1. **State the problem:** We have a large right triangle with a hypotenuse of 50 and a base segment of 45. Inside it, a smaller right triangle is formed by a perpendicular, with legs labeled $x$ and $y$. We need to solve for $x$ and $y$. 2. **Identify the triangles and relationships:** The large triangle has sides 45 (base), unknown height, and hypotenuse 50. The smaller right triangle inside has legs $x$ and $y$. 3. **Use the Pythagorean theorem for the large triangle:** $$45^2 + h^2 = 50^2$$ $$2025 + h^2 = 2500$$ $$h^2 = 2500 - 2025 = 475$$ $$h = \sqrt{475} = \sqrt{25 \times 19} = 5\sqrt{19}$$ 4. **Use similarity of triangles:** The smaller right triangle is similar to the large one because they share an angle and both have right angles. 5. **Set up ratios from similarity:** $$\frac{x}{45} = \frac{y}{h} = \frac{\text{hypotenuse of small}}{50}$$ 6. **Given $x=15$, find $y$ using the ratio:** $$\frac{15}{45} = \frac{y}{5\sqrt{19}}$$ $$\frac{1}{3} = \frac{y}{5\sqrt{19}}$$ $$y = \frac{5\sqrt{19}}{3}$$ 7. **Simplify $y$ if possible:** $y = \frac{5\sqrt{19}}{3}$ is already simplified. **Final answers:** $$x = 15$$ $$y = \frac{5\sqrt{19}}{3}$$