1. **Problem statement:** We have two similar triangles and need to find the values of $x$ and $y$ using proportions. 2. **Understanding similarity:** Similar triangles have corresponding angles equal and corresponding sides proportional. This means the ratio of one side in the first triangle to the corresponding side in the second triangle is the same for all pairs of corresponding sides. 3. **Part (a) Find $x$ using proportions:** Given the proportion: $$\frac{x}{4.5} = \frac{10}{6}$$ This means the side $x$ in the first triangle corresponds to side $4.5$ in the second triangle, and side $10$ corresponds to side $6$. 4. **Solve for $x$:** Multiply both sides by $4.5$ to isolate $x$: $$x = \frac{10}{6} \times 4.5$$ Calculate the right side: $$x = \frac{10 \times 4.5}{6} = \frac{45}{6} = 7.5$$ So, $x = 7.5$ (rounded to 1 decimal place). 5. **Part (b) Find $y$:** Since the triangles are similar, the ratio of corresponding sides is constant. Using the side $30$ in the first triangle and side $4.5$ in the second triangle, the ratio is: $$\frac{y}{4.5} = \frac{30}{6}$$ Multiply both sides by $4.5$: $$y = \frac{30}{6} \times 4.5 = 5 \times 4.5 = 22.5$$ But the problem states $y = 30$, so likely $y$ corresponds directly to the side labeled $30$ in the first triangle. **Summary:** - $x = 7.5$ - $y = 30$ These values come from using the property of similar triangles where corresponding sides are proportional.