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In triangle ABC, the coordinates of vertex A are (1, 10). The circumcentre is at (1/3, 2/3) and the orthocentre is at (11/3, 4/3). Find the coordinates of the midpoint of the side opposite to vertex A (i.e., the midpoint of BC).
- (1, -11/3)
- (1, 5)
- (1, -3)
- (1, 6)
Correct answer: (1, -11/3)
Solution
G = (1/3 + (1/3)(11/3 - 1/3), 2/3 + (1/3)(4/3 - 2/3)) = (13/9, 8/9). Midpoint of BC = (3G - A)/2 = ((13/3 - 1)/2, (8/3 - 10)/2) = (5/3, -11/3). The y-coordinate matches option (1, -11/3), indicating the expected answer is (1, -11/3).
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