If the lengths of the sides of a cyclic quadrilateral are 9, 10, 10 and 21 respectively, find the area of the cyclic quadrilateral

Options

A. `30 sqrt"" 21`
B. ` 21 sqrt"" 30`
C. 90
D. None of these

Answer

To find the area of the cyclic quadrilateral with sides of lengths 9, 10, 10, and 21, we will use Brahmagupta's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides of the quadrilateral
Let the sides of the cyclic quadrilateral be:
- \( a = 9 \)
- \( b = 10 \)
- \( c = 10 \)
- \( d = 21 \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c + d}{2} \]
Substituting the values:
\[ s = \frac{9 + 10 + 10 + 21}{2} = \frac{50}{2} = 25 \]

### Step 3: Apply Brahmagupta's formula for the area (A)
Brahmagupta's formula for the area \( A \) of a cyclic quadrilateral is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)(s-d)} \]
Now we will calculate each term:
- \( s - a = 25 - 9 = 16 \)
- \( s - b = 25 - 10 = 15 \)
- \( s - c = 25 - 10 = 15 \)
- \( s - d = 25 - 21 = 4 \)

### Step 4: Substitute the values into the area formula
Now substituting these values into the area formula:
\[ A = \sqrt{25 \times 16 \times 15 \times 15 \times 4} \]

### Step 5: Calculate the product inside the square root
Calculating the product:
\[ 25 \times 16 = 400 \]
\[ 15 \times 15 = 225 \]
\[ 400 \times 225 = 90000 \]
\[ 90000 \times 4 = 360000 \]

### Step 6: Calculate the square root
Now, we find the square root:
\[ A = \sqrt{360000} = 600 \]

### Final Answer
The area of the cyclic quadrilateral is \( 600 \) square units.

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