Find the range of possible measures of the third side. \[ 8,17 \]

To find the range of possible measures for the third side of a triangle where the other two sides are 8 and 17, we can apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. Let the third side be \( x \). We need to satisfy the following inequalities: 1. \( 8 + 17 > x \) --> \( 25 > x \) --> \( x < 25 \) 2. \( 8 + x > 17 \) --> \( x > 9 \) 3. \( 17 + x > 8 \) --> \( x > -9 \) (This is always true since \( x \) must be positive) Combining the relevant inequalities, we find: \( 9 < x < 25 \) So, the range for the third side is \( (9, 25) \).