https://file.aiquickdraw.com/mathbot/file/1727379094149ylilq73m.jpg

Review what you have learned so far about similarity as you solve for x.

In this image, two rectangles are shown, one smaller and one larger. The dimensions for both rectangles are given, with the smaller rectangle having a height of 3 and a width \( x \), while the larger rectangle has a width of 21 and a perimeter of 60.

### To solve for \( x \), we will use the concept of similarity, as the rectangles have the same shape (i.e., their corresponding sides are proportional).

#### Step 1: Use the perimeter to find the height of the larger rectangle.
The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2 \times (\text{length} + \text{width})
\]
For the larger rectangle, the perimeter is 60 and the width is 21. Let the height of the larger rectangle be \( h \).

\[
60 = 2 \times (h + 21)
\]

Solving for \( h \):
\[
60 = 2h + 42
\]
\[
60 - 42 = 2h
\]
\[
18 = 2h
\]
\[
h = 9
\]

So, the height of the larger rectangle is 9.

#### Step 2: Set up a proportion using the similarity of the rectangles.
Since the rectangles are similar, the ratio of their corresponding sides must be equal. The corresponding sides are the heights and widths of the rectangles.

The ratio of the heights is:
\[
\frac{3}{9} = \frac{1}{3}
\]

Therefore, the ratio of the widths must also be:
\[
\frac{x}{21} = \frac{1}{3}
\]

#### Step 3: Solve for \( x \).
Now, solve the proportion:
\[
\frac{x}{21} = \frac{1}{3}
\]

Cross-multiply to find \( x \):
\[
3x = 21
\]
\[
x = \frac{21}{3} = 7
\]

So, \( x = 7 \).

### Final Answer:
The value of \( x \) is 7.

Would you like more details or have any other questions?

Here are some related questions to deepen your understanding:

1. How do we know the rectangles in this problem are similar?
2. How can you use perimeter to find the dimensions of a rectangle?
3. What other properties of similar shapes could you apply to solve for unknown values?
4. How would the solution change if the perimeter of the larger rectangle were different?
5. Can this method be used to solve problems involving triangles or other polygons?

**Tip:** When dealing with similar shapes, always check that the ratios of corresponding sides are consistent across the entire shape.

Learn how to solve for the unknown side length x in similar rectangles using proportions. The problem explores key geometry concepts like the perimeter formula and the proportionality theorem. Suitable for Grades 7-9.

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