8) Determine if the lengths 3,9 , and 10 form an acute, right, or obtuse triangle, or if they do-not form a triangle at all. not a triangle acute right obtuse

To determine whether the lengths **3, 9, and 10** form an acute, right, or obtuse triangle (or do not form a triangle at all), follow these steps: ### 1. **Check the Triangle Inequality Theorem** A set of three lengths can form a triangle if the sum of any two sides is greater than the third side. - **3 + 9 > 10?** \( 12 > 10 \) → **True** - **3 + 10 > 9?** \( 13 > 9 \) → **True** - **9 + 10 > 3?** \( 19 > 3 \) → **True** Since all three conditions are true, the lengths **3, 9, and 10** can form a triangle. ### 2. **Determine the Type of Triangle** To determine whether the triangle is acute, right, or obtuse, use the **Pythagorean Theorem**: - Let **c** be the longest side. In this case, \( c = 10 \). - Let **a** and **b** be the other two sides. Here, \( a = 3 \) and \( b = 9 \). Compute \( c^2 \) and \( a^2 + b^2 \): - \( c^2 = 10^2 = 100 \) - \( a^2 + b^2 = 3^2 + 9^2 = 9 + 81 = 90 \) **Compare the two values:** - If \( c^2 = a^2 + b^2 \), the triangle is **right-angled**. - If \( c^2 < a^2 + b^2 \), the triangle is **acute**. - If \( c^2 > a^2 + b^2 \), the triangle is **obtuse**. Here, \( 100 > 90 \), so \( c^2 > a^2 + b^2 \). ### **Conclusion:** The triangle with sides **3, 9, and 10** is an **obtuse** triangle. **Answer:** **obtuse**