Sarah is creating a triangular birthday card for her teacher. In order to glue lace around the outside of the card, Sarah needs to know the length measure 4 inches and 9 inches. If the length of the third side is \( x \) inches, which inequality is true? \( 5 \leq x \leq 13 \) \( 5

Ask by Mckenzie Medina. in the United States

Jan 23,2025

To determine the possible lengths for the third side \( x \) of the triangle, 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 remaining side. Given the two sides of the triangle are 4 inches and 9 inches, we'll apply the triangle inequality theorem as follows: 1. **Sum of the first two sides greater than the third side:** \[ 4 + 9 > x \implies 13 > x \implies x < 13 \] 2. **Sum of the first side and the third side greater than the second side:** \[ 4 + x > 9 \implies x > 5 \] 3. **Sum of the second side and the third side greater than the first side:** \[ 9 + x > 4 \] This simplifies to: \[ x > -5 \] Since \( x \) represents a length, it must be positive, so this condition is always true and doesn’t provide additional restrictions beyond \( x > 5 \). Combining these inequalities, we find: \[ 5 < x < 13 \] **Answer:** \( 5 < x < 13 \)