iven two sides of a triangle, find a range of po \( 4 \mathrm{~cm}, 17 \mathrm{~cm} \)

To find the range of possible lengths for the third side of a triangle when given two sides, we can use the triangle inequality theorem. The 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 the two given sides be \( a = 4 \, \text{cm} \) and \( b = 17 \, \text{cm} \). We will denote the length of the third side as \( c \). According to the triangle inequality theorem, we have the following conditions: 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) Now, let's apply these inequalities step by step: 1. From \( a + b > c \): \[ 4 + 17 > c \implies 21 > c \implies c < 21 \] 2. From \( a + c > b \): \[ 4 + c > 17 \implies c > 17 - 4 \implies c > 13 \] 3. From \( b + c > a \): \[ 17 + c > 4 \implies c > 4 - 17 \implies c > -13 \] (This condition is always satisfied since \( c \) must be positive.) Combining the results from the first two inequalities, we find: \[ 13 < c < 21 \] Thus, the range of possible lengths for the third side \( c \) is: \[ (13 \, \text{cm}, 21 \, \text{cm}) \]