Question

The circumcentre of the triangle with vertices \( (8, 6), (8, -2) \) and \( (2, -2) \) is at the point:

• A. \( (2, -1) \)
• B. \( (1, -2) \)
• C. \( (5, 2) \)
• D. \( (2, 5) \)
• E. \( (4, 5) \)

Hint:

If you see coordinates sharing the same x-values and same y-values (e.g., $(x_1, y_1), (x_1, y_2), (x_2, y_2)$), it's a right triangle! The circumcenter is just the midpoint of the two points that don't share a common coordinate.

Answer 1

The Correct Option is C

Concept: The circumcentre of a right-angled triangle is the midpoint of the hypotenuse. We should first check if the given triangle is right-angled by looking at the coordinates.

Step 1: Analyze the vertices.
The vertices are \( A(8, 6) \), \( B(8, -2) \), and \( C(2, -2) \).
• Line \( AB \) is vertical (\( x = 8 \)).
• Line \( BC \) is horizontal (\( y = -2 \)). Since a vertical line and a horizontal line are perpendicular, the triangle has a right angle at vertex \( B(8, -2) \).

Step 2: Identify the hypotenuse and find its midpoint.
The hypotenuse is the side opposite the right angle, which is the segment \( AC \) joining \( (8, 6) \) and \( (2, -2) \). The circumcentre is the midpoint of \( AC \): \[ \text{Midpoint} = \left( \frac{8 + 2}{2}, \frac{6 + (-2)}{2} \right) \] \[ \text{Midpoint} = \left( \frac{10}{2}, \frac{4}{2} \right) = (5, 2) \]