(a) One triangle has side lengths 12,14 , and 10 . Compute the sum of the squares of the shorter lengths \( 10^{2}+12^{2}=\square \) Compute the square of the longest length. \( 14^{2}=\square \) What kind of triangle is it? Acute triangle Right triangle Obtuse triangle

Let's analyze the given triangle with side lengths 10, 12, and 14. ### 1. Sum of the Squares of the Shorter Lengths Compute \( 10^{2} + 12^{2} \): \[ 10^{2} + 12^{2} = 100 + 144 = 244 \] ### 2. Square of the Longest Length Compute \( 14^{2} \): \[ 14^{2} = 196 \] ### 3. Determining the Type of Triangle To determine the type of triangle, compare the sum of the squares of the two shorter sides with the square of the longest side: - If \( a^{2} + b^{2} > c^{2} \), the triangle is **acute**. - If \( a^{2} + b^{2} = c^{2} \), the triangle is **right**. - If \( a^{2} + b^{2} < c^{2} \), the triangle is **obtuse**. In this case: \[ 244 > 196 \] Since \( 10^{2} + 12^{2} > 14^{2} \), the triangle is an **acute triangle**. **Answer:** \[ 10^{2} + 12^{2} = 244 \] \[ 14^{2} = 196 \] **The triangle is an acute triangle.**