1. **Problem statement:** Calculate the exact length of the unknown side in the right triangle with hypotenuse $\sqrt{85}$ and one leg $\sqrt{68}$. Then estimate the unknown base length of an isosceles triangle with equal sides 11 and altitude 8. 2. **Right triangle side length:** Use the Pythagorean theorem: $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse, and $a,b$ are legs. 3. Substitute known values: $$\sqrt{85}^2 = \sqrt{68}^2 + x^2 \implies 85 = 68 + x^2$$ 4. Solve for $x^2$: $$x^2 = 85 - 68 = 17$$ 5. Take the square root: $$x = \sqrt{17}$$ which is the exact length of the unknown leg. 6. **Isosceles triangle base length:** The altitude splits the base into two equal segments, each of length $\frac{b}{2}$. 7. Use the Pythagorean theorem on one right triangle formed by the altitude: $$11^2 = 8^2 + \left(\frac{b}{2}\right)^2$$ 8. Substitute values: $$121 = 64 + \frac{b^2}{4}$$ 9. Solve for $b^2$: $$\frac{b^2}{4} = 121 - 64 = 57$$ 10. Multiply both sides by 4: $$b^2 = 228$$ 11. Take the square root: $$b = \sqrt{228}$$ 12. Estimate $b$ to the nearest tenth: $$\sqrt{228} \approx 15.1$$ **Final answers:** - Unknown leg in right triangle: $\sqrt{17}$ - Base of isosceles triangle (nearest tenth): 15.1