5.5 triangle inequality theorem
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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. It can be written as inequalities comparing the different side lengths, such as AB + AC > BC. The converse is also true - the side opposite the largest angle will be the longest side. Examples demonstrate using the triangle inequality theorem to determine if a set of lengths could form a triangle or find the possible range of values for the third side of a triangle given two side lengths.
5.5 triangle inequality theorem
- 1. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side
- 2. Inequalities in One Triangle They have to be able to reach!! 3 2 4 3 6 3 3 6 6 Note that there is only one situation that you can have a triangle; when the sum of two sides of the triangle are greater than the third.
- 3. Triangle Inequality Theorem A AB + AC > BC AB + BC > AC AC + BC > AB B C
- 4. Triangle Inequality Theorem Biggest Side Opposite Biggest Angle A Medium Side Opposite Medium Angle Smallest Side Opposite Smallest Angle 3 5 B C m<B is greater than m<C
- 5. Triangle Inequality Theorem Converse is true also A Biggest Angle Opposite _____________ Medium Angle Opposite ______________ Smallest Angle Opposite _______________ 65 30 C Angle A > Angle B > Angle C So CB >AC > AB B
- 6. Example: List the measures of the sides of the triangle, in order of least to greatest. B <A = 2x + 1 <B = 4x <C = 4x -11 Solving for x: A C 2x +1 + 4x + 4x - 11 =180 Note: Picture is not to scale Plugging back into our Angles: <A = 39o; <B = 76; <C = 65 10x - 10 = 180 10x = 190 X = 19 Therefore, BC < AB < AC
- 7. Using the Exterior Angle Inequality Example: Solve the inequality if AB + AC > BC C (x+3) + (x+ 2) > 3x - 2 x+3 2x + 5 > 3x - 2 x<7 3x - 2 A x+2 B
- 8. Example: Determine if the following lengths are legs of triangles A) 4, 9, 5 B) 9, 5, 5 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: 4+5 ? 9 5+5 ? 9 9>9 10 > 9 Since the sum is not greater than the third side, this is not a triangle Since the sum is greater than the third side, this is a triangle
- 9. Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? B 12 6 A C X=? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18
- 10. Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? B 12 6 A C X=? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18