1. **State the problem:** We have a right triangle with hypotenuse 50 ft and legs 30 ft and 40 ft. A height $h$ is drawn from the right angle to the hypotenuse, dividing the base into two segments $x$ and $y$. We want to find procedures to calculate $x$, $y$, and $h$. 2. **Recall the Pythagorean theorem:** For a right triangle with legs $a$, $b$ and hypotenuse $c$, the relation is $$a^2 + b^2 = c^2$$ 3. **Use similarity of triangles:** The height $h$ divides the original triangle into two smaller right triangles, each similar to the original triangle and to each other. This gives the relations: $$\frac{h}{x} = \frac{30}{50} \quad \text{and} \quad \frac{h}{y} = \frac{40}{50}$$ 4. **Express $x$ and $y$ in terms of $h$:** From the similarity ratios, $$x = \frac{50}{30}h \quad \text{and} \quad y = \frac{50}{40}h$$ 5. **Use the fact that $x + y = 50$ (the hypotenuse):** $$\frac{50}{30}h + \frac{50}{40}h = 50$$ 6. **Simplify the equation:** $$50\left(\frac{1}{30} + \frac{1}{40}\right)h = 50$$ 7. **Calculate the sum inside parentheses:** $$\frac{1}{30} + \frac{1}{40} = \frac{4}{120} + \frac{3}{120} = \frac{7}{120}$$ 8. **Substitute back:** $$50 \times \frac{7}{120} h = 50$$ 9. **Simplify:** $$\frac{350}{120} h = 50$$ 10. **Divide both sides by $\frac{350}{120}$:** $$h = \frac{50}{\frac{350}{120}} = 50 \times \frac{120}{350}$$ 11. **Simplify the fraction:** $$h = 50 \times \frac{12}{35} = \frac{600}{35} = \frac{120}{7} \approx 17.14$$ 12. **Find $x$ and $y$ using $h$:** $$x = \frac{50}{30} \times \frac{120}{7} = \frac{50 \times 120}{30 \times 7} = \frac{6000}{210} = \frac{200}{7} \approx 28.57$$ $$y = \frac{50}{40} \times \frac{120}{7} = \frac{50 \times 120}{40 \times 7} = \frac{6000}{280} = \frac{75}{7} \approx 10.71$$ **Final answers:** $$h = \frac{120}{7} \approx 17.14, \quad x = \frac{200}{7} \approx 28.57, \quad y = \frac{75}{7} \approx 10.71$$