To solve the problem regarding rectangles $A$ and $B$ where the areas are equal and we need the perimeter of rectangle $A$, let's proceed step by step.

1. Problem Analysis

• Given:

  • Rectangle $A$: Height $6$, Width $2x - 5$.
  • Rectangle $B$: Height $2x - 1$, Width $4$.
  • Areas of rectangles $A$ and $B$ are equal.

• Find: Perimeter of rectangle $A$.

2. Equating the Areas

The area of a rectangle is given by: $\text{Area} = \text{Height} \times \text{Width}$

For rectangle $A$: $\text{Area}_A = 6 \times (2x - 5)$

For rectangle $B$: $\text{Area}_B = 4 \times (2x - 1)$

Since the areas are equal: $6(2x - 5) = 4(2x - 1)$

3. Solve for $x$

Expand both sides: $12x - 30 = 8x - 4$

Combine like terms: $12x - 8x = 30 - 4$ $4x = 26$

Solve for $x$: $x = \frac{26}{4} = 6.5$

4. Find Dimensions of Rectangle $A$

• Height of $A = 6$ (given).
• Width of $A = 2x - 5$: $2(6.5) - 5 = 13 - 5 = 8$

So, dimensions of rectangle $A$ are:

• Height = $6$
• Width = $8$

5. Perimeter of Rectangle $A$

The perimeter of a rectangle is given by: $\text{Perimeter} = 2 \times (\text{Height} + \text{Width})$

Substitute the values: $\text{Perimeter} = 2 \times (6 + 8) = 2 \times 14 = 28 \, \text{cm}$

Final Answer:

The perimeter of rectangle $A$ is 28 cm.

Do you need further clarification or a breakdown of any step? Let me know!

Here are 5 related questions to expand your understanding:

1. How would the problem change if the height of rectangle $B$ were $3x$ instead of $2x - 1$?
2. What is the perimeter of rectangle $B$ using the same value of $x$?
3. If the areas were not equal but $\text{Area}_A = 2 \times \text{Area}_B$, what would $x$ be?
4. What happens if the width of rectangle $A$ becomes $3x - 4$ instead of $2x - 5$?
5. Can you derive a general formula for the perimeter of a rectangle if you only know its area and one side?

Tip: Always write down known values and formulas first when solving word problems involving geometry.