Find the range of possible measures of the third side. $$ 8,17 $$

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To find the range of possible measures for the third side of a triangle when the lengths of the other two sides are given as $$ a = 8 $$ and $$ b = 17 $$, we can use the triangle inequality theorem. This 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. Let's denote the length of the third side as $$ c $$. According to the triangle inequality, we have the following conditions: 1. $$ a + b > c $$ 2. $$ a + c > b $$ 3. $$ b + c > a $$ Substituting the known values into these inequalities: 1. $$ 8 + 17 > c $$ 2. $$ 8 + c > 17 $$ 3. $$ 17 + c > 8 $$ Now, let's simplify each inequality: 1. $$ 25 > c $$ or $$ c < 25 $$ 2. $$ c > 17 - 8 $$ or $$ c > 9 $$ 3. $$ c > 8 - 17 $$ or $$ c > -9 $$ (This condition is always satisfied since $$ c $$ must be positive.) From the first two inequalities, we can combine the results: - From $$ c < 25 $$ - From $$ c > 9 $$ Thus, the range of possible measures for the third side $$ c $$ is: $$ 9 < c < 25 $$ In interval notation, this can be expressed as: $$ (9, 25) $$ So, the range of possible measures of the third side is $$ (9, 25) $$. The third side must be greater than 9 and less than 25.

Find the range of possible measures of the third side.

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