The length of a rectangle is 16 cm and the length of its diagonal is 20 cm. The area of the rectangle is

Options

A. `320cm^(2)`
B. `160cm^(2)`
C. `192cm^(2)`
D. `156cm^(2)`

Answer

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

### Step 1: Identify the given values
We know the following:
- Length (l) of the rectangle = 16 cm
- Diagonal (d) of the rectangle = 20 cm

### Step 2: Use the Pythagorean theorem
In a rectangle, the diagonal can be calculated using the Pythagorean theorem:
\[ d^2 = l^2 + b^2 \]
where \( b \) is the breadth (width) of the rectangle.

### Step 3: Substitute the known values into the equation
Substituting the known values into the equation:
\[ 20^2 = 16^2 + b^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 400 = 256 + b^2 \]

### Step 5: Rearrange the equation to solve for \( b^2 \)
Now, rearranging the equation to isolate \( b^2 \):
\[ b^2 = 400 - 256 \]

### Step 6: Perform the subtraction
Calculating the right side:
\[ b^2 = 144 \]

### Step 7: Find the breadth \( b \)
Taking the square root of both sides to find \( b \):
\[ b = \sqrt{144} = 12 \text{ cm} \]

### Step 8: Calculate the area of the rectangle
Now that we have both the length and breadth, we can find the area \( A \) of the rectangle using the formula:
\[ A = l \times b \]
Substituting the values:
\[ A = 16 \times 12 \]

### Step 9: Perform the multiplication
Calculating the area:
\[ A = 192 \text{ cm}^2 \]

Thus, the area of the rectangle is **192 cm²**.

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