Answer the following questions based on the information given below.
In a kite, the shorter diagonal intersects the longer diagonal in the ratio of 6:7. and the longer diagonal of the kite is 52 cm and perimeter of the kite is 208 cm.
Find the area of the circle inscribed in this kite.

Options

A. 2080 sq. cm
B. 1263 sq. cm
C. 1591 sq. cm
D. 2340 sq. cm

Answer

To find the area of the circle inscribed in the kite, we can follow these steps:

### Step 1: Understand the Kite's Properties
In a kite, the diagonals intersect at right angles, and the shorter diagonal divides the longer diagonal into two segments in a given ratio. Here, the shorter diagonal intersects the longer diagonal in the ratio of 6:7.

### Step 2: Determine the Lengths of the Diagonals
Given that the longer diagonal is 52 cm, we can denote the segments of the longer diagonal as:
- \( 6x \) and \( 7x \)

The total length of the longer diagonal is:
\[ 6x + 7x = 13x \]
Setting this equal to 52 cm:
\[ 13x = 52 \]
\[ x = \frac{52}{13} = 4 \]

Now, we can find the lengths of the segments:
- Shorter diagonal segment 1: \( 6x = 6 \times 4 = 24 \) cm
- Shorter diagonal segment 2: \( 7x = 7 \times 4 = 28 \) cm

Thus, the length of the shorter diagonal is:
\[ 24 + 28 = 52 \text{ cm} \]

### Step 3: Calculate the Perimeter and Side Lengths
The perimeter of the kite is given as 208 cm. Since the kite has two pairs of equal sides, we can denote the lengths of the sides as \( a \) and \( b \):
\[ 2a + 2b = 208 \]
\[ a + b = 104 \]

### Step 4: Use the Pythagorean Theorem
To find the lengths of the sides \( a \) and \( b \), we can use the diagonals. The diagonals intersect at right angles, so we can form right triangles. The half lengths of the diagonals are:
- Half of the shorter diagonal: \( \frac{24}{2} = 12 \) cm
- Half of the longer diagonal: \( \frac{52}{2} = 26 \) cm

Using the Pythagorean theorem:
\[ a^2 = 12^2 + 26^2 \]
\[ a^2 = 144 + 676 \]
\[ a^2 = 820 \]
\[ a = \sqrt{820} \approx 28.7 \text{ cm} \]

### Step 5: Calculate the Semi-perimeter
The semi-perimeter \( s \) is half of the perimeter:
\[ s = \frac{208}{2} = 104 \text{ cm} \]

### Step 6: Calculate the Area of the Kite
The area \( A \) of the kite can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Here, both diagonals are 52 cm:
\[ A = \frac{1}{2} \times 52 \times 24 = 624 \text{ cm}^2 \]

### Step 7: Calculate the Radius of the Inscribed Circle
The radius \( r \) of the inscribed circle can be found using the formula:
\[ r = \frac{A}{s} \]
Substituting the values we found:
\[ r = \frac{624}{104} = 6 \text{ cm} \]

### Step 8: Calculate the Area of the Inscribed Circle
The area \( A_c \) of the circle is given by:
\[ A_c = \pi r^2 \]
Substituting the radius:
\[ A_c = \pi \times (6)^2 = 36\pi \approx 113.1 \text{ cm}^2 \]

### Final Answer
The area of the circle inscribed in the kite is approximately \( 113.1 \text{ cm}^2 \).