The diagonal of a rectangle field is 17 m and its perimeter is 46 m. What is the area of the field ?

Options

A. 112 sq m
B. 120 sq m
C. 132 sq m
D. 289 sq m

Answer

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

### Step 1: Define Variables
Let the length of the rectangle be \( l \) meters and the breadth be \( b \) meters.

### Step 2: Use the Perimeter Formula
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(l + b) \]
We know the perimeter is 46 meters, so we can set up the equation:
\[ 2(l + b) = 46 \]
Dividing both sides by 2 gives:
\[ l + b = 23 \quad \text{(Equation 1)} \]

### Step 3: Use the Diagonal Formula
The diagonal \( d \) of a rectangle can be found using the Pythagorean theorem:
\[ d^2 = l^2 + b^2 \]
Given that the diagonal is 17 meters, we can write:
\[ l^2 + b^2 = 17^2 \]
Calculating \( 17^2 \) gives:
\[ l^2 + b^2 = 289 \quad \text{(Equation 2)} \]

### Step 4: Express \( l^2 + b^2 \) in Terms of \( l + b \)
From Equation 1, we can square both sides:
\[ (l + b)^2 = 23^2 \]
This expands to:
\[ l^2 + 2lb + b^2 = 529 \]
Now, substituting \( l^2 + b^2 \) from Equation 2:
\[ 289 + 2lb = 529 \]

### Step 5: Solve for \( lb \)
Rearranging the equation gives:
\[ 2lb = 529 - 289 \]
Calculating the right side:
\[ 2lb = 240 \]
Dividing by 2:
\[ lb = 120 \]

### Step 6: Calculate the Area
The area \( A \) of the rectangle is given by:
\[ A = l \times b \]
From our calculation, we found:
\[ A = 120 \text{ square meters} \]

### Conclusion
Thus, the area of the rectangle field is \( \boxed{120} \) square meters.

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