To solve the problem, we use the fact that triangles $\triangle ABC$ and $\triangle DEF$ are similar. Similar triangles have proportional corresponding sides. Let us solve each part step-by-step.

Part (a): Find the length of segment $BC$

Given:

• $AB = 3$, $AC = 7$, and $DE = 2$, $DF = \frac{10}{3}$.
• Since the triangles are similar, the ratios of corresponding sides are equal: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}.$

1. Calculate the ratio using $AB$ and $DE$: $\text{Scale factor} = \frac{AB}{DE} = \frac{3}{2}.$

2. Use the scale factor to find $BC$: Since $BC$ corresponds to $EF$, and the scale factor is $\frac{3}{2}$, we have: $BC = EF \cdot \text{Scale factor}.$ Substitute $EF = 5$ (since $EF = \frac{10}{3} \cdot \frac{3}{2} = 5$) and solve: $BC = 5 \cdot \frac{3}{2} = 7.5.$

Thus, $BC = 7.5$.

Part (b): Find the length of segment $DF$

Given:

• $AC = 7$, $DF = \frac{10}{3}$, and $\frac{AB}{DE} = \frac{3}{2}$.

1. Since $AC$ corresponds to $DF$, use the scale factor: $\frac{AC}{DF} = \frac{3}{2}.$

2. Solve for $DF$: $DF = AC \cdot \frac{DE}{AB}.$ Substitute $AC = 7$, $DE = 2$, and $AB = 3$: $DF = 7 \cdot \frac{2}{3} = \frac{14}{3}.$

Thus, $DF = \frac{14}{3}$ or approximately $4.67$.

Final Answers:

• (a) $BC = 7.5$
• (b) $DF = \frac{14}{3}$ or $4.67$.

Would you like further details or clarifications?

Related Questions:

1. How do we know two triangles are similar?
2. What is the importance of proportionality in solving for missing sides?
3. Can you find the perimeter of each triangle?
4. How do we calculate the area of these similar triangles?
5. Could you generalize this process to any pair of similar triangles?

Tip: Always verify the correspondence of sides when working with similar triangles.