https://file.aiquickdraw.com/mathbot/file/1726844342741uy7sq3pa.jpg

The image shows two similar triangles, \(\triangle ABC\) (the smaller one) and \(\triangle DEF\) (the larger one). The sides of \(\triangle ABC\) are given as follows:

- \(AC = 3\),
- \(AB = \frac{7}{3}\),
- \(BC = \frac{3}{2}\).

For \(\triangle DEF\), one side is given:
- \(DF = 5\).

Since the triangles are similar, we can use the proportionality of corresponding sides to determine the lengths of \(DE\) and \(EF\).

### Part a: Length of side \(DE\)
The corresponding sides of \(\triangle ABC\) and \(\triangle DEF\) are proportional. Let's set up the ratio using the given sides:

\[
\frac{DF}{AC} = \frac{DE}{AB}
\]

Substituting the known values:

\[
\frac{5}{3} = \frac{DE}{\frac{7}{3}}
\]

Now solve for \(DE\):

\[
DE = \frac{5 \times \frac{7}{3}}{1} = \frac{35}{3}
\]

So, the length of side \(DE\) is \(\frac{35}{3}\), or approximately 11.67 units.

### Part b: Length of side \(EF\)
Similarly, we can use the ratio of corresponding sides to find the length of side \(EF\). The corresponding sides are:

\[
\frac{DF}{AC} = \frac{EF}{BC}
\]

Substitute the known values:

\[
\frac{5}{3} = \frac{EF}{\frac{3}{2}}
\]

Now solve for \(EF\):

\[
EF = \frac{5 \times \frac{3}{2}}{1} = \frac{15}{2}
\]

So, the length of side \(EF\) is \(\frac{15}{2}\), or 7.5 units.

---

Would you like more details or clarifications on these calculations? 

Here are five related questions to consider:
1. What are the properties of similar triangles?
2. How do you know which sides of the triangles correspond to each other?
3. What is the general formula for solving proportions in similar triangles?
4. How can you check if two triangles are similar?
5. Can you use other methods, like the Pythagorean theorem, to verify the side lengths?

**Tip:** When working with similar triangles, always make sure to set up ratios between corresponding sides properly.

a. What is the length of side DE? b. What is the length of side EF?

Learn how to calculate the lengths of sides DE and EF in similar triangles using proportionality and similarity theorems. This problem provides detailed steps for students in Grades 8-10, focusing on the proportional sides of similar triangles.

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