 240 = ? 80 320 47.5 190 100 400 Substitute the length of the sides. 1 4 = ? Simplify each ratio. Since the ratios of the corresponding sides are equivalent, the figures are similar.

ADDITIONAL EXAMPLE 1 Tell whether the triangles are similar. Yes, the triangles are similar.

ADDITIONAL EXAMPLE 2 Molly made two sizes of tiles. Are the shapes of the tiles similar? No, the shapes are not similar.

4.1 LESSON QUIZ 7.5.A Tell whether the shapes are similar. 1. a rectangle with height 7 inches and length 12 inches, and a rectangle with height 8.75 inches and length 15 inches similar

Tell whether the shapes are similar. 2. not similar

Tell whether the shapes are similar. 3. not similar 4. similar

12 ft; since the triangles are similar, their In the figure to the right, ΔABC is similar to ΔDAC. What must the length of AC be? Explain. 12 ft; since the triangles are similar, their corresponding sides are in proportion. so AC = 12.

How can you use ratios to determine if two figures are similar? Sample answer: If the measures of the corresponding angles are equal, check the ratios of the lengths of the corresponding sides. If the ratios are proportional, the shapes are similar.