The side of a rhombus is 5 cm and one of its diagonal is 8 cm. What is the area of the rhombus ?

Options

A. a)`30 cm^(2)`
B. b)`20 cm^(2)`
C. c)`40 cm^(2)`
D. d)`24 cm^(2)`

Answer

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

### Step 1: Understand the properties of the rhombus
A rhombus has all four sides equal, and its diagonals bisect each other at right angles (90 degrees).

### Step 2: Identify the given values
- Side length (s) = 5 cm
- One diagonal (d1) = 8 cm

### Step 3: Calculate half of the diagonal
Since the diagonals bisect each other, half of the diagonal d1 will be:
\[ \frac{d1}{2} = \frac{8}{2} = 4 \text{ cm} \]

### Step 4: Use the Pythagorean theorem to find the other diagonal
Let the other diagonal be \(d2\). In the right triangle formed by half of the diagonals and the side of the rhombus, we can apply the Pythagorean theorem:
\[ s^2 = \left(\frac{d1}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + \left(\frac{d2}{2}\right)^2 \]
\[ 25 = 16 + \left(\frac{d2}{2}\right)^2 \]
\[ 25 - 16 = \left(\frac{d2}{2}\right)^2 \]
\[ 9 = \left(\frac{d2}{2}\right)^2 \]
Taking the square root:
\[ \frac{d2}{2} = 3 \implies d2 = 6 \text{ cm} \]

### Step 5: Calculate the area of the rhombus
The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d1 \times d2 \]
Substituting the values of the diagonals:
\[ A = \frac{1}{2} \times 8 \times 6 \]
\[ A = \frac{1}{2} \times 48 = 24 \text{ cm}^2 \]

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

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