The perimeter of a rhombus is 100 cm and one of its diagonal is 40 cm. Then its area is :

Question

Accepted Answer

To determine the area of the rhombus, we need to use the given information: the perimeter of the rhombus is 100 cm, and one of its diagonals is 40 cm.

1. **Understanding the properties of a rhombus:**
   - A rhombus is a type of quadrilateral where all sides are of equal length.
   - The perimeter of a rhombus is the sum of the lengths of all its sides. Since all sides are equal, if the perimeter is 100 cm, each side of the rhombus is 100 cm / 4 = 25 cm.

2. **Using the diagonal to find the area:**
   - The diagonals of a rhombus bisect each other at right angles.
   - Let the diagonals be $d_1$ and $d_2$. We know $d_1 = 40$ cm.
   - The formula for the area of a rhombus is given by $\text{Area} = \frac{1}{2} \times d_1 \times d_2$.

3. **Finding the second diagonal:**
   - Since the diagonals bisect each other at right angles, they form right-angled triangles with the sides of the rhombus.
   - Using the Pythagorean theorem in one of these triangles, where the hypotenuse is the side of the rhombus (25 cm), and the legs are half of the diagonals ($\frac{d_1}{2} = 20$ cm and $\frac{d_2}{2}$):
     $$
     \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = \text{side}^2
     $$
     $$
     20^2 + \left(\frac{d_2}{2}\right)^2 = 25^2
     $$
     $$
     400 + \left(\frac{d_2}{2}\right)^2 = 625
     $$
     $$
     \left(\frac{d_2}{2}\right)^2 = 225
     $$
     $$
     \frac{d_2}{2} = 15 \quad \Rightarrow \quad d_2 = 30 \text{ cm}
     $$

4. **Calculating the area:**
   - Substitute the values of the diagonals into the area formula:
     $$
     \text{Area} = \frac{1}{2} \times 40 \times 30 = 600 \text{ cm}^2
     $$

5. **Justification for Option 2:**
   - The calculated area of the rhombus is 600 cm², which matches Option 2.

6. **Explanation for why other options are incorrect:**
   - **Option 1 (300 cm²):** This is incorrect because the calculated area using the correct diagonal lengths does not match this value.
   - **Option 3 (780 cm²):** This is incorrect as it exceeds the calculated area based on the given dimensions.
   - **Option 4 (900 cm²):** This is incorrect as it significantly exceeds the calculated area and does not align with the properties of the rhombus given the side and diagonal lengths.

Therefore, the correct answer is Option 2, with the area of the rhombus being 600 cm².