If the diagonal of a rectangle is 17 cm long and its perimeter is 46 cm, the area of the rectangle is

Options

A. `100cm^(2)`
B. `110cm^(2)`
C. `120cm^(2)`
D. `150cm^(2)`

Answer

To find the area of the rectangle given its diagonal and perimeter, we can follow these steps:

### Step 1: Write down the given information
- Diagonal (d) = 17 cm
- Perimeter (P) = 46 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.

### Step 3: Set up the equation for the perimeter
Substituting the given perimeter into the formula:
\[ 46 = 2(l + b) \]

### Step 4: Solve for \( l + b \)
Dividing both sides by 2:
\[ l + b = \frac{46}{2} = 23 \]
This gives us our first equation:
\[ l + b = 23 \quad \text{(Equation 1)} \]

### Step 5: Use the Pythagorean theorem
The diagonal of the rectangle can be related to its sides using the Pythagorean theorem:
\[ d^2 = l^2 + b^2 \]
Substituting the value of the diagonal:
\[ 17^2 = l^2 + b^2 \]
This simplifies to:
\[ 289 = l^2 + b^2 \quad \text{(Equation 2)} \]

### Step 6: Square Equation 1
Now, we square Equation 1:
\[ (l + b)^2 = 23^2 \]
This expands to:
\[ l^2 + 2lb + b^2 = 529 \quad \text{(Equation 3)} \]

### Step 7: Substitute Equation 2 into Equation 3
From Equation 2, we know that:
\[ l^2 + b^2 = 289 \]
Substituting this into Equation 3:
\[ 289 + 2lb = 529 \]

### Step 8: Solve for \( lb \)
Now, we can solve for \( 2lb \):
\[ 2lb = 529 - 289 \]
\[ 2lb = 240 \]
Dividing both sides by 2 gives:
\[ lb = \frac{240}{2} = 120 \]

### Step 9: Find the area of the rectangle
The area \( A \) of the rectangle is given by:
\[ A = l \times b \]
Thus, the area is:
\[ A = 120 \, \text{cm}^2 \]

### Final Answer
The area of the rectangle is \( 120 \, \text{cm}^2 \).

---