# Finding the centroid of the triangular region

## Homework Statement

Find the coordinates of the centroid G of the triangular region with vertices (0,0),(a,0),(b,c).

## Homework Equations

for the centroid x = (1 / area) * double integral ( x dA)
y = (1 / area) * double integral ( y dA)

## The Attempt at a Solution

Ok, what i did so far for this was try to get an equation for the lines on the left and right of the triangle. i got x = by/c and x = -y(a-b)/c + a (both were found using point slope)

Then I integrated with respects to x first and used the above equations as my limits of integration, and then integrated with respects to y and used 0 and c as limits of integration.

I want to know if that sounds like the right method of going into this problem. I get a really long mess of a's and b's for x and c canceled out. It feels like I'm missing something.

The whole point of the assignment was to try to prove that the three medians of a triangle intersect the centroid, but if I'm already going in the right direction, I'm sure I can figure the rest out.

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vsage
Without actually seeing the math you did, it looks like your idea of what to do is right.

centroid is given by coordinates $$x= \frac{x_1+x_2+x_3}{3} , \ y= \frac{y_1+y_2+y_3}{3}$$

Last edited:
$$G ( {x=x_1+x_2+x_3/3} , {y=y_1+y_2+y_3}/3 )$$

Can be proved by using midpoint theorem and the fact that medians bisect each other in ratio 2:1

HallsofIvy
Homework Helper
Of course, that's assuming what the OP was asked to show: that the medians intersect at the centroid.

Chumatha87, your basic idea is correct. Integrating with respect to x, you will want to divide the integral in two parts: 0 to b and b to a.

Don't forget to divide by the area which is (1/2)ac.

coordinates are G(a+b/3,c/3)

for points (0,0),(a,0),(b,c).

HallsofIvy