1. **State the problem:** We have a right triangle with sides labeled 5, 6, $y-1$, $x-2$, and 10, and we want to find the unknown values $x$ and $y$ using proportions. 2. **Set up proportions:** In similar triangles or right triangles with altitudes, corresponding sides are proportional. The problem suggests using the given side lengths to write proportions. 3. **Write the proportions:** From the diagram and labels, the proportions are: $$\frac{5}{y-1} = \frac{6}{10}$$ and $$\frac{x-2}{5} = \frac{6}{10}$$ 4. **Solve for $y$:** Multiply both sides of the first proportion by $y-1$: $$5 = \frac{6}{10} (y-1)$$ Multiply both sides by 10 to clear the denominator: $$10 \times 5 = 6 (y-1)$$ $$50 = 6 (y-1)$$ Divide both sides by 6: $$\frac{50}{6} = y-1$$ Simplify fraction: $$\frac{50}{6} = \frac{25}{3} \approx 8.333$$ Add 1 to both sides: $$y = 8.333 + 1 = 9.333$$ 5. **Solve for $x$:** Multiply both sides of the second proportion by 5: $$x-2 = \frac{6}{10} \times 5$$ Simplify right side: $$x-2 = 3$$ Add 2 to both sides: $$x = 5$$ **Final answers:** $$x = 5, \quad y \approx 9.333$$