The perimeter of a rhombus is 52 cm. If one diagonal is 24 cm, find :
its area.

Answer

To find the area of the rhombus given its perimeter and one diagonal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Perimeter of the rhombus (P) = 52 cm
- One diagonal (BD) = 24 cm

2. **Calculate the Length of One Side of the Rhombus:**
- The formula for the perimeter of a rhombus is:
\[ P = 4s \]
where \(s\) is the length of one side.
- Rearranging the formula to find \(s\):
\[ s = \frac{P}{4} = \frac{52}{4} = 13 \text{ cm} \]

3. **Use the Properties of the Rhombus:**
- The diagonals of a rhombus bisect each other at right angles. Let the intersection point of the diagonals be \(O\).
- Therefore, \(OB = \frac{BD}{2} = \frac{24}{2} = 12 \text{ cm}\).

4. **Apply the Pythagorean Theorem:**
- In triangle \(AOB\), we can apply the Pythagorean theorem:
\[ AO^2 + OB^2 = AB^2 \]
- We know \(AB = s = 13 \text{ cm}\) and \(OB = 12 \text{ cm}\), so:
\[ AO^2 + 12^2 = 13^2 \]
- Substituting the values:
\[ AO^2 + 144 = 169 \]
- Solving for \(AO^2\):
\[ AO^2 = 169 - 144 = 25 \]
- Thus, \(AO = \sqrt{25} = 5 \text{ cm}\).

5. **Calculate the Length of the Other Diagonal (AC):**
- Since \(AO\) is half of diagonal \(AC\):
\[ AC = 2 \times AO = 2 \times 5 = 10 \text{ cm} \]

6. **Calculate the Area of the Rhombus:**
- The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \(d_1\) and \(d_2\) are the lengths of the diagonals.
- Substituting the values we found:
\[ A = \frac{1}{2} \times AC \times BD = \frac{1}{2} \times 10 \times 24 \]
- Calculating the area:
\[ A = \frac{1}{2} \times 240 = 120 \text{ cm}^2 \]

### Final Answer:
The area of the rhombus is \(120 \text{ cm}^2\).

---