If the orthocenter of a triangle is (1, 1) an | vedclass

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Question

(A) The centroid $G$ divides the line segment joining the orthocenter $H$ and the circumcenter $O$ in the ratio $2:1$. Let $H = (1, 1)$ and $O = \left

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) The centroid $G$ divides the line segment joining the orthocenter $H$ and the circumcenter $O$ in the ratio $2:1$. Let $H = (1, 1)$ and $O = \left( \frac{3}{2}, \frac{3}{4} ight)$. The centroid $G(x, y)$ is given by the section formula: $x = \frac{2 	imes \frac{3}{2} + 1 	imes 1}{2 + 1} = \frac{3 + 1}{3} = \frac{4}{3}$ $y = \frac{2 	imes \frac{3}{4} + 1 	imes 1}{2 + 1} = \frac{\frac{3}{2} + 1}{3} = \frac{\frac{5}{2}}{3} = \frac{5}{6}$ Thus,the centroid is $\left( \frac{4}{3}, \frac{5}{6} ight)$.

If the orthocenter of a triangle is $(1, 1)$ and the circumcenter is $\left( \frac{3}{2}, \frac{3}{4} \right)$,find its centroid.

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