1. **Problem Statement:** We want to determine if triangle $\triangle ABC$ with vertices $A(5,5)$, $B(-3,-1)$, and $C(1,-3)$ is a right triangle by calculating the lengths of its sides. 2. **Formula for distance between two points:** The distance between points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula comes from the Pythagorean theorem applied to the horizontal and vertical differences. 3. **Calculate length $AB$:** $$AB = \sqrt{(-3 - 5)^2 + (-1 - 5)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$$ 4. **Calculate length $BC$:** $$BC = \sqrt{(1 - (-3))^2 + (-3 - (-1))^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20}$$ 5. **Calculate length $AC$:** $$AC = \sqrt{(1 - 5)^2 + (-3 - 5)^2} = \sqrt{(-4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80}$$ 6. **Check Pythagorean theorem:** For a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. $$ (\sqrt{20})^2 + (\sqrt{80})^2 = 10^2 $$ $$ 20 + 80 = 100 $$ $$ 100 = 100 $$ 7. **Explanation of LS and RS:** LS means Left Side of the equation and RS means Right Side. Since LS = RS, the Pythagorean theorem holds, confirming $\triangle ABC$ is a right triangle. This is how the lengths were found using the distance formula and how the final check was done using the Pythagorean theorem.