The side of a rhombus are 10 cm and one of its diagonals is 16 cm . The area of the rhombus is :

Options

A. `96 cm^(2) `
B. `95 cm^(2) `
C. `94 cm^(2) `
D. `93 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 a rhombus
A rhombus has two diagonals that bisect each other at right angles. This means that if we know one diagonal and the length of the sides, we can use the Pythagorean theorem to find the other diagonal.

### Step 2: Identify the given values
- Side length of the rhombus (AD) = 10 cm
- Length of diagonal AC = 16 cm

### Step 3: Find half of the diagonal AC
Since the diagonals bisect each other, we can find half of diagonal AC:
- AO = OC = AC / 2 = 16 cm / 2 = 8 cm

### Step 4: Use the Pythagorean theorem to find half of diagonal BD
Let OD be half of diagonal BD. According to the Pythagorean theorem in triangle AOD:
\[ AD^2 = AO^2 + OD^2 \]
Substituting the known values:
\[ 10^2 = 8^2 + OD^2 \]
\[ 100 = 64 + OD^2 \]
\[ OD^2 = 100 - 64 = 36 \]
\[ OD = \sqrt{36} = 6 \text{ cm} \]

### Step 5: Find the full length of diagonal BD
Since OD is half of diagonal BD:
\[ BD = 2 \times OD = 2 \times 6 = 12 \text{ cm} \]

### Step 6: Calculate the area of the rhombus
The area (A) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times AC \times BD \]
Substituting the values we found:
\[ A = \frac{1}{2} \times 16 \times 12 \]
\[ A = 8 \times 12 = 96 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is **96 cm²**.

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