A circle is circumscribed by the rhombus which in turn is made up by joining the mid-points of a rectangle whose sides, 12 cm and16 cm respectively. What is the area of the circle ?

Options

A. `(625)/(26)pi`
B. `(676)/(25)pi`
C. `(576)/(25)pi`
D. can't be determined

Answer

To find the area of the circle circumscribed by the rhombus formed by joining the midpoints of a rectangle with sides 12 cm and 16 cm, we can follow these steps:

### Step 1: Identify the dimensions of the rectangle
The rectangle has a length of 16 cm and a breadth of 12 cm.

### Step 2: Determine the midpoints of the rectangle
The midpoints of the rectangle will be:
- Midpoint of length (16 cm): \( \frac{16}{2} = 8 \) cm
- Midpoint of breadth (12 cm): \( \frac{12}{2} = 6 \) cm

### Step 3: Form the rhombus
The rhombus is formed by connecting the midpoints of the rectangle. The diagonals of the rhombus are equal to the lengths of the sides of the rectangle:
- One diagonal (length) = 16 cm
- Other diagonal (breadth) = 12 cm

### Step 4: Calculate the lengths of the diagonals of the rhombus
Let’s denote the diagonals of the rhombus as \( d_1 \) and \( d_2 \):
- \( d_1 = 16 \) cm (horizontal diagonal)
- \( d_2 = 12 \) cm (vertical diagonal)

### Step 5: Find the radius of the circle
The radius of the circle (which is also the distance from the center of the rhombus to a vertex) can be calculated using the relationship between the diagonals of the rhombus. The diagonals bisect each other at right angles.

Using the formula for the radius \( R \) of the circle inscribed in the rhombus:
\[ R = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} \]
Substituting the values:
\[ R = \sqrt{\left(\frac{16}{2}\right)^2 + \left(\frac{12}{2}\right)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ cm} \]

### Step 6: Calculate the area of the circle
The area \( A \) of the circle is given by the formula:
\[ A = \pi R^2 \]
Substituting the radius:
\[ A = \pi (10)^2 = 100\pi \text{ cm}^2 \]

### Final Answer
The area of the circle is \( 100\pi \text{ cm}^2 \).

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