Triangle Similarity 09599C

1. **Problem Statement:** We have two similar right triangles, a smaller one embedded inside a larger one. Given some side lengths and angles, we need to find the values of the pronumerals $x$, $y$, $z$, and $w$ in the smaller triangle. 2. **Similarity Rule:** Since the triangles are similar, corresponding sides are proportional. This means: $$\frac{\text{side in smaller triangle}}{\text{corresponding side in larger triangle}} = k$$ where $k$ is the scale factor. 3. **Identify Corresponding Sides:** - Larger triangle sides: $10$, $7.5$, and hypotenuse (not given explicitly). - Smaller triangle sides: $x$, $y$, $z$, $w$ (some given as $2$ and $1$). 4. **Calculate Scale Factor:** Given smaller triangle side $2$ corresponds to larger triangle side $7.5$, so: $$k = \frac{2}{7.5} = \frac{2}{7.5} = \frac{4}{15} \approx 0.2667$$ 5. **Find $x$:** If $x$ corresponds to $10$ in the larger triangle: $$x = k \times 10 = \frac{4}{15} \times 10 = \frac{40}{15} = \frac{8}{3} \approx 2.67$$ 6. **Find $y$:** If $y$ corresponds to $7.5$ in the larger triangle: $$y = k \times 7.5 = \frac{4}{15} \times 7.5 = 2$$ 7. **Find $z$ and $w$:** Given $z = 1$ and $w$ corresponds to the hypotenuse of the smaller triangle. Since the triangles are right angled and similar, the hypotenuse scales by the same factor $k$. Calculate hypotenuse of larger triangle using Pythagoras: $$\text{hypotenuse} = \sqrt{10^2 + 7.5^2} = \sqrt{100 + 56.25} = \sqrt{156.25} = 12.5$$ Then: $$w = k \times 12.5 = \frac{4}{15} \times 12.5 = \frac{50}{15} = \frac{10}{3} \approx 3.33$$ 8. **Summary of values:** - $x = \frac{8}{3} \approx 2.67$ - $y = 2$ - $z = 1$ - $w = \frac{10}{3} \approx 3.33$ These values satisfy the similarity and proportionality of the triangles.