One side of a rhombus is 10 cms and one of its diagonals is 12 cms . The area of the rhombus is -----

Options

A. 120 sq. cm
B. 96 sq.cm
C. 80 sq.cm
D. 60 sq.cm

Answer

To find the area of the rhombus, we can use the formula:

\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]

where \(d_1\) and \(d_2\) are the lengths of the diagonals of the rhombus.

### Step 1: Identify the given values
We know one side of the rhombus is 10 cm and one diagonal \(d_1\) is 12 cm. We need to find the length of the other diagonal \(d_2\).

### Step 2: Use the properties of the rhombus
In a rhombus, the diagonals bisect each other at right angles. Therefore, we can use the Pythagorean theorem to find the length of the second diagonal.

Let \(d_1 = 12 \, \text{cm}\) and \(d_2\) be the unknown diagonal. Since the diagonals bisect each other, each half of \(d_1\) is:

\[ \frac{d_1}{2} = \frac{12}{2} = 6 \, \text{cm} \]

Now, we can form a right triangle with one side being half of \(d_1\), half of \(d_2\), and the hypotenuse being the side of the rhombus (10 cm).

### Step 3: Set up the equation using Pythagorean theorem
Using the Pythagorean theorem:

\[ \left(\frac{d_2}{2}\right)^2 + 6^2 = 10^2 \]

### Step 4: Solve for \(d_2\)
Substituting the values:

\[ \left(\frac{d_2}{2}\right)^2 + 36 = 100 \]

Subtract 36 from both sides:

\[ \left(\frac{d_2}{2}\right)^2 = 64 \]

Taking the square root of both sides:

\[ \frac{d_2}{2} = 8 \]

Thus, multiplying both sides by 2 gives:

\[ d_2 = 16 \, \text{cm} \]

### Step 5: Calculate the area of the rhombus
Now we have both diagonals: \(d_1 = 12 \, \text{cm}\) and \(d_2 = 16 \, \text{cm}\).

Using the area formula:

\[ \text{Area} = \frac{1}{2} \times 12 \times 16 \]

Calculating the area:

\[ \text{Area} = \frac{1}{2} \times 192 = 96 \, \text{cm}^2 \]

### Final Answer
The area of the rhombus is \(96 \, \text{cm}^2\).

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