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.,CSOC=21
Where C-Coordinate of centroid
O-Coordinate of orthocenter
S--Coordinate of circumcenter
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., length of CSlength of OC=12 where ⎧⎪⎨⎪⎩C−centroidO−OrthocenterS−circumcenter⎫⎪⎬⎪⎭
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.