What is the area of a rectangle if its diagonal is 51 cm and one of its sides is 24 cm

Options

A. 540
B. 810
C. 1080
D. 360

Answer

To find the area of a rectangle when given the length of one side and the length of the diagonal, we can use the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Diagonal (D) = 51 cm
- One side (B) = 24 cm

### Step 2: Use the Pythagorean theorem
In a rectangle, the relationship between the sides and the diagonal can be expressed using the Pythagorean theorem:
\[ D^2 = L^2 + B^2 \]
Where:
- \( D \) is the diagonal,
- \( L \) is the length,
- \( B \) is the breadth (or one side).

### Step 3: Rearrange the formula to find the length
We need to find the length \( L \):
\[ L^2 = D^2 - B^2 \]

### Step 4: Substitute the known values
Substituting the values we have:
\[ L^2 = 51^2 - 24^2 \]

### Step 5: Calculate the squares
Calculate \( 51^2 \) and \( 24^2 \):
- \( 51^2 = 2601 \)
- \( 24^2 = 576 \)

### Step 6: Perform the subtraction
Now, substitute these values back into the equation:
\[ L^2 = 2601 - 576 \]
\[ L^2 = 2025 \]

### Step 7: Find the length by taking the square root
Now, take the square root of \( L^2 \) to find \( L \):
\[ L = \sqrt{2025} = 45 \, \text{cm} \]

### Step 8: Calculate the area of the rectangle
The area \( A \) of the rectangle is given by:
\[ A = L \times B \]
Substituting the values we found:
\[ A = 45 \, \text{cm} \times 24 \, \text{cm} \]
\[ A = 1080 \, \text{cm}^2 \]

### Final Answer
The area of the rectangle is \( 1080 \, \text{cm}^2 \).

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