What is the centroid of a triangle with corners at #(4 , 1 )#, #(3 , 2 )#, and #(5 , 0 )#?

Question

What is the centroid of a triangle with corners at #(4  ,  1   )#, #(3  , 2   )#, and #(5 , 0   )#?

Noah Atkinson · Accepted Answer

A triangle is formed by three non-collinear points.

However, since the provided points are collinear, no triangle with these coordinates can exist, rendering the question meaningless.

I'll explain how I found out that the points provided are collinear if you have that question.

Let #A(x_1,y_1), B(x_2,y_2) and C(x_3,y_3)# be three points then the condition for these three points to be collinear is that

#(y_2-y_1)/(x_2-x_1)=(y_3-y_1)/(x_3-x_1)#

Here let #A=(4,1), B=(3,2) and C=(5,0)#

#implies (2-1)/(3-4)=(0-1)/(5-4)#

#implies 1/-1=-1/1#

#implies -1=-1#

The given points are collinear since the condition has been verified.

If the man who asked you the question insists that you locate the centroid, follow the steps listed below to find the centroid.

If #A(x_,y_1), B(x_2,y_2) and C(x_3,y_3)# are the three vertices of a triangle it's centroid is given by

#G=((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)#

Where #G# is the centroid

Here let #A=(4,1), B=(3,2) and C=(5,0)#

#implies G=((4+3+5)/3,(1+2+0)/3)#

#implies G=(12/3,3/3)#

#implies G=(4,1)#

Therefore, the centroid is #(4,1)#.

Violet Aldrich · Answer

undefined To find the centroid of a triangle, you take the average of the x-coordinates of the vertices to find the x-coordinate of the centroid, and the average of the y-coordinates of the vertices to find the y-coordinate of the centroid.

Given the vertices (4, 1), (3, 2), and (5, 0), the x-coordinate of the centroid is (4 + 3 + 5) / 3 = 4 and the y-coordinate of the centroid is (1 + 2 + 0) / 3 = 1.

Therefore, the centroid of the triangle is (4, 1).