First slide

Cartesian plane

Question

If a triangle has its orthocentre at (1, 1) and circumcentre at $(\frac{3}{2}, \frac{3}{4})$ then the coordinates of the centroid of
the triangle are

Moderate

A

$(\frac{4}{3}, - \frac{5}{6})$
B

$(\frac{4}{3}, \frac{5}{6})$
C

$(- \frac{4}{3}, \frac{5}{6})$
D

$(- \frac{4}{3}, - \frac{5}{6})$

Solution

Since the centroid divides the line joining the orthocentre and circumcentre in the ratio 2 : 1 internally, therefore, if the centroid is (x, y), then
$x = \frac{2 . \frac{3}{2} + 1.1}{2 + 1} = \frac{4}{3} a n d y = \frac{2 . \frac{3}{4} + 1.1}{2 + 1} = \frac{5}{6}$
$∴$ Coordinates of centroid are $(\frac{4}{3}, \frac{5}{6})$

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