One diagonal of a rhombus is 24 cm whose side is 13 cm. Find the area of the rhombus.

Options

A. 25 sq. cm
B. 312 sq. cm.
C. 125 sq. cm.
D. 120 sq. cm.

Answer

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

### Step 1: Identify the given values
We are given:
- One diagonal (AC) = 24 cm
- Side of the rhombus (AB) = 13 cm

### Step 2: Use the properties of the rhombus
In a rhombus, the diagonals bisect each other at right angles. Let's denote the other diagonal as BD. Since AC is 24 cm, each half of this diagonal (AO) will be:
\[ AO = \frac{AC}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Step 3: Apply the Pythagorean theorem
We can use the Pythagorean theorem to find the length of half of the other diagonal (BO). According to the theorem:
\[ AB^2 = AO^2 + BO^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + BO^2 \]
\[ 169 = 144 + BO^2 \]

### Step 4: Solve for BO
Now, rearranging the equation to find \( BO^2 \):
\[ BO^2 = 169 - 144 = 25 \]
Taking the square root:
\[ BO = \sqrt{25} = 5 \text{ cm} \]

### Step 5: Find the length of the second diagonal (BD)
Since BO is half of the diagonal BD, we can find BD:
\[ BD = 2 \times BO = 2 \times 5 = 10 \text{ cm} \]

### Step 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. Here, \( D_1 = 24 \text{ cm} \) and \( D_2 = 10 \text{ cm} \):
\[ A = \frac{1}{2} \times 24 \times 10 = \frac{240}{2} = 120 \text{ cm}^2 \]

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

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