Finding the combined centroid of two regions given the centroids of both regions

[theBEAST]

Homework Statement

The Attempt at a Solution

I tried to do this problem by taking the average of the y-bar centroid values but that gave me the wrong answer. I am only interesting in knowing why this method is incorrect.

Thanks!

 

[SteamKing]

What is the definition of centroid? Your method would be valid if the areas of the two regions were equal.

 

[HallsofIvy]

It can be shown that the centroid of the union of two (non-overlapping) regions is the weighted average of the two centroids, weighted by the areas of the regions. That is, if the two regions have centroid $(x_1, y_1)$ and $(x_2, y_2)$ and have areas $A_1$ and $A_2$, respectively, then the centroid of the combined regions is at
$$\left(\frac{A_1x_1+ A_2x_2}{A_1+ A_2}, \frac{A_1y_1+ A_2y_2}{A_1+ A_2}\right)$$

 

[haruspex]

I tried to do this problem by taking the average of the y-bar centroid values but that gave me the wrong answer.

As your grandmother should have taught you, don't take an average of averages. If an elephant and a flea get on opposite ends of a symmetric seesaw, will they balance? So where's their combined centroid?