If the perimeter and diagonal of a rectangle are 14 and 5 cms respectively, find its area.

Options

A. `12 cm^(2)`
B. `16 cm^(2)`
C. `20 cm^(2)`
D. `24 cm^(2) `

Answer

To solve the problem, we need to find the area of a rectangle given its perimeter and diagonal. Here are the steps to solve it:

### Step 1: Understand the given information
- Perimeter (P) of the rectangle = 14 cm
- Diagonal (d) of the rectangle = 5 cm

### Step 2: Use the formula for the perimeter of a rectangle
The formula for the perimeter of a rectangle is:
\[ P = 2(L + B) \]
where \( L \) is the length and \( B \) is the breadth.

Setting the perimeter equal to 14 cm:
\[ 2(L + B) = 14 \]

### Step 3: Simplify to find \( L + B \)
Dividing both sides by 2:
\[ L + B = 7 \quad \text{(1)} \]

### Step 4: Use the Pythagorean theorem
The diagonal of a rectangle can be found using the Pythagorean theorem:
\[ d^2 = L^2 + B^2 \]
Given that the diagonal is 5 cm:
\[ 5^2 = L^2 + B^2 \]
\[ 25 = L^2 + B^2 \quad \text{(2)} \]

### Step 5: Express \( L^2 + B^2 \) in terms of \( L + B \)
We can use the identity:
\[ (L + B)^2 = L^2 + B^2 + 2LB \]
Substituting \( L + B = 7 \):
\[ 7^2 = L^2 + B^2 + 2LB \]
\[ 49 = L^2 + B^2 + 2LB \]

### Step 6: Substitute \( L^2 + B^2 \) from equation (2)
From equation (2), we know:
\[ L^2 + B^2 = 25 \]
Substituting this into the equation:
\[ 49 = 25 + 2LB \]
\[ 49 - 25 = 2LB \]
\[ 24 = 2LB \]

### Step 7: Solve for \( LB \)
Dividing both sides by 2:
\[ LB = 12 \quad \text{(3)} \]

### Step 8: Find the area of the rectangle
The area (A) of the rectangle is given by:
\[ A = L \times B \]
From equation (3), we have:
\[ A = LB = 12 \, \text{cm}^2 \]

### Final Answer
The area of the rectangle is \( 12 \, \text{cm}^2 \).

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