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The problem asks to find \(x\), given that the figures are similar and the ratio of their perimeters is 7:10. One of the corresponding side lengths of the rectangles is 12 (in the larger figure), and the corresponding side length of the smaller rectangle is \(x\).

### Step-by-Step Solution:

Since the figures are similar, the ratio of their corresponding side lengths will be the same as the ratio of their perimeters.

Given:
\[
\frac{\text{Perimeter of Small Figure}}{\text{Perimeter of Large Figure}} = \frac{7}{10}
\]

Therefore, the corresponding sides of the figures will also be in the ratio:
\[
\frac{x}{12} = \frac{7}{10}
\]

Now, solve for \(x\):
\[
x = 12 \times \frac{7}{10}
\]

Let's compute that.The value of \(x\) is approximately 8.4.

Would you like more details on this, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the relationship between the areas of similar figures?
2. How would the ratio of areas compare to the ratio of the sides?
3. Can the ratio of side lengths be used to determine other properties of similar shapes?
4. What happens to the ratio of perimeters if the figures are scaled differently?
5. How can you calculate the area of the smaller rectangle given this information?

**Tip:** In similar figures, the ratio of areas is the square of the ratio of side lengths.

Find x. The figures are similar. The ratio of the perimeters is 7:10.

Find the value of x by solving a proportionality problem involving similar figures with a given perimeter ratio of 7:10. This solution explores how to apply the similarity theorem and ratio rules to find the unknown side length. Suitable for Grades 6-8 students.

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