Question

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

A
(3,5)
B
(3,1)
C
(3,1)
D
(9,5)

Solution

## The correct option is A $$(-3, 5)$$In any triangle, orthocentre, centroid and circumcentre are collinear and centroid divides the line joining orthocentre and circumcenter in the ratio $$2 : 1.$$Let the orthocentre be $$(x,y)$$.Using the section formula, $$\left( \cfrac { m{ x }_{ 2 } + n{ x }_{ 1 } }{ m + n } ,\cfrac { m{ y }_{ 2} + n{ y }_{ 1 } }{ m + n } \right)$$Substituting $$({ x }_{ 1 },{ y }_{ 1 }) = (x,y)$$ and $$({x }_{ 2 },{ y}_{ 2 }) = (6,2)$$  and $$m = 2, n = 1$$ in the section formula, we get the centroid $$= \left( \cfrac { 2(6) + 1(x) }{ 2 +1 } ,\cfrac { 2(2) +1(y) }{ 2 + 1 } \right) = \left(\cfrac { x + 12 }{ 3 } ,\cfrac { y + 4 }{ 3} \right)$$Given centroid $$= (3,3)$$$$=> \left(\cfrac { x + 12 }{ 3 } ,\cfrac { y + 4 }{ 3} \right) = (3,3)$$$$=> \cfrac { x + 12 }{ 3 } = 3 ; \cfrac { y + 4 }{ 3} = 3$$$$=> x + 12 = 9 ; y + 4 = 9$$$$=> x = -3 ; y = 5$$Hence, orthocentre $$= (-3,5)$$Maths

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