1. **Problem Statement:** Given right triangle $\triangle RST$ with right angle at $S$, and sides $RS = ST$. The hypotenuse $RT$ is 18 inches. Find the area of $\triangle RST$. 2. **Formula and Important Rules:** - Since $\triangle RST$ is right-angled at $S$ and $RS = ST$, it is an isosceles right triangle. - In an isosceles right triangle, the legs are equal, and the hypotenuse $c$ relates to each leg $a$ by the formula: $$c = a\sqrt{2}$$ - The area of a triangle is: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Find the length of legs $RS$ and $ST$:** Given $RT = 18$, and $RT = a\sqrt{2}$, solve for $a$: $$18 = a\sqrt{2}$$ Divide both sides by $\sqrt{2}$: $$a = \frac{18}{\sqrt{2}}$$ Rationalize the denominator: $$a = \frac{18}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{18\sqrt{2}}{2} = 9\sqrt{2}$$ 4. **Calculate the area:** Since legs are equal, area is: $$\text{Area} = \frac{1}{2} \times a \times a = \frac{1}{2} a^2$$ Substitute $a = 9\sqrt{2}$: $$\text{Area} = \frac{1}{2} (9\sqrt{2})^2 = \frac{1}{2} \times 81 \times 2 = \frac{1}{2} \times 162 = 81$$ 5. **Final answer:** The area of $\triangle RST$ is $81$ square inches. **Answer choice:** D 81 in.$^2$