If the orthocenter and centroid of a triangle | vedclass

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Question

(A) Let the orthocenter be $O(-3, 5)$,the centroid be $G(3, 3)$,and the circumcenter be $P(x, y)$.The centroid $G$ divides the line segment joining th

Vedclass · Accepted Answer

$(6, 2)$

Vedclass · Answer

(A) Let the orthocenter be $O(-3, 5)$,the centroid be $G(3, 3)$,and the circumcenter be $P(x, y)$.The centroid $G$ divides the line segment joining the orthocenter $O$ and the circumcenter $P$ in the ratio $2:1$.Using the section formula:$3 = \frac{2x + 1(-3)}{2+1} \implies 3 = \frac{2x-3}{3} \implies 9 = 2x-3 \implies 2x = 12 \implies x = 6$.$3 = \frac{2y + 1(5)}{2+1} \implies 3 = \frac{2y+5}{3} \implies 9 = 2y+5 \implies 2y = 4 \implies y = 2$.Thus,the circumcenter $P$ is $(6, 2)$.

If the orthocenter and centroid of a triangle are $(-3, 5)$ and $(3, 3)$ respectively,find its circumcenter.

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