Question

# If the orthocentre, circumcentre of a triangle are $$(-3,5,2),(6,2,5)$$ respectively, then the centroid of the triangle is

A
(3,3,4)
B
(32,72,92)
C
(9,9,12)
D
(92,32,32)

Solution

## The correct option is A $$(3,3,4)$$Circumcentre $$= \left( 6, 2, 5 \right)$$Orthocenter $$= \left( -3, 5 , 2 \right)$$As we know that centroid divides the line joining orthocentre and circumcentre in ratio $$2 : 1$$.Therefore, on applying section formula,Centroid $$= \left( \cfrac{1 \times \left( -3 \right) + 2 \times 6}{2 + 1}, \cfrac{1 \times 5 + 2 \times 2}{2 + 1}, \cfrac{1 \times 2 + 2 \times 5}{2 + 1} \right) \\ = \left( \cfrac{-3 + 12}{3}, \cfrac{5 + 4}{3}, \cfrac{2 + 10}{3} \right) \\ = \left( 3, 3, 4 \right)$$Mathematics

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