Question

If $$(\alpha \, , \, \beta) \, , \, (\bar{x} \, , \,\bar{y})$$ and (p , q) are the co - ordinates of the circumcentre, centroid and orthocentre of a triangle, then $$3\bar{x} \, = \,2\alpha +p \, \, \, and \, \, \, 3\bar{y} \, = \, 2\beta \, + \, q$$

A
True
B
False

Solution

The correct option is A True Let, H,OandG be the orthocentre, circumcentre and centroidof any triangle.Then, these points are collinear.Further, G divides the line segment HO from H in the ratio $$2:1$$Here,$$H(p,q), O(\alpha, \beta)$$ and $$G(\bar x, \bar y)$$Then by the section formula:$$\implies \bar y=\dfrac{2\beta +q}{3}\implies3\bar y=2\beta+q$$ and $$\implies \bar x=\dfrac{2\alpha+p}{3}\implies 3\bar x=2\alpha +p$$ Mathematics

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