If the opposite sides of a cyclic quadrilateral are equal and the sum of two of the adjacent sides is 10 cm, find the maximum possible area of the quadrilateral (in sq. cm).

Options

A. A)100
B. B)25
C. C)24
D. D)data insufficient

Answer

To solve the problem step by step, we need to find the maximum possible area of a cyclic quadrilateral where the opposite sides are equal and the sum of two adjacent sides is 10 cm.

### Step 1: Understand the properties of the cyclic quadrilateral
A cyclic quadrilateral is a quadrilateral whose vertices lie on a single circle. The opposite sides of a cyclic quadrilateral are equal, which means if we denote the sides as \( L \) and \( B \), we have:
- \( AB = CD \) and \( AD = BC \)

### Step 2: Set up the equation for the sides
We are given that the sum of two adjacent sides is 10 cm. Let's denote the two adjacent sides as \( L \) and \( B \):
\[ L + B = 10 \]

### Step 3: Express one side in terms of the other
From the equation \( L + B = 10 \), we can express \( B \) in terms of \( L \):
\[ B = 10 - L \]

### Step 4: Write the area formula
The area \( A \) of a rectangle (which is a type of cyclic quadrilateral) can be calculated using the formula:
\[ A = L \times B \]
Substituting \( B \) from the previous step:
\[ A = L \times (10 - L) \]
\[ A = 10L - L^2 \]

### Step 5: Find the maximum area
To find the maximum area, we need to find the vertex of the quadratic equation \( A = -L^2 + 10L \). The maximum value of a quadratic \( ax^2 + bx + c \) occurs at \( L = -\frac{b}{2a} \):
Here, \( a = -1 \) and \( b = 10 \):
\[ L = -\frac{10}{2 \times -1} = 5 \]

### Step 6: Calculate the corresponding value of \( B \)
Now, substituting \( L = 5 \) back into the equation for \( B \):
\[ B = 10 - L = 10 - 5 = 5 \]

### Step 7: Calculate the maximum area
Now that we have both sides:
\[ A = L \times B = 5 \times 5 = 25 \text{ cm}^2 \]

### Conclusion
The maximum possible area of the cyclic quadrilateral is \( 25 \text{ cm}^2 \).

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