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I need help with this finding a centroid of a triangle

by rock.freak667
Tags: centroid, triangle
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rock.freak667
#1
Oct19-08, 10:06 PM
HW Helper
P: 6,202
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.
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Dick
#2
Oct19-08, 11:04 PM
Sci Advisor
HW Helper
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P: 25,228
I think everyone's latex is failing tonight. Must have to do with the server migration. Let's just try this again later.
HallsofIvy
#3
Oct20-08, 06:45 AM
Math
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PF Gold
P: 39,500
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!


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