The perimeter of a rectangle having area equal to $144 cm^2$ and sides in the ratio 4 : 9 is :

Question

Accepted Answer

To determine the perimeter of the rectangle with an area of $144 \, \text{cm}^2$ and sides in the ratio 4:9, we need to follow these steps:

1. **Understanding the Ratio**: The sides of the rectangle are in the ratio 4:9. Let's denote the sides as $4x$ and $9x$.

2. **Calculating the Area**: The area of the rectangle is given by the formula:
   $$
   \text{Area} = \text{length} \times \text{width} = 4x \times 9x = 36x^2
   $$
   We know the area is $144 \, \text{cm}^2$, so:
   $$
   36x^2 = 144
   $$

3. **Solving for $x$**: Divide both sides by 36 to find $x^2$:
   $$
   x^2 = \frac{144}{36} = 4
   $$
   Taking the square root of both sides gives:
   $$
   x = 2
   $$

4. **Finding the Dimensions**: Substitute $x = 2$ back into the expressions for the sides:
   $$
   \text{Length} = 9x = 9 \times 2 = 18 \, \text{cm}
   $$
   $$
   \text{Width} = 4x = 4 \times 2 = 8 \, \text{cm}
   $$

5. **Calculating the Perimeter**: The perimeter $P$ of a rectangle is given by:
   $$
   P = 2(\text{Length} + \text{Width}) = 2(18 + 8) = 2 \times 26 = 52 \, \text{cm}
   $$

Thus, the perimeter of the rectangle is $52 \, \text{cm}$, which corresponds to Option 1.

**Explanation for Incorrect Options**:

- **Option 2 (56 cm)**: This option is incorrect because the calculated perimeter using the correct dimensions (18 cm and 8 cm) is 52 cm, not 56 cm.

- **Option 3 (60 cm)**: This option is incorrect for the same reason. The perimeter calculation based on the given area and side ratio does not result in 60 cm.

- **Option 4 (64 cm)**: Similarly, this option is incorrect because the perimeter derived from the correct dimensions is 52 cm, not 64 cm.

In conclusion, the correct perimeter of the rectangle, based on the given conditions, is indeed 52 cm, which is why Option 1 is the correct answer.