https://file.aiquickdraw.com/mathbot/file/1733681756488euwittnm.jpg

To solve the problem, let’s analyze the rectangles step by step:

### Step 1: Understand the dimensions and given conditions
- Rectangle **ABCD** has dimensions:
  - Length = \(3x\)
  - Width = \(x\)

- Rectangle **EFGH** has dimensions:
  - Length = \(6x - 6\)
  - Width = \(2x\)

The areas of the two rectangles are **equal**. We are tasked with finding the **perimeter of rectangle ABCD** as a simplified algebraic expression.

---

### Step 2: Set up the area equality
The area of a rectangle is given by:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

For rectangle ABCD:
\[
\text{Area}_{ABCD} = 3x \cdot x = 3x^2
\]

For rectangle EFGH:
\[
\text{Area}_{EFGH} = (6x - 6) \cdot 2x = 2x(6x - 6) = 12x^2 - 12x
\]

Since the areas are equal:
\[
3x^2 = 12x^2 - 12x
\]

---

### Step 3: Simplify the equation
Rearrange the terms to solve for \(x\):
\[
0 = 12x^2 - 12x - 3x^2
\]
\[
0 = 9x^2 - 12x
\]

Factorize:
\[
0 = 3x(3x - 4)
\]

Set each factor to 0:
1. \(3x = 0 \implies x = 0\) (not valid since \(x = 0\) would make the dimensions zero)
2. \(3x - 4 = 0 \implies x = \frac{4}{3}\)

Thus, \(x = \frac{4}{3}\).

---

### Step 4: Calculate the perimeter of rectangle ABCD
The perimeter of a rectangle is given by:
\[
\text{Perimeter} = 2(\text{Length} + \text{Width})
\]

Substitute the dimensions of rectangle ABCD:
\[
\text{Length} = 3x, \quad \text{Width} = x
\]
\[
\text{Perimeter} = 2(3x + x) = 2(4x) = 8x
\]

Substitute \(x = \frac{4}{3}\):
\[
\text{Perimeter} = 8 \cdot \frac{4}{3} = \frac{32}{3}
\]

---

### Final Answer:
The perimeter of rectangle ABCD is:
\[
\boxed{\frac{32}{3}}
\]

---

Would you like further clarification on any step or additional questions? Here are 5 related questions to extend your understanding:

1. How would the solution change if the areas of the rectangles were not equal?
2. What is the perimeter of rectangle EFGH using the same value of \(x\)?
3. How can the quadratic equation be solved using a graphing method?
4. What are the units of the perimeter, and how do they relate to \(x\)?
5. How can you verify the solution by substituting \(x = \frac{4}{3}\) into both areas?

**Tip:** Always check whether values for variables are reasonable (e.g., \(x > 0\)) when solving real-world geometry problems.

What is the perimeter of rectangle ABCD if its length is 3x and width is x, and the area of rectangle ABCD is equal to the area of rectangle EFGH, where the length of EFGH is (6x - 6) and width is 2x?

Learn how to solve the problem involving two rectangles with equal areas, where one rectangle has dimensions 3x by x and the other has dimensions (6x - 6) by 2x. The solution guides you through algebraic steps to find the perimeter of rectangle ABCD. Suitable for students in Grades 9-11.

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