Find the area of a rectangular plot one side of which is 48 m and its diagonal 50 m.

Answer

To find the area of the rectangular plot, we will follow these steps:

### Step 1: Identify the given values
We are given:
- One side (length) of the rectangle, \( AB = 48 \) m
- The diagonal \( BD = 50 \) m

### Step 2: Use the Pythagorean theorem
In a rectangle, the diagonal forms a right triangle with the two sides. According to the Pythagorean theorem:
\[ BD^2 = AB^2 + AD^2 \]
Where:
- \( BD \) is the diagonal (hypotenuse)
- \( AB \) is one side (base)
- \( AD \) is the other side (perpendicular)

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 50^2 = 48^2 + AD^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 2500 = 2304 + AD^2 \]

### Step 5: Rearrange the equation to solve for \( AD^2 \)
Now, rearranging the equation to isolate \( AD^2 \):
\[ AD^2 = 2500 - 2304 \]
\[ AD^2 = 196 \]

### Step 6: Take the square root to find \( AD \)
Taking the square root of both sides to find \( AD \):
\[ AD = \sqrt{196} = 14 \text{ m} \]

### Step 7: Calculate the area of the rectangle
The area \( A \) of a rectangle is given by:
\[ A = \text{length} \times \text{breadth} \]
Substituting the values we have:
\[ A = 48 \times 14 \]
\[ A = 672 \text{ m}^2 \]

### Final Answer:
The area of the rectangular plot is \( 672 \text{ m}^2 \).

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