In a triangle `ABC, angleB=90^(@), D` is the midpoint of `AC, BD=sqrt117`. Sum of sides of AB & BC is 30 cm. Find the area of triangle ABC.

Answer

To solve the problem step by step, we will follow the information given and apply the necessary mathematical principles.

### Step 1: Identify the given information
We have a right triangle \( ABC \) where:
- \( \angle B = 90^\circ \)
- \( D \) is the midpoint of \( AC \)
- \( BD = \sqrt{117} \)
- The sum of sides \( AB + BC = 30 \) cm

### Step 2: Find the length of \( AC \)
Since \( D \) is the midpoint of \( AC \), we can express \( AC \) in terms of \( BD \):
\[ AC = 2 \times BD = 2 \times \sqrt{117} \]
Calculating this gives:
\[ AC = 2\sqrt{117} \text{ cm} \]

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting \( AC \):
\[ AB^2 + BC^2 = (2\sqrt{117})^2 = 4 \times 117 = 468 \]
This gives us our first equation:
\[ AB^2 + BC^2 = 468 \quad \text{(1)} \]

### Step 4: Use the sum of sides equation
We know from the problem statement that:
\[ AB + BC = 30 \quad \text{(2)} \]

### Step 5: Square equation (2)
Squaring both sides of equation (2):
\[ (AB + BC)^2 = 30^2 \]
Expanding this:
\[ AB^2 + 2AB \cdot BC + BC^2 = 900 \]
Substituting \( AB^2 + BC^2 \) from equation (1):
\[ 468 + 2AB \cdot BC = 900 \]

### Step 6: Solve for \( AB \cdot BC \)
Rearranging gives:
\[ 2AB \cdot BC = 900 - 468 = 432 \]
Thus:
\[ AB \cdot BC = \frac{432}{2} = 216 \quad \text{(3)} \]

### Step 7: Find the area of triangle \( ABC \)
The area \( A \) of triangle \( ABC \) can be calculated using:
\[ A = \frac{1}{2} \times AB \times BC \]
Substituting \( AB \cdot BC \) from equation (3):
\[ A = \frac{1}{2} \times 216 = 108 \text{ cm}^2 \]

### Final Answer
The area of triangle \( ABC \) is \( 108 \text{ cm}^2 \).

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