If the points A (2,9), B (a,5) and C (5,5) are the vertices of a `triangleABC`. Right-angled at B, then find the values of a and hence the area of `triangleABC`.

Explanation

To solve the problem, we need to find the value of \( a \) for the triangle \( ABC \) where \( A(2, 9) \), \( B(a, 5) \), and \( C(5, 5) \) is right-angled at \( B \). We will then calculate the area of the triangle.

### Step 1: Use the Pythagorean Theorem
Since triangle \( ABC \) is right-angled at \( B \), we can apply the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]

...