I need help with this finding a centroid of a triangle

rock.freak667

1. The problem statement, all variables and given/known data

Well I just need to understand how to find the centroid of a triangle, I know it's 2/3 from the vertex, but I need to know how finding it is done.


2. Relevant equations

[tex]\overline{x}= \frac{\int x dA}{\int dA}[/tex]

[tex]\overline{y}=\frac{\int y dA}{\int dA}[/tex]


3. The attempt at a solution

Firstly I drew a triangle using the equation y=hx/b.



Then I considered a small rectangular element, whose height is and width is [itex]\delta x[/itex].

the area of this element is

The Area of this small element is [itex]\delta A=y \delta x[/itex]

Now the sum of all the infinitesmal areas is given by

[tex]dA=\sum_{x=0}^{x=b} y \delta x[/tex]

as [itex]\delta x \rightarrow 0[/itex]

[tex]\int dA=\int_0 ^b y dx[/tex]
1. The problem statement, all variables and given/known data

Well I just need to understand how to find the centroid of a triangle, I know it's 2/3 from the vertex, but I need to know how finding it is done.


2. Relevant equations

[tex]\overline{x}= \frac{\int x dA}{\int dA}[/tex]

[tex]\overline{y}=\frac{\int y dA}{\int dA}[/tex]


3. The attempt at a solution

Firstly I drew a triangle using the equation y=hx/b.



Then I considered a small rectangular element, whose height is and width is [itex]\delta x[/itex].

the area of this element is

The Area of this small element is [itex]\delta A=y \delta x[/itex]

Now the sum of all the infinitesmal areas is given by

[tex]dA=\sum_{x=0} ^{x=b} y \delta x[/tex]

as [itex]\delta x \rightarrow 0[/itex]

[tex]\int dA=\int_0 ^b y dx[/tex]

So the x-coordinate of the centroid is

[tex]\overline{x}=\frac{\int_0 ^b \frac{h}{b}x^2}{\int_0 ^b \frac{h}{b}x}[/tex]
So the x-coordinate of the centroid is

[tex]\overline{x}=\frac{\int_0 ^b \frac{h}{b}x^2}{\int_0 ^b \frac{h}{b}x}[/tex]

This is correct so far I assume, but what I do not understand is how to get the y-coordinate which should be the same answer.

EDIT: If my latex is wrong, I will type it over, so far the preview is only showing latex which I have typed for previous questions and not what I actually typed in the post, yet when I post the message it says my latex code is invalid.

Dick

I think everyone's latex is failing tonight. Must have to do with the server migration. Let's just try this again later.

HallsofIvy

The centroid of a triangle is simply the average of its three points. I searched through the code for your LaTex but I could find nowhere that you actually state what triangle you are talking about!