Similar Triangles and the Pythagorean Theorem

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Similar Triangles and the Pythagorean Theorem Similar Triangles Two triangles are similar if they contain angles of the same measure. Similar triangles have the same shape but may be different in size. Also, the ratios of corresponding side lengths of the triangles are equal. SIMILAR TRIANGLE FACTS If two triangles have three angles of the same measure, the triangles are similar. Similar triangles always have the same shape If two triangles are similar, the sides are in direct proportion. 6 in 10 in Example: In the above triangles, 3 in 5 in 8 in 4 in 2. Example 1 One triangle has angles of 45, 45, 90 and another triangle has angles of 40, 50, and 90. Are they similar? Since these triangles have different angle measures, they are not similar.

Example 2 Are the triangles shown here similar? Since these triangles clearly have a different shape, they are not similar. Example 3 Are the triangles shown here similar? 12 cm 6 cm The ratio of the longest side lengths is = 2. 6 cm 3 cm The ratio of the shortest side lengths is = 2. 9 cm 4.5 cm The ratio of the second longest side lengths is = 2. Since the ratios are all the same, the triangles are similar. Solving for Side Lengths of Similar Triangles Many applications involve solving for side lengths of similar triangles. Since the ratio of corresponding side lengths is constant in similar triangles, a proportion may be set up and solved and an unknown side length may be determined. PROCEDURE TO SOLVE FOR A SIDE LENGTH OF A SIMILAR TRIANGLE 1. Write the ratios of corresponding side lengths. Use (?) or some other variable for the unknown side length. 2. Set two ratios equal to each other to make a proportion. Solve the proportion for the unknown as was done in Section 5.1.

Example 4 For the given similar triangles, find the unknown side length. 2 ft 2.1 ft 5 ft (?) 2.1 ft 5 ft 2 ft 2.1 ft 5 ft 2 ft The ratios of the side lengths are and. Set these two ratios equal to each other to obtain =. Solve this proportion for the unknown by setting the cross products equal and dividing. 2.1 2 = 5 (?) 4.2 = 5 (?) 4.2 5 = (?) = 0.84 ft Example 5 For the given similar triangles, find the unknown side length. 3.5 ft 5.2 ft 1.4 ft (?) 3.5 ft 1.4 ft 5.2 ft The proportion consists of =. The unknown is determined by cross-multiplying and dividing. 1.4 5.2 = 3.5 (?) 7.28 = 3.5 (?) 7.28 3.5 = (?) = 2.08 ft

Example 6 A building casts a 32 ft shadow. A 4 foot tall boy standing near the building casts a shadow 6 ft in length. Using similar triangles, determine the height of the building. The two triangles illustrated are similar. 32 ft 6 ft 4 ft The proportion consists of =. Solve this proportion for the unknown by making the cross products equal and then dividing. 32 4 = 6 (?) 128 = 6 (?) 128 6 = (?) = 21.33 ft (rounded to a hundredth of a foot) Example 7 For the given similar triangles, find the unknown side length. 7.5 ft 4.4 ft 8.2 ft The proportion consists of =. 4.4 8.2 = 7.5 (?) 36.08 = 7.5 (?) 36.08 7.5 = (?) = 4.81 ft (rounded to a hundredth of a foot)

Right Triangles and the Pythagorean Theorem A triangle is a right triangle if it contains a 90 degree angle. The Pythagorean Theorem states the following relationship between the side lengths. THE PYTHAGOREAN THEOREM For any given right triangle with side lengths A, B, and C, where C is the longest side, the following is always true. C 2 A 2 B 2 C B 90 A The lengths of any of the sides may be determined by using the following formulas. Note that these formulas involve use of the square root () function on a calculator. C A 2 B 2 A C 2 B 2 B C 2 A 2 The longest side C is known as the hypotenuse of the triangle. Example 8 Given the following right triangle, find the missing side length. (?) in 5 in 12 in Substituting A = 5 and B = 12 into the formula C A 2 B 2 results in C 5 2 12 2 25 144 = 169 = 13 The square root may be determined by entering the keystrokes 169 on a calculator.

Example 8 Given the following right triangle, find the missing side length. Round the result to the nearest tenth of an inch. 9 in 5 in (?) in Substituting A = 5 and C = 9 into the formula B 9 2 5 2 81 25 = 56 = 7.48 7.5 inches. B C 2 A 2 results in The square root may be determined by entering the keystrokes 56 on a calculator. MATH HINT When calculating the side length of a right triangle, remember the following. LONGEST SIDE (SHORT SIDE) 2 (OTHER SHORT SIDE) 2 SHORT SIDE (LONGEST SIDE) 2 (OTHER SHORT SIDE) 2 Remember to square terms and add or subtract them before taking the square root. Example 9 The cross-section of a roof is shown below. What is the distance from the peak of the roof to the edge of the roof? 8 ft 20 ft 20 ft 40 ft This cross-section consists of two right triangles, each with short sides of length 8 feet and 20 feet. The longest side = (?) = 8 2 20 2 464 21.54 feet.

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