Triangle Inequalities

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Presentation transcript:

Triangle Inequalities

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a + b > c a + c > b b + c > a

Triangle Inequality Theorem: Can you make a triangle? Yes!

Triangle Inequality Theorem: Can you make a triangle? NO because 4 + 5 < 12

6, 4, 5 State if the three numbers can be the measures of the sides of a triangle.

3, 4, 5 State if the three numbers can be the measures of the sides of a triangle.

Finding the range of the third side: Example Given a triangle with sides of length a and b, find the range of possible values for the third side. Solution Let x be the length of the third side of the triangle. The maximum value: x < a + b The minimum value: x > |a – b| So |a – b|< x < a + b (x is between |a – b| and a + b.) x < a + b x > |a – b|

Finding the range of the third side: Given The lengths of two sides of a triangle Find the maximum value by adding the two sides. Find the minimum value by subtracting the two sides. Difference < Third Side < Sum

Finding the range of the third side: Example Given a triangle with sides of length 3 and 7, find the range of possible values for the third side. Solution Let x be the length of the third side of the triangle. The maximum value: Add the two sides The minimum value: Subtract the two sides So x is between 4 and 10 4 < x < 10 x < 10 x x > 4 x

In a Triangle: The smallest angle is opposite the smallest side. The largest angle is opposite the largest side.