Multiple choice

If the orthocenter and centroid of a traingle are (-3,5) and (3,3) then its circumcenter is

  1. (6,2)

  2. (3,-1)

  3. (-3,5)

  4. (-3,1)
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A Correct answer
Explanation

The centroid G divides the line segment joining the orthocenter H and circumcenter O in the ratio 2:1. Let O be (x,y). Then (2x + (-3))/3 = 3 and (2y + 5)/3 = 3. Solving gives 2x - 3 = 9 => x = 6, and 2y + 5 = 9 => y = 2.

AI explanation

The centroid of a triangle divides the line segment joining the orthocentre and the circumcentre in the ratio 2:1. Using the section formula, the centroid G(3,3) is calculated as G = (2O + H)/3, where O is the circumcentre and H(-3,5) is the orthocentre. Substituting the values gives 3 = (2x - 3)/3, which solves to x = 6, and 3 = (2y + 5)/3, which solves to y = 2. The coordinates of the circumcentre are therefore (6,2).

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