If the coordinates of the orthocenter and the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the orthocenter be $H(4, -1)$ and the centroid be $G(2, 1)$.Let the circumcenter be $O(x, y)$.We know that the centroid $G$ divides the line s

Vedclass · Accepted Answer

None of these

Vedclass · Answer

(D) Let the orthocenter be $H(4, -1)$ and the centroid be $G(2, 1)$.Let the circumcenter be $O(x, y)$.We know that the centroid $G$ divides the line segment joining the orthocenter $H$ and the circumcenter $O$ in the ratio $2:1$.Using the section formula,$G = \left( \frac{1 \cdot x_H + 2 \cdot x_O}{1+2}, \frac{1 \cdot y_H + 2 \cdot y_O}{1+2} \right)$.Substituting the values: $(2, 1) = \left( \frac{4 + 2x}{3}, \frac{-1 + 2y}{3} \right)$.Equating the $x$-coordinates: $2 = \frac{4 + 2x}{3} \implies 6 = 4 + 2x \implies 2x = 2 \implies x = 1$.Equating the $y$-coordinates: $1 = \frac{-1 + 2y}{3} \implies 3 = -1 + 2y \implies 2y = 4 \implies y = 2$.Thus,the coordinates of the circumcenter are $(1, 2)$.Since $(1, 2)$ is not among the given options,the correct answer is $D$.

If the coordinates of the orthocenter and the centroid of a triangle are $(4, -1)$ and $(2, 1)$ respectively,then what are the coordinates of the circumcenter of the triangle?

Similar Questions

If $A(2, 2)$,$B(-4, -4)$,and $C(5, -8)$ are the vertices of a triangle,then the length of the median passing through vertex $C$ is:

The coordinates of the orthocentre of the triangle bounded by the lines $4x - 7y + 10 = 0$,$x + y = 5$,and $7x + 4y = 15$ are:

Two vertices of a triangle are $(4, -3)$ and $(-2, 5)$. If the orthocentre of the triangle is at $(1, 2)$,then the third vertex is

If $\left( \frac{3}{2}, 0 \right)$,$\left( \frac{3}{2}, 6 \right)$,and $(-1, 6)$ are the midpoints of the sides of a triangle,find the incenter of the triangle.

$P(a, b)$ is the mid-point of a line segment between the axes. Show that the equation of the line is $\frac{x}{a} + \frac{y}{b} = 2$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module