1. **State the problem:** We are given three geometric figures with unknown side lengths labeled as $x$. We need to find $x$ in each case. 2. **Problem 8 (Rectangle with diagonal):** Given a rectangle with sides 12 and $x$, and diagonal 15, use the Pythagorean theorem: $$12^2 + x^2 = 15^2$$ 3. **Solve for $x$ in problem 8:** $$144 + x^2 = 225$$ $$x^2 = 225 - 144$$ $$x^2 = 81$$ $$x = \sqrt{81} = 9$$ 4. **Problem 10 (Isosceles triangle with equal sides 5, base 8, height $x$):** Use the Pythagorean theorem on one right triangle formed by height $x$: Half base = $\frac{8}{2} = 4$ $$x^2 + 4^2 = 5^2$$ 5. **Solve for $x$ in problem 10:** $$x^2 + 16 = 25$$ $$x^2 = 25 - 16$$ $$x^2 = 9$$ $$x = \sqrt{9} = 3$$ 6. **Problem 11 (Inverted isosceles triangle with base 6, equal sides $x$, height 6):** Use the Pythagorean theorem on one right triangle formed by height 6: Half base = $\frac{6}{2} = 3$ $$6^2 + 3^2 = x^2$$ 7. **Solve for $x$ in problem 11:** $$36 + 9 = x^2$$ $$x^2 = 45$$ $$x = \sqrt{45} = 3\sqrt{5}$$ **Final answers:** - Problem 8: $x = 9$ - Problem 10: $x = 3$ - Problem 11: $x = 3\sqrt{5}$