1. **Problem statement:** Given triangles $\triangle ABC \sim \triangle ADE$, find: a) the scale factor from $\triangle ABC$ to $\triangle ADE$, b) the perimeter of $\triangle ABC$ if the perimeter of $\triangle ADE$ is 42, c) the area of $\triangle ADE$ if the area of $\triangle ABC$ is 64. 2. **Step 1: Find the scale factor (7a)** The scale factor between similar triangles is the ratio of corresponding sides. Given sides: $AB = 10$, $BC = 12$, $DE = 21$. Since $\triangle ABC \sim \triangle ADE$, side $AB$ corresponds to side $AD$, and side $BC$ corresponds to side $DE$. We have $DE = 21$, but $AD$ is not given directly. However, since $DE$ corresponds to $BC$, we use these sides to find the scale factor. Scale factor $k = \frac{DE}{BC} = \frac{21}{12} = \frac{7}{4}$. 3. **Step 2: Find the perimeter of $\triangle ABC$ (7b)** The perimeter scales by the scale factor between triangles. Given perimeter of $\triangle ADE = 42$, and scale factor $k = \frac{7}{4}$. Since $k = \frac{\text{side in } ADE}{\text{side in } ABC}$, the perimeter of $\triangle ADE = k \times \text{perimeter of } ABC$. So, $$42 = \frac{7}{4} \times \text{perimeter of } ABC$$ Divide both sides by $\frac{7}{4}$: $$\text{perimeter of } ABC = 42 \times \cancel{\frac{4}{7}} = 42 \times \frac{4}{7}$$ $$= 6 \times 4 = 24$$ 4. **Step 3: Find the area of $\triangle ADE$ (7c)** The area scales by the square of the scale factor. Given area of $\triangle ABC = 64$, and scale factor $k = \frac{7}{4}$. Area of $\triangle ADE = k^2 \times \text{area of } ABC = \left(\frac{7}{4}\right)^2 \times 64$$ Calculate: $$\left(\frac{7}{4}\right)^2 = \frac{49}{16}$$ So, $$\text{area of } ADE = \frac{49}{16} \times 64 = 49 \times \cancel{\frac{64}{16}} = 49 \times 4 = 196$$ **Final answers:** - 7a) Scale factor = \(\frac{7}{4}$ - 7b) Perimeter of $\triangle ABC = 24$ - 7c) Area of $\triangle ADE = 196$