Question

Let \( A(0,0) \) and \( B(8,0) \) be two vertices of a right angled triangle whose hypotenuse is \( BC \). If the circumcentre is \( (4,2) \), then the point \( C \) is:

• A. \( (2,4) \)
• B. \( (0,8) \)
• C. \( (0,4) \)
• D. \( (0,6) \)
• E. \( (2,8) \)

Hint:

For right triangles, the circumcentre always lies at the midpoint of the hypotenuse — a very useful shortcut.

Answer 1

The Correct Option is C

Concept:
• In a right-angled triangle, the circumcentre lies at the midpoint of the hypotenuse.
• Midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Step 1: Using the property of circumcentre in a right triangle.
Since \( BC \) is the hypotenuse, the circumcentre is the midpoint of \( BC \). Let \( C = (x,y) \), and \( B = (8,0) \) \[ \left( \frac{8 + x}{2}, \frac{0 + y}{2} \right) = (4,2) \]

Step 2: Equating coordinates.
\[ \frac{8 + x}{2} = 4 \Rightarrow x = 0 \] \[ \frac{y}{2} = 2 \Rightarrow y = 4 \] Thus, \( C = (0,4) \)

Step 3: Verification.
Check side lengths: \[ AB = 8, \quad AC = 4, \quad BC = \sqrt{80} \] Largest side is \( BC \), so it is the hypotenuse. Hence, the answer is \( (0,4) \)