let`ABC` be a right angled triangle at `C. `If the inscribed circle touches the side `AB `at `D` and `(AD) (BD)=11,` then find the area of triangle `ABC.`.

Answer

To find the area of triangle \( ABC \) given that it is a right-angled triangle at \( C \) and that the inscribed circle touches side \( AB \) at point \( D \) with the condition \( AD \cdot BD = 11 \), we can follow these steps:

### Step-by-Step Solution:

1. **Label the Triangle**:
Let \( AD = z \), \( BD = y \), and \( CD = x \). Since \( D \) is the point where the incircle touches \( AB \), we have:
\[ AB = AD + BD = z + y \]

2. **Using the Tangent Properties**:
The lengths of the tangents drawn from a point outside the circle to the circle are equal. Therefore:
- \( AE = AD = z \)
- \( BF = BD = y \)
- \( CE = CF = x \)

3. **Applying the Pythagorean Theorem**:
Since triangle \( ABC \) is right-angled at \( C \), we can apply the Pythagorean theorem:
\[ AC^2 + BC^2 = AB^2 \]
Here, \( AC = AE + CE = z + x \) and \( BC = BF + CF = y + x \). Thus:
\[ (z + x)^2 + (y + x)^2 = (z + y)^2 \]

4. **Expanding the Equation**:
Expanding both sides:
\[ (z^2 + 2zx + x^2) + (y^2 + 2xy + x^2) = (z^2 + 2zy + y^2) \]
This simplifies to:
\[ z^2 + y^2 + 2zx + 2xy + 2x^2 = z^2 + y^2 + 2zy \]
Cancel \( z^2 \) and \( y^2 \) from both sides:
\[ 2zx + 2xy + 2x^2 = 2zy \]
Dividing by 2:
\[ zx + xy + x^2 = zy \]

5. **Substituting the Given Condition**:
We know that \( AD \cdot BD = z \cdot y = 11 \). Therefore, we can substitute \( zy = 11 \) into our equation:
\[ zx + xy + x^2 = 11 \]

6. **Finding the Area of Triangle \( ABC \)**:
The area \( A \) of triangle \( ABC \) can be expressed as:
\[ A = \frac{1}{2} \times BC \times AC = \frac{1}{2} \times (y + x) \times (z + x) \]
Expanding this:
\[ A = \frac{1}{2} \times (xy + xz + yx + x^2) = \frac{1}{2} \times (xy + zx + x^2) \]
From our earlier relation, we know \( zx + xy + x^2 = 11 \):
\[ A = \frac{1}{2} \times 11 = \frac{11}{2} \]

7. **Final Area Calculation**:
Thus, the area of triangle \( ABC \) is:
\[ A = 11 \text{ square units} \]