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The two triangles \(\triangle ABC\) and \(\triangle DEF\) are given as similar (\(\triangle ABC \sim \triangle DEF\)). From the properties of similar triangles, we know that corresponding sides are proportional.

The corresponding sides of the triangles are:

- \(AB\) corresponds to \(DE\)
- \(AC\) corresponds to \(DF\)
- \(BC\) corresponds to \(EF\)

We can now set up ratios of corresponding sides to solve for \(x\) and \(y\).

### Step 1: Set up the proportion for side \(AC\) and \(DF\)

Given:
- \(AC = 4y - 2\)
- \(DF = 12\)

Using the ratio \( \frac{AC}{DF} = \frac{AB}{DE} \), we can use corresponding sides:
\[
\frac{4y - 2}{12} = \frac{21}{14}
\]
Simplify \( \frac{21}{14} \) to \( \frac{3}{2} \):
\[
\frac{4y - 2}{12} = \frac{3}{2}
\]
Cross-multiply:
\[
2(4y - 2) = 3 \times 12
\]
Simplify:
\[
8y - 4 = 36
\]
Solve for \(y\):
\[
8y = 36 + 4
\]
\[
8y = 40
\]
\[
y = \frac{40}{8} = 5
\]

### Step 2: Set up the proportion for side \(AB\) and \(DE\)

Now, to solve for \(x\), we use the ratio \( \frac{AB}{DE} = \frac{BC}{EF} \).

Given:
- \(AB = 21\)
- \(DE = 14\)
- \(BC = 24\)
- \(EF = 16\)

Use the ratio:
\[
\frac{AB}{DE} = \frac{BC}{EF}
\]
Simplify \( \frac{AB}{DE} = \frac{21}{14} = \frac{3}{2} \):
\[
\frac{3}{2} = \frac{24}{16}
\]
Since the ratio holds, the corresponding sides are proportional.

For side \(AD = x\), we know that:
\[
x = 16 \, \text{(as corresponding side)}.
\]

### Final Answers:
- \(y = 5\)
- \(x = 16\)

Would you like further details or explanations?

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Here are some related questions:
1. What is the relationship between the angles in similar triangles?
2. How do you calculate the area of similar triangles if one side length is known?
3. How does the ratio of corresponding sides affect the perimeter of similar triangles?
4. What happens to the ratio of the areas when two triangles are similar?
5. Can two triangles be similar if their corresponding angles are not equal?

**Tip:** In similar triangles, the ratio of any two corresponding sides is always constant.

Given two similar triangles, ABC and DEF, find the value of x and y.

In this problem, learn how to solve for x and y in two similar triangles using the properties of proportional sides. The step-by-step solution demonstrates how to apply the theorem of similar triangles and cross-multiply to find the missing values. Suitable for students in Grades 8-10.

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