Section 5.5 Notes: The Triangle Inequality

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Section 5-5 Inequalities for One Triangle

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Section 5.5 Notes: The Triangle Inequality EQ: How are any two sides of a triangle related to the third side?

Triangle Inequality Theorem Triangle Inequality Theorem The sum of the lengths of any two sides must be greater than the third side. PQ + QR > PR QR + PR > PQ PR + PQ > QR Q P R

Example 1 a) Is it possible to form a triangle with side lengths of 6.5, 6.5, and 14.5? If not, explain why not. b) Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not. 6.5 + 6.5 = 13 > 14.5 (not true) Since 13 is not greater than 14.5, it is not possible to form a triangle with those side lengths. 6.8 + 7.2 = 14 > 5.1 (true) 6.8 + 5.1 = 11.9 > 7.2 (true) 7.2 + 5.1 = 12.3 > 6.8 (true) Since all of the sums of two sides is greater than the third side, it is possible.

Example 2 In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? a) 7 b) 9 c) 11 d) 13 7.2 + 5.2 = 12.4 Any value greater than 12.4 cannot be PR. Therefore d) 13 cannot be the length of PR

Do the you try before looking at the answer.

YOU TRY! Decide whether it is possible to construct a triangle with the given side lengths. Explain your reasoning. 1. 4 ft, 9 ft, 10 ft 2. 8 m , 9 m , 18 m 3. 5 cm, 7 cm, 12 cm 4 + 9 = 13 > 10 (true) 9 + 10 = 19 > 4 (true) 4 + 10 = 14 > 9 (true) Since all of the sums of two sides is greater than the third side, it is possible. 8 + 9 = 17 > 18 (not true) Since it is not true, these measurements do not create a triangle 5 + 7 = 12 > 12 (not true) Since 12 is not greater than 12 these measurements do not create a triangle.

Example 4 The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side. a. 4, 8 b. 13, 8 c. 10, 15 To solve for these problems you have to create an inequality in order to solve for the missing side. See examples. 4 + x > 8  x > 4 8 + x > 4  x > -4 8 + 4 > x  12 > x Range = 4 < x < 12 13 + x > 8  x > -5 8 + x > 13  x > 5 13 + 8 > x  21 > x Range = 5 < x < 21 10 + x > 5  x > -5 5 + x > 10  x > 5 10 + 5 > x  15 > x Range = 5 < x < 15

Do the you try before looking at the answer.

You Try! Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. 1. 5 inches, 12 inches 2. 3 feet, 40 inches 5 + x > 12  x > 7 12 + x > 5  x > -7 12 + 5 > x  17 > x Range: 7 < x < 17 3 feet = 36 inches 36 + x > 40  x > 4 40 + x > 36  x > -4 36 + 40 > x  76 > x Range: 4 < x < 76