A rectangle with one side of length 4 cm. is inscribed in a circle of diameter 5 cm. Find, the area of the rectangle.

Options

A. `21 cm^(2)`
B. `12 cm^(2)`
C. `4 cm^(2)`
D. `3 cm^(2)`

Answer

To find the area of the rectangle inscribed in a circle, we will follow these steps:

### Step 1: Understand the dimensions of the rectangle and the circle.
We know that one side of the rectangle (length) is 4 cm, and the diameter of the circle is 5 cm.

### Step 2: Calculate the radius of the circle.
The radius (r) of the circle is half of the diameter.
\[ r = \frac{diameter}{2} = \frac{5 \, \text{cm}}{2} = 2.5 \, \text{cm} \]

### Step 3: Use the Pythagorean theorem to find the other side (breadth) of the rectangle.
In a rectangle inscribed in a circle, the diagonal of the rectangle is equal to the diameter of the circle. We can denote the length of the rectangle as \( l = 4 \, \text{cm} \) and the breadth as \( b \). The relationship can be expressed using the Pythagorean theorem:
\[ l^2 + b^2 = d^2 \]
where \( d \) is the diameter of the circle. Substituting the known values:
\[ 4^2 + b^2 = 5^2 \]
\[ 16 + b^2 = 25 \]

### Step 4: Solve for the breadth (b).
Rearranging the equation gives:
\[ b^2 = 25 - 16 \]
\[ b^2 = 9 \]
Taking the square root of both sides, we find:
\[ b = 3 \, \text{cm} \]

### Step 5: Calculate the area of the rectangle.
The area (A) of the rectangle is given by the formula:
\[ A = l \times b \]
Substituting the values we found:
\[ A = 4 \, \text{cm} \times 3 \, \text{cm} = 12 \, \text{cm}^2 \]

### Final Answer:
The area of the rectangle is \( 12 \, \text{cm}^2 \).

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