1. **State the problem:** Find the lengths of the legs and the hypotenuse of the right triangle formed by points $(-5, 3)$ and $(-8, 6)$. 2. **Formula used:** The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ For a right triangle with horizontal and vertical legs, the legs are the absolute differences in $x$ and $y$ coordinates, and the hypotenuse is the distance between the points. 3. **Calculate the legs:** - Horizontal leg length: $$|x_2 - x_1| = |-8 - (-5)| = |-8 + 5| = | -3 | = 3$$ - Vertical leg length: $$|y_2 - y_1| = |6 - 3| = 3$$ 4. **Calculate the hypotenuse:** $$\text{hypotenuse} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-8 + 5)^2 + (6 - 3)^2} = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}$$ 5. **Simplify the hypotenuse:** $$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$$ **Final answer:** - Leg 1 (horizontal) = 3 - Leg 2 (vertical) = 3 - Hypotenuse = $3\sqrt{2}$