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I'm having trouble understanding the motivation between finding the centroid of a quadrilateral.

Q: Find the centroid of a quadrilateral with vertices at (-8,12), (7,15) (13,-9), and (-2,-3).

I've solved the problem, using the procedure described @How can I construct the centroid of a quadrilateral? (by finding the intersection of the 4 centroids of the triangles formed by the 2 diagonals of a quadrilateral). However, I've been struggling to understand why this method gives the centroid of a quadrilateral (perhaps due to a theorem?).

** To clarify, I found the centroid which divides the quadrilateral into 4 equal areas

DarkRunner
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1 Answers1

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The key property, which can be derived by the definition, is that the centroid of a system of $2$ objects lies on the line which connects the centroid of each single object.

Then dividing the quadrilateral by a diagonal we find a first segment that contains the centroid of the quadrilateral and dividing by other diagonal we find a second segment that contains the centroid of the quadrilateral. Therefore the centroid coincides with the intersection of the two segments.

user
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  • Is there a reason why the centroid of a system can't lie outside said segment? – DarkRunner May 18 '18 at 01:51
  • Note that that $$\vec x_c=\frac{A_1}{A_1+A_2}\vec x_1+\frac{A_2}{A_1+A_2}\vec x_2=\frac{1}{1+A_2/A_1}\vec x_1+\frac{A_2/A_1}{1+A_2/A_1}\vec x_2=\frac{1}{1+a}\vec x_1+\frac{a}{1+a}\vec x_2=\frac{1}{1+a}\vec x_1+\frac{a}{1+a}\vec x_2=\frac{1+a-a}{1+a}\vec x_1+\frac{a}{1+a}\vec x_2=\vec x_1+\frac{a}{1+a}(\vec x_2-\vec x_1)$$ and for $a\in (0,\infty)$ it describe a line segment from the two centroids. – user May 18 '18 at 06:22