The problem shown in the image involves two similar triangles, $\triangle ABC$ and $\triangle ADEF$, and some given measurements:

• $AB = 6.3 \, \text{cm}$
• $BC = 5.6 \, \text{cm}$
• $AC = 8.4 \, \text{cm}$

We are asked to construct the triangles and solve the ratio $\frac{AC}{DF} = \frac{7}{5}$. This means that the sides of triangle $ADEF$ are proportional to the corresponding sides of triangle $ABC$ by a ratio of $\frac{7}{5}$.

To solve this problem, follow these steps:

1. Construct $\triangle ABC$ using the given measurements for $AB$, $BC$, and $AC$.

2. Set up the ratio between the sides of $\triangle ABC$ and $\triangle ADEF$. Since the triangles are similar, the corresponding sides will have the same ratio.

   • $\frac{AC}{DF} = \frac{7}{5}$

3. Find the length of side $DF$: Since $AC = 8.4 \, \text{cm}$ and the ratio is $\frac{7}{5}$, we can find $DF$ by: $DF = \frac{5}{7} \times 8.4 = 6 \, \text{cm}$

4. Find other sides of $\triangle ADEF$: Similarly, use the ratio $\frac{7}{5}$ to find $DE$ and $EF$, proportional to the sides $AB$ and $BC$ respectively.

Would you like a detailed explanation of how to construct the triangles, or do you have any specific questions? Here are some related questions to expand on:

1. How do you determine if two triangles are similar?
2. What is the significance of the ratio of corresponding sides in similar triangles?
3. How can you calculate other corresponding sides using proportions?
4. How would you approach constructing these triangles geometrically?
5. What real-world applications rely on the properties of similar triangles?

Tip: When dealing with similar triangles, always remember that the angles remain the same, and the sides are proportional.