The hypotenuse of a right triangle is 10 ft long. The longer leg is 2 ft longer than the shorter leg. Find the side lengths of the triangle Length of the shorter leg: Length of the longer leg: Length of the hypotenuse:

Let's solve the problem step by step. **Given:** - Hypotenuse of the right triangle = 10 ft - Longer leg is 2 ft longer than the shorter leg. **Let:** - Shorter leg = \( x \) ft - Longer leg = \( x + 2 \) ft **Using the Pythagorean theorem:** \[ x^2 + (x + 2)^2 = 10^2 \] \[ x^2 + x^2 + 4x + 4 = 100 \] \[ 2x^2 + 4x + 4 = 100 \] \[ 2x^2 + 4x - 96 = 0 \] \[ x^2 + 2x - 48 = 0 \quad \text{(dividing by 2)} \] **Solving the quadratic equation:** \[ x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-48)}}{2(1)} \] \[ x = \frac{-2 \pm \sqrt{4 + 192}}{2} \] \[ x = \frac{-2 \pm \sqrt{196}}{2} \] \[ x = \frac{-2 \pm 14}{2} \] **Possible solutions:** \[ x = \frac{12}{2} = 6 \quad \text{(valid, since length cannot be negative)} \] \[ x = \frac{-16}{2} = -8 \quad \text{(invalid)} \] **Therefore:** - **Length of the shorter leg:** 6 ft - **Length of the longer leg:** \( 6 + 2 = 8 \) ft - **Length of the hypotenuse:** 10 ft **Final Answer:** - **Length of the shorter leg:** 6 ft - **Length of the longer leg:** 8 ft - **Length of the hypotenuse:** 10 ft