Find the area of a rectangle whose length is 8 cm and diagonal 10 m.

Options

A. `24 m^(2)`
B. `48 m^(2)`
C. `56 m^(2)`
D. `36 m^(2)`

Answer

To find the area of a rectangle whose length is 8 cm and diagonal is 10 m, we can follow these steps:

### Step 1: Convert the units
Since the length is given in centimeters and the diagonal in meters, we need to convert the length to meters for consistency.
- Length (L) = 8 cm = 0.08 m

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

### Step 3: Substitute the known values
We know:
- Diagonal (D) = 10 m
- Length (L) = 0.08 m

Substituting these values into the equation:
\[ 10^2 = (0.08)^2 + B^2 \]

### Step 4: Calculate the squares
Calculating the squares:
- \( 10^2 = 100 \)
- \( (0.08)^2 = 0.0064 \)

So the equation becomes:
\[ 100 = 0.0064 + B^2 \]

### Step 5: Solve for breadth (B)
Rearranging the equation to find B:
\[ B^2 = 100 - 0.0064 \]
\[ B^2 = 99.9936 \]

Now, take the square root to find B:
\[ B = \sqrt{99.9936} \approx 9.99968 \text{ m} \]

### Step 6: Calculate the area of the rectangle
The area (A) of the rectangle can be calculated using the formula:
\[ A = L \times B \]
Substituting the values:
\[ A = 0.08 \text{ m} \times 9.99968 \text{ m} \]
\[ A \approx 0.7999744 \text{ m}^2 \]

### Final Answer
The area of the rectangle is approximately \( 0.8 \text{ m}^2 \).

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