Area of triangle of sides 24 cm, 25 cm and 7 cm is equal to the area of a rectangle of length 14 cm, then perimeter of the rectangle is:

Question

Accepted Answer

To determine the perimeter of the rectangle, we first need to calculate the area of the triangle with sides 24 cm, 25 cm, and 7 cm. We can use Heron's formula to find the area of the triangle.

1. **Calculate the semi-perimeter (s) of the triangle:**

$$
   s = \frac{24 + 25 + 7}{2} = 28 \text{ cm}
   $$

2. **Apply Heron's formula to find the area (A) of the triangle:**

$$
   A = \sqrt{s(s-a)(s-b)(s-c)}
   $$

where $ a = 24 \text{ cm}, b = 25 \text{ cm}, c = 7 \text{ cm} $.

$$
   A = \sqrt{28(28-24)(28-25)(28-7)}
   $$

$$
   A = \sqrt{28 \times 4 \times 3 \times 21}
   $$

$$
   A = \sqrt{28 \times 4 \times 3 \times 21} = \sqrt{7056} = 84 \text{ cm}^2
   $$

The area of the triangle is 84 cm². According to the problem, this area is equal to the area of a rectangle with a length of 14 cm.

3. **Calculate the width of the rectangle:**

Let the width of the rectangle be $ w $.

$$
   \text{Area of rectangle} = \text{length} \times \text{width} = 14 \times w = 84
   $$

$$
   w = \frac{84}{14} = 6 \text{ cm}
   $$

4. **Calculate the perimeter of the rectangle:**

The perimeter $ P $ of a rectangle is given by:

$$
   P = 2(\text{length} + \text{width}) = 2(14 + 6) = 2 \times 20 = 40 \text{ cm}
   $$

Therefore, the perimeter of the rectangle is 40 cm, which corresponds to Option 4.

**Explanation for Incorrect Options:**

- **Option 1 (20 cm):** This option is incorrect because it suggests a perimeter that is too small. The calculated perimeter based on the given dimensions is 40 cm.

- **Option 2 (18 cm):** This option is incorrect because it is even smaller than Option 1 and does not match the calculated perimeter of 40 cm.

- **Option 3 (36 cm):** This option is incorrect because it is close but still less than the calculated perimeter of 40 cm.

In conclusion, the correct perimeter of the rectangle, based on the given conditions, is 40 cm, which is why Option 4 is the correct answer.