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We are given that the area of rectangle A is three times the area of triangle B, and we are asked to find the area of rectangle A. Here's how we can solve this:

### Step 1: Find the area of triangle B
The formula for the area of a triangle is:

\[
\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]

From the diagram, the base of triangle B is 3 cm and the height is \(2x\) cm. Therefore, the area of triangle B is:

\[
\text{Area of Triangle B} = \frac{1}{2} \times 3 \times 2x = 3x \, \text{cm}^2
\]

### Step 2: Express the area of rectangle A
The area of rectangle A is the product of its length and width. From the diagram, the length is \(2x - 2\) cm and the width is 6 cm. So, the area of rectangle A is:

\[
\text{Area of Rectangle A} = 6 \times (2x - 2) = 6(2x - 2) = 12x - 12 \, \text{cm}^2
\]

### Step 3: Set up the equation
We are told that the area of rectangle A is three times the area of triangle B. So we can set up the equation:

\[
\text{Area of Rectangle A} = 3 \times \text{Area of Triangle B}
\]

Substitute the expressions for the areas:

\[
12x - 12 = 3 \times 3x
\]

### Step 4: Solve for \(x\)
Simplify the equation:

\[
12x - 12 = 9x
\]

Subtract \(9x\) from both sides:

\[
3x - 12 = 0
\]

Add 12 to both sides:

\[
3x = 12
\]

Divide both sides by 3:

\[
x = 4
\]

### Step 5: Find the area of rectangle A
Now that we know \(x = 4\), substitute this value into the expression for the area of rectangle A:

\[
\text{Area of Rectangle A} = 12x - 12 = 12(4) - 12 = 48 - 12 = 36 \, \text{cm}^2
\]

Thus, the area of rectangle A is **36 cm²**.

Would you like any further details or explanations? 

Here are 5 related questions for further exploration:
1. How would the problem change if the rectangle's width was variable instead of constant?
2. What would happen if the area of rectangle A was only twice the area of triangle B?
3. How could we determine the perimeter of triangle B using the value of \(x\)?
4. What other dimensions could satisfy the condition that rectangle A's area is three times triangle B's?
5. How would this problem change if we had a different shape, like a parallelogram, instead of a triangle?

**Tip:** When solving geometry problems involving relationships between areas, always remember to express all areas in terms of a single variable before setting up your equations.

The area of rectangle A is three times the area of triangle B. Work out the area of rectangle A. All lengths are in cm.

Learn how to find the area of rectangle A given that it is three times the area of triangle B. This problem involves key geometry concepts like area formulas for triangles and rectangles, and algebraic manipulation. Suitable for students in Grades 8-10.

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