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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.,CSOC=21

Where C-Coordinate of centroid

O-Coordinate of orthocenter

S--Coordinate of circumcenter


A

Only 1

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B

Only 2

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C

Both 1 & 2

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D

None of these

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Solution

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., length of CSlength of OC=12 where CcentroidOOrthocenterScircumcenter

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.


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