What is the area (in `cm^(2)`) of the rhombus having side as 5 cm and one of the diagonal as 8 cm?

Options

A. 25
B. 24
C. 26
D. 23

Answer

To find the area of a rhombus when given the length of one side and one diagonal, we can use the following steps:

### Step 1: Understand the properties of a rhombus
A rhombus is a type of quadrilateral where all four sides are of equal length. The diagonals of a rhombus bisect each other at right angles.

### Step 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.

### Step 3: Identify the given values
In this problem, we know:
- The length of one side \( s = 5 \, \text{cm} \)
- One diagonal \( d_1 = 8 \, \text{cm} \)

### Step 4: Find the length of the second diagonal
To find the length of the second diagonal \( d_2 \), we can use the relationship between the sides and the diagonals in a rhombus. The diagonals bisect each other at right angles, so we can use the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the known values:
\[ 5^2 = \left(\frac{8}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 25 = 4^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 25 = 16 + \left(\frac{d_2}{2}\right)^2 \]
\[ 25 - 16 = \left(\frac{d_2}{2}\right)^2 \]
\[ 9 = \left(\frac{d_2}{2}\right)^2 \]
Taking the square root of both sides:
\[ \frac{d_2}{2} = 3 \]
Thus, multiplying by 2 gives:
\[ d_2 = 6 \, \text{cm} \]

### Step 5: Calculate the area of the rhombus
Now that we have both diagonals, we can substitute \( d_1 \) and \( d_2 \) into the area formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
\[ A = \frac{1}{2} \times 8 \times 6 \]
\[ A = \frac{1}{2} \times 48 \]
\[ A = 24 \, \text{cm}^2 \]

### Final Answer
The area of the rhombus is \( 24 \, \text{cm}^2 \).

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