1. **Stating the problem:** We have a right triangle with a hypotenuse of length 15 and one leg of length 10. We need to find the area of the triangle and the length of the other leg. 2. **Formula used:** For a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse, and $a$, $b$ are the legs. The area $A$ of a right triangle is: $$A = \frac{1}{2} \times a \times b$$ 3. **Find the missing leg:** Let the unknown leg be $b$. Given $a=10$, $c=15$: $$10^2 + b^2 = 15^2$$ $$100 + b^2 = 225$$ $$b^2 = 225 - 100 = 125$$ $$b = \sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}$$ 4. **Calculate the area:** $$A = \frac{1}{2} \times 10 \times 5\sqrt{5} = 5 \times 5\sqrt{5} = 25\sqrt{5}$$ 5. **Final answers:** - The other leg length is $5\sqrt{5}$. - The area of the triangle is $25\sqrt{5}$. These results give the exact values for the triangle's dimensions and area.