Question

Find the centroid of the triangle whose mid points of side are d(5,2,-3) e(3,0,1p) and f(-1,1,-4)

Answer 1

The question is actually pretty easy to solve. It's just equation solving.

First, I will give you the general method and then the easy method which can be used for this specific case.

The centroid of any triangle can be found if you know its vertices. So you need to find the vertices. Consider the vertices as (x1, y1, z1) ; (x2, y2, z2) ; and (x3, y3, z3).

now the mid point is $(x_1 + x_2)/2$

so the equations are $(x_1 + x_2)/2= -1\\(y_1 + y_2)/2= 1\\(z_1 + z_2)/2 = -4$

and similarly the other.

Now solving the equations for x1, y1 and z1 you should get x1 = 1, x2= -3 and x3 = 9.

Then y1 = 3, y2 = -1 and y3 = 1.

And z1 = -8, z2 = 0 and z3 = 2.

Now you can just apply the formula for the centroid when you know the vertices. Which is : $(x_1 + x_2 + x_3)/3$

so if you take the coordinates of the centroid as (a,b,c)

a= (x1+x2+x3)/3 = (1-3+9)/3 = 7/3

b= (y1+y2+y3)/3 = (3-1+1)/3 = 1

c = (z1+z2+z3)/3 = (-8+0+2)/3 = -2

So your answer is (7/3, 1, -2)

Now to the simpler method. You don't necessarily need to use this but in a triangle, the centroid of a smaller triangle formed by joining the mid points of the sides is the same as the centroid of the original triangle. So you can just use the centroid formula for the coordinates of the mid points and the answer will be the same.

Thank you.