If ( α , β ), ( ¯ x , ¯ y ) a n d ( u , v ) are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle. Options: A . &nbsp 3 ¯ x = 2 α + u a n d 3 ¯ y = 2 β + v B . &nbsp 3 ¯ x = 2 α − u a n d 3 ¯ y = 2 β − v C . &nbsp 3 ¯ x = 2 α − u a n d 3 ¯ y = 2 β + v D . &nbsp 3 ¯ x = 2 α + u a n d 3 ¯ y = 2 β − v

If (α, β), (¯x , ¯y) and

Question

If ( $α, β$ ), $(¯ x, ¯ y) a n d (u, v)$ are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle.

Options:

A . 3

¯ x = 2 α + u a n d 3 ¯ y = 2 β + v

B . 3

¯ x = 2 α - u a n d 3 ¯ y = 2 β - v

C . 3

¯ x = 2 α - u a n d 3 ¯ y = 2 β + v

D . 3

¯ x = 2 α + u a n d 3 ¯ y = 2 β - v

Answer: Option C
:
C
We know that, the centroid of a triangle divides the segment joining the orthocentre and circumcentre internally in the ratio 2 : 1. Therefore,

¯ x = \frac{2 α + u}{2 + 1} a n d ¯ y = \frac{2 β + v}{2 + 1}

\Rightarrow 3 ¯ x = 2 α + u a n d ¯ y = 2 β + v

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