The area of a rhombus, one of whose diagonals measures 8 cm and the side is 5 cm, is :

Options

A. 25 `cm^(2)`
B. 24 `cm^(2)`
C. 24.5 `cm^(2)`
D. 26 `cm^(2)`

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Values**:
- One diagonal (d1) = 8 cm
- Side length (s) = 5 cm

2. **Use the Formula for the Area of a 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.

3. **Find the Length of the Second Diagonal (d2)**:
To find the second diagonal, we can use the properties of the rhombus. The diagonals bisect each other at right angles. Therefore, we can form two right triangles with the diagonals and the sides of the rhombus.

Let:
- Half of diagonal 1 (d1/2) = 8 cm / 2 = 4 cm
- Half of diagonal 2 (d2/2) = y cm

Using the Pythagorean theorem in one of the triangles formed:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Plugging in the values:
\[ 5^2 = 4^2 + y^2 \]
\[ 25 = 16 + y^2 \]
\[ y^2 = 25 - 16 = 9 \]
\[ y = 3 \text{ cm} \]
Therefore, the second diagonal \(d_2 = 2y = 2 \times 3 = 6 \text{ cm}\).

4. **Calculate the Area**:
Now that we have both diagonals:
- \(d_1 = 8 \text{ cm}\)
- \(d_2 = 6 \text{ cm}\)

Plugging these values into the area formula:
\[ A = \frac{1}{2} \times 8 \times 6 \]
\[ A = \frac{1}{2} \times 48 = 24 \text{ cm}^2 \]

5. **Final Answer**:
The area of the rhombus is \(24 \text{ cm}^2\).