1. **Problem 1: Find the values of $x$ and $y$ in similar triangular prisms.** Given two similar right triangles with sides: - Larger triangle: 14 cm (vertical), 8 cm (base), 20 cm (hypotenuse) - Smaller triangle: $x=6$ cm (base), $y$ cm (vertical), hypotenuse unknown Since the triangles are similar, corresponding sides are proportional: $$\frac{14}{y} = \frac{8}{6} = \frac{20}{\text{hypotenuse}}$$ We use the ratio of bases to find $y$: 2. Set up the proportion: $$\frac{14}{y} = \frac{8}{6}$$ 3. Cross multiply: $$14 \times 6 = 8 \times y$$ $$84 = 8y$$ 4. Solve for $y$: $$y = \frac{84}{8}$$ 5. Simplify the fraction by canceling common factors: $$y = \frac{\cancel{84}^{12 \times 7}}{\cancel{8}^{4 \times 2}} = \frac{42}{4} = \frac{21}{2} = 10.5$$ So, $y = 10.5$ cm. 6. Given the note $y = 2x$, check consistency: $$y = 2 \times 6 = 12$$ This conflicts with the proportional calculation, so the correct $y$ from similarity is $10.5$ cm. --- 7. **Problem 2: Find the perimeter of the actual garden from the scale drawing.** Given: - Scale: $\frac{1}{4}$ inch = 3 feet - Garden dimensions on drawing: 1/2 inch (vertical side), 1 3/4 inch (horizontal side) 8. Convert each side to actual length: Vertical side: $$\text{actual} = \frac{1}{2} \text{ inch} \times \frac{3 \text{ feet}}{\frac{1}{4} \text{ inch}} = \frac{1}{2} \times \frac{3}{\frac{1}{4}} = \frac{1}{2} \times 12 = 6 \text{ feet}$$ Horizontal side: $$\text{actual} = \frac{7}{4} \text{ inch} \times \frac{3 \text{ feet}}{\frac{1}{4} \text{ inch}} = \frac{7}{4} \times 12 = 21 \text{ feet}$$ 9. Calculate perimeter of the rectangular garden: $$P = 2 \times (\text{length} + \text{width}) = 2 \times (6 + 21) = 2 \times 27 = 54 \text{ feet}$$ **Final answers:** - $x = 6$ cm (given) - $y = 10.5$ cm - Perimeter of actual garden = 54 feet