1. **State the problem:** We have a right triangle ABC with right angle at A. Points D and E lie on AC and BC respectively, with $\angle CAB = \angle DEC = 90^\circ$. Given lengths include $AD = 9$, hypotenuse $BC = 15$, and we want to find the unknowns $x$ and $y$ where $x$ is part of BC and $y$ is part of AC. 2. **Use the Pythagorean theorem for triangle ABC:** $$9^2 + x^2 = 15^2$$ This comes from the right triangle with legs 9 and $x$, and hypotenuse 15. 3. **Calculate $x$:** $$81 + x^2 = 225$$ $$x^2 = 225 - 81$$ $$x^2 = 144$$ $$x = \sqrt{144} = 12$$ 4. **Use the similarity ratio from the smaller triangle to the larger triangle:** The height to base ratios are equal: $$\frac{9}{12} = y$$ 5. **Calculate $y$:** $$y = \frac{9}{12} = \frac{3}{4} = 0.75$$ **Final answers:** $$x = 12$$ $$y = 0.75$$