Question

If the orthocentre, circumcentre of a triangle are (-3,5,2) and (6,2,5) then the centroid

Answer 1

The centroid divides the distance between the orthocenter and the circumcenter in the ratio 2 is to 1. Using Section formula we will get the centroid's coordinates as (3,3,4)

Answer 2

Given:

The orthocentre, circumcentre of a triangle are (-3,5,2) and (6,2,5).

To Find:

Find the centroid of triangle.

Solution:

The orthocentre, circumcentre of a triangle are:

$\mathbf{(X_{1},Y_{1},Z_{1})= (-3,5,2)}$

$\mathbf{(X_{2},Y_{2},Z_{2})= (6,2,5)}$

Let centroid of triangle are:

Centroid of triangle = (X, Y, Z)

We know that centroid of triangle divide the orthocentre, circumcentre of a triangle in 2:1, so by formula:

$\mathbf{(X,Y,Z)=\left ( \dfrac{mX_{2}+nX_{1}}{m+n},\dfrac{mY_{2}+nY_{1}}{m+n},\dfrac{mZ_{2}+nZ_{1}}{m+n} \right )}$      ...1)

Where:

m = 2

n = 1

On putting respective value in equation 1):

$\mathbf{(X,Y,Z)=\left ( \dfrac{2\times 6+1\times (-3)}{2+1},\dfrac{2\times 2+1\times 5}{2+1},\dfrac{2\times 5+1\times 2}{2+1} \right )}$

On solving, we get:

(X, Y, Z) = (3, 3, 4)

If the orthocentre, circumcentre of a triangle are (-3,5,2) and (6,2,5) then the centroid of triangle is (3 , 3 , 4)