Question

If $A(1,2,3),B(2,3,1)$ and $C(3,1,2$) are the vertices of the triangle. Find the coordinates of its orthocenter $\left(O\right)$ and In center $\left(I\right)$.

• A. O(1, 1, 1), I(1, 1, 1)
• B. O((3, 3, 3), I(3, 3, 2)
• C. O(2, 2, 3), I(2, 2, 2)
• D. O(4, 4, 4), I(1, 2, 3)

Answer 1

The correct option is C

O(2, 2, 3), I(2, 2, 2)

Let's find the length of each side of triangle so that

we can find the type of triangle which will help us to find

the orthocenter and incenter of the triangle.

$AB=\sqrt{(2-1{)}^2+(3-2{)}^2+(1-3{)}^2}$

$=\sqrt{1+1+4}=\sqrt{6}$
$BC=\sqrt{(3-2{)}^2+(1-3{)}^2+(2-1{)}^2}=\sqrt{1+4+1}=\sqrt{6}$

$CA=\sqrt{(3-1{)}^2+(1-2{)}^2+(2-3{)}^2}=\sqrt{4+1+1}=\sqrt{6}$

Here ,AB=BC=CA

$△ABC$ ia an equilateral triangle and in equilateral triangle centroid,

Incentre,orthocenter and circum center,all are same.

It will be easy to calculate the coordinate of centroid which is

same as coordinate of orthocenter and In center in equilateral triangle

Cooridnate centroid G$(\frac{1+2+3}{3},\frac{2+3+1}{3},\frac{3+1+2}{3})$

Here, coordinates of orthocenter is G(2,2,2) and coordinates of In center is I(2,2,2).

Option C is correct