Question

Which of the following statements are correct?

1. For triangle orthocenter,circumcenter and

centroid are collinear.

2. Centroid divides the line joining the circumcenter $\&$ orthocenter

in the ratio of $2:1$ i.e.,$\frac{CS}{OC}=\frac{2}{1}$

Where C-Coordinate of centroid

O-Coordinate of orthocenter

S--Coordinate of circumcenter

• A. Only 1
• B. Only 2
• C. Both 1 & 2
• D. None of these

Answer 1

The correct option is A

Only 1

Definition of Centroid: Centroid of a triangle is the point of concurrency of medians. The centroid G of a triangle ABC divides the median AD in the

ratio 2:1

Circumcenter: is the point of concurrency of perpendicular bisectors of the sides of the triangle.

Orthocenter: is the point of concurrency of altitudes of a triangle.

If we draw these three coordinates' orthocenter, centroid and circumcentre for a triangle, we find that orthocenter, centroid, circumcenter of a triangle lies in a straight line or these three points are collinear.

For equilateral triangle orthocenter, centroid &circumcenter is a same point.

Statement (1) is correct.

We also find that centroid of a triangle divides the line joining the circumcenter and orthocenter in the ratio 1:2.

i.e., $\frac{lengthofCS}{lengthofOC}=\frac{1}{2}where\left\{\begin{array}{c}C-centroid\\ O-Orthocenter\\ S-circumcenter\end{array}\right\}$

But in the statement it is given that centroid divides the line joining the circumcenter& orthocenter in the ratio of 2:1 which is NOT correct.

Centroid divides the line joining circumcenter and orthocenter in the ratio 1:2

Given statement is false.