If the area of a rhombus is `96 cm^(2)` and one of its diagonals is 16 cm, find its perimeter

Options

A. 70cm
B. 40cm
C. 50cm
D. 60cm

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given information
We are given:
- Area of the rhombus (A) = 96 cm²
- One diagonal (d₁) = 16 cm

### Step 2: Use the formula for the area of a rhombus
The area of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d₁ \times d₂ \]
where \( d₁ \) and \( d₂ \) are the lengths of the diagonals.

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 96 = \frac{1}{2} \times 16 \times d₂ \]

### Step 4: Solve for the second diagonal (d₂)
First, simplify the equation:
\[ 96 = 8 \times d₂ \]
Now, divide both sides by 8:
\[ d₂ = \frac{96}{8} = 12 \text{ cm} \]

### Step 5: Find the lengths of half the diagonals
Since the diagonals bisect each other at right angles in a rhombus:
- Half of diagonal 1 (d₁/2) = \( \frac{16}{2} = 8 \) cm
- Half of diagonal 2 (d₂/2) = \( \frac{12}{2} = 6 \) cm

### Step 6: Use the Pythagorean theorem to find the side length
In triangle AOB, where O is the intersection of the diagonals:
- AO = 6 cm
- BO = 8 cm

Using the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
\[ AB^2 = 6^2 + 8^2 \]
\[ AB^2 = 36 + 64 \]
\[ AB^2 = 100 \]
Taking the square root of both sides:
\[ AB = \sqrt{100} = 10 \text{ cm} \]

### Step 7: Calculate the perimeter of the rhombus
The perimeter (P) of a rhombus is given by:
\[ P = 4 \times \text{side length} \]
Substituting the side length we found:
\[ P = 4 \times 10 = 40 \text{ cm} \]

### Final Answer
The perimeter of the rhombus is **40 cm**.

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