I with this finding a centroid of a triangle

In summary, the conversation discusses the process of finding the centroid of a triangle and the equations used to do so, with a focus on the x-coordinate. The speaker is confused about how to find the y-coordinate and is seeking clarification.
  • #1
rock.freak667
Homework Helper
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Homework Statement



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.


Homework Equations



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

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


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]

Homework Statement



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.


Homework Equations



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

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


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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  • #2
I think everyone's latex is failing tonight. Must have to do with the server migration. Let's just try this again later.
 
  • #3
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!
 

1. What is a centroid of a triangle?

A centroid of a triangle is the point at which the three medians of a triangle intersect. A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side.

2. How do you find the centroid of a triangle?

To find the centroid of a triangle, you can use the formula (x1 + x2 + x3)/3 and (y1 + y2 + y3)/3, where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the triangle's vertices.

3. What is the significance of the centroid in a triangle?

The centroid is significant because it is the center of mass for a triangle. This means that if the triangle were cut out of a sheet of uniform material, the centroid would be the point where the triangle could balance on a needle.

4. Can a centroid be located outside of a triangle?

No, a centroid will always be located within the triangle. This is because the medians will always intersect within the triangle, and the centroid is the point of intersection.

5. How does the centroid relate to the other parts of a triangle?

The centroid divides each median into two segments, with the ratio of 2:1. This means that the distance from the centroid to the midpoint of each side is two-thirds of the length of the median. Additionally, the centroid is also the center of gravity for the triangle, meaning that it balances the triangle if it were suspended at that point.

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