Finding the centroid of the triangular region

[Chumatha87]

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.

 

[vsage]

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

 

[f(x)]

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

 

[.ultimate]

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

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

 

[HallsofIvy]

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.

 

[.ultimate]

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

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

 

[HallsofIvy]

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

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

All you are telling us is that you do not understand what the original question was.

 

[Chumatha87]

alright i got the answer as ( (a+b)/3, c/3 ) for the centroid using double integrals, it seems that I divided by the area in one part of the equation, but neglected to do it in another part, so things didn't cancel out the first time. Thanks for the help guys.