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The image presents a mathematical activity that involves calculating the value of "x" and the length of the sides of two rectangles based on their perimeters.

Let's break down the problem:

### a. Perimeter = 32 cm
- One side is given as \( 8 \, \text{cm} \), and the other side is marked as \( x \).
- The perimeter formula for a rectangle is \( P = 2L + 2W \), where \( L \) is the length and \( W \) is the width.

Using the given perimeter of 32 cm:
\[
32 = 2(8) + 2(x)
\]
Simplifying:
\[
32 = 16 + 2x
\]
\[
2x = 16
\]
\[
x = 8 \, \text{cm}
\]
Thus, for this rectangle, the unknown side \( x \) is 8 cm.

### b. Perimeter = 120 cm
- One side is given as \( 32 + 4x \, \text{cm} \), and the other side is marked as \( 10 \, \text{cm} \).
- Using the perimeter formula \( P = 2L + 2W \), where \( L = 32 + 4x \) and \( W = 10 \), we have:

\[
120 = 2(32 + 4x) + 2(10)
\]
Simplifying:
\[
120 = 2(32 + 4x) + 20
\]
\[
120 = 64 + 8x + 20
\]
\[
120 = 84 + 8x
\]
\[
8x = 36
\]
\[
x = 4.5
\]
Thus, \( x = 4.5 \) for this rectangle.

### Summary of Results:
- For rectangle **a**, \( x = 8 \, \text{cm} \).
- For rectangle **b**, \( x = 4.5 \).

Would you like more details on any part of this solution or further clarification?

### Related Questions:
1. How do you calculate the perimeter of a square?
2. What would happen if the width in problem a were doubled?
3. How can you verify the result of problem b using substitution?
4. What is the relationship between the length and width in a rectangle if the perimeter is constant?
5. How does changing the perimeter affect the value of "x" in rectangle b?

### Tip:
When solving perimeter problems, always ensure all variables are isolated step by step to avoid confusion.

Calculate the value of x and the length of the sides of two rectangles based on their given perimeters.

Explore how to calculate the value of x and determine the side lengths of two rectangles using their given perimeters. This problem covers key algebraic and geometric concepts and is suitable for students in Grades 6-8.

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