(a) One triangle has side lengths \( \mathbf{1 6}, \mathbf{6} \), and 13. Compute the sum of the squares of the shorter lengths. \[ 6^{2}+13^{2}=\square \] Compute the square of the longest length. \[ 16^{2}=\square \] What kind of triangle is it? Acute triangle Right triangle Obtuse triangle (b) The other triangle has side lengths 12,15 , and 9. 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

Let's analyze both triangles step by step. --- ### **(a) Triangle with Side Lengths 6, 13, and 16** 1. **Sum of the Squares of the Shorter Lengths:** \[ 6^{2} + 13^{2} = 36 + 169 = \boxed{205} \] 2. **Square of the Longest Length:** \[ 16^{2} = \boxed{256} \] 3. **Determining the Type of Triangle:** - **Comparison:** \[ 6^{2} + 13^{2} = 205 < 16^{2} = 256 \] - **Conclusion:** Since the sum of the squares of the shorter sides is less than the square of the longest side, the triangle is **Obtuse**. **Answer:** - Sum of squares of shorter lengths: **205** - Square of longest length: **256** - **Type of Triangle:** Obtuse triangle --- ### **(b) Triangle with Side Lengths 9, 12, and 15** 1. **Sum of the Squares of the Shorter Lengths:** \[ 9^{2} + 12^{2} = 81 + 144 = \boxed{225} \] 2. **Square of the Longest Length:** \[ 15^{2} = \boxed{225} \] 3. **Determining the Type of Triangle:** - **Comparison:** \[ 9^{2} + 12^{2} = 225 = 15^{2} = 225 \] - **Conclusion:** Since the sum of the squares of the shorter sides equals the square of the longest side, the triangle is **Right**. **Answer:** - Sum of squares of shorter lengths: **225** - Square of longest length: **225** - **Type of Triangle:** Right triangle ---