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Triangle Inequality Theorem

Triangle Inequality Theorem. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Inequalities in One Triangle. They have to be able to reach!!. 3. 2. 4. 3. 6. 6. 3. 3.

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Triangle Inequality Theorem

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  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 6 3 3 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. 6

  3. Triangle Inequality Theorem • AB + BC > AC A • AB + AC > BC • AC + BC > AB B C

  4. Triangle Inequality Theorem • Biggest Side Opposite Biggest Angle • Medium Side Opposite Medium Angle • Smallest Side Opposite Smallest Angle A 3 5 B C m<B is greater than m<C

  5. A 65 30 B C Triangle Inequality Theorem • Converse is true also • Biggest Angle Opposite _____________ • Medium Angle Opposite ______________ • Smallest Angle Opposite _______________ Angle A > Angle B > Angle C So CB >AC > AB

  6. Example: List the measures of the sides of the triangle, in order of least to greatest. <B = 4x <A = 2x + 1 <C = 4x -11 Solving for x: 2x +1 + 4x + 4x - 11 =180 Note: Picture is not to scale 10x - 10 = 180 Plugging back into our Angles: <A = 39o; <B = 76; <C = 65 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 3x - 2 A x < 7 x + 2 B

  8. Example: Determine if the following lengths are legs of triangles B) 9, 5, 5 A) 4, 9, 5 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: 4 + 5 ? 9 9 > 9 5 + 5 ? 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. 12 6 X = ? Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18

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