11. If a triangle has lengths of 27 m and 11 m , check all the possible lengths for the third side. a 39 m 17 m 35 m a 16 m a

To determine the possible lengths for the third side of a triangle given two sides, 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. Given: - Side 1: \( a = 27 \, m \) - Side 2: \( b = 11 \, m \) - Side 3: \( c \) (unknown) According to the triangle inequality theorem, the following conditions must hold: 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) Let's apply these inequalities step by step. 1. **From the first inequality**: \[ 27 + 11 > c \implies 38 > c \implies c < 38 \] 2. **From the second inequality**: \[ 27 + c > 11 \implies c > 11 - 27 \implies c > -16 \] (This condition is always satisfied since \( c \) must be a positive length.) 3. **From the third inequality**: \[ 11 + c > 27 \implies c > 27 - 11 \implies c > 16 \] Combining the valid inequalities, we find: \[ 16 < c < 38 \] Now, let's check the provided options: - 39 m: Not valid (greater than 38) - 17 m: Valid (between 16 and 38) - 35 m: Valid (between 16 and 38) - 16 m: Not valid (not greater than 16) Thus, the possible lengths for the third side are: - 17 m - 35 m