(b) The other triangle has side lengths 15,9 , and 12 . Compute the sum of the squares of the shorter lengths. \[ 9^{2}+12^{2}=\square \] Compute the square of the longest length. \( 15^{2}=\square \) What kind of triangle is it? Acute triangle Right triangle Obtuse triangle

To determine the type of triangle based on the side lengths 15, 9, and 12, we will follow these steps: 1. **Compute the sum of the squares of the shorter lengths (9 and 12)**: \[ 9^2 + 12^2 \] 2. **Compute the square of the longest length (15)**: \[ 15^2 \] 3. **Compare the results to classify the triangle**: - If \(9^2 + 12^2 > 15^2\), the triangle is acute. - If \(9^2 + 12^2 = 15^2\), the triangle is right. - If \(9^2 + 12^2 < 15^2\), the triangle is obtuse. Let's perform the calculations step by step. ### Step 1: Compute \(9^2 + 12^2\) \[ 9^2 = 81 \] \[ 12^2 = 144 \] Now, sum these values: \[ 9^2 + 12^2 = 81 + 144 = 225 \] ### Step 2: Compute \(15^2\) \[ 15^2 = 225 \] ### Step 3: Compare the results Now we compare: \[ 9^2 + 12^2 = 225 \] \[ 15^2 = 225 \] Since \(9^2 + 12^2 = 15^2\), the triangle is a right triangle. ### Final Answers: - The sum of the squares of the shorter lengths is \(225\). - The square of the longest length is \(225\). - The triangle is a **Right triangle**.