Find the area of a rectangular plot, one side of which measures `35 m` and the diagonal is `37 m.`

Options

A. `820m^(2)`
B. `420m^(2)`
C. `920m^(2)`
D. `480m^(2)`

Answer

To find the area of a rectangular plot where one side measures 35 m and the diagonal measures 37 m, we can follow these steps:

### Step 1: Identify the known values
We know:
- One side (length) of the rectangle, \( a = 35 \, m \)
- The diagonal of the rectangle, \( d = 37 \, m \)

### Step 2: Use the Pythagorean theorem
In a rectangle, the relationship between the sides and the diagonal can be expressed using the Pythagorean theorem:
\[ d^2 = a^2 + b^2 \]
where \( b \) is the other side (breadth) of the rectangle.

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 37^2 = 35^2 + b^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 37^2 = 1369 \]
\[ 35^2 = 1225 \]
So the equation becomes:
\[ 1369 = 1225 + b^2 \]

### Step 5: Solve for \( b^2 \)
Rearranging the equation to solve for \( b^2 \):
\[ b^2 = 1369 - 1225 \]
\[ b^2 = 144 \]

### Step 6: Find \( b \) by taking the square root
Taking the square root of both sides to find \( b \):
\[ b = \sqrt{144} = 12 \, m \]

### Step 7: Calculate the area of the rectangle
Now that we have both dimensions of the rectangle (length \( a = 35 \, m \) and breadth \( b = 12 \, m \)), we can calculate the area \( A \):
\[ A = a \times b = 35 \times 12 \]
\[ A = 420 \, m^2 \]

### Final Answer
The area of the rectangular plot is \( 420 \, m^2 \).

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