The side and one of the diagonals of a rhombus are 13cm and 24cm respectively. The area of the rhombus (in sq.cm.) is

Options

A. a. 156
B. b. 240
C. c. 120
D. d. 130

Answer

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

### Step-by-Step Solution:

1. **Identify Given Values:**
- Side of the rhombus (s) = 13 cm
- One diagonal (d1) = 24 cm

2. **Draw the Rhombus:**
- Label the rhombus as ABCD, where AC and BD are the diagonals.
- Since the diagonals bisect each other at right angles in a rhombus, let O be the intersection point of the diagonals.

3. **Determine Half of the Diagonal:**
- Since diagonal AC = 24 cm, then AO = OC = 12 cm (because diagonals bisect each other).

4. **Apply the Pythagorean Theorem:**
- In triangle AOB, we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
- Here, AB = 13 cm (side of the rhombus), AO = 12 cm, and BO is unknown.
- Substitute the known values:
\[ 13^2 = 12^2 + BO^2 \]
\[ 169 = 144 + BO^2 \]

5. **Solve for BO:**
- Rearranging gives:
\[ BO^2 = 169 - 144 \]
\[ BO^2 = 25 \]
\[ BO = \sqrt{25} = 5 \text{ cm} \]

6. **Find the Length of the Other Diagonal:**
- Since BO = 5 cm, then the full length of diagonal BD = 2 * BO = 2 * 5 = 10 cm.

7. **Calculate the Area of the Rhombus:**
- The area (A) of a rhombus can be calculated using the formula:
\[ A = \frac{d1 \times d2}{2} \]
- Substitute the values of the diagonals:
\[ A = \frac{24 \times 10}{2} \]
\[ A = \frac{240}{2} = 120 \text{ sq.cm} \]

### Final Answer:
The area of the rhombus is **120 sq.cm**.

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