Multiple choice

If the co-ordinates of orthocentre and the centroid of a triangle ABC are (-5, 7) and (5, 5), then the circumcentre of the triangle ABC is:

  1. (25, 1)

  2. (10, 4)

  3. (-5, 2)

  4. (0, 6)
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The centroid G divides the line segment joining the orthocentre H and circumcentre O in the ratio 2:1. Let O = (x, y). Then (2x + 1*(-5))/3 = 5 and (2y + 1*7)/3 = 5. Solving gives x = 10, y = 4.

AI explanation

Euler's line theorem states that the centroid divides the distance from the orthocentre to the circumcentre in a 2:1 ratio. The coordinates of the centroid are calculated as the weighted average of the orthocentre and circumcentre, using the formula G = (O + 2C) / 3. We substitute the known coordinates into 5 = (-5 + 2x) / 3 and 5 = (7 + 2y) / 3 to solve for the circumcentre coordinates of C. Solving these linear equations gives x = 10 and y = 4, making the circumcentre (10, 4).

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