# Determine coordinates $x_c$ and $y_c$

Answer: $x_c=18.2$ and $y_c=9.5$.

Who can solve with explanation?

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so it means that $x_c$ and $y_c$ are coordinates of center of small circle? – dato datuashvili Dec 8 '13 at 13:08
no it means Area of triangle+Area of rectangle-Area of circle with own midlle poiont of each other – Anzor Bukhrashvili Dec 8 '13 at 13:11

Given the distributive property of the centroid, calling $A_i, i=1,2,3$ the areas of the triangle, rectangle and circle, respectively, and $P_i=(x_i,y_i)$ the respective centroids, we have

\begin{align} x_c&=\frac{A_1x_1+A_2x_2-A_3x_3}{A_1+A_2-A_3}\\ &=\frac{a^2\cdot\frac{2}{3}a+4a^2\cdot2a-\pi r^2\cdot2a}{a^2+4a^2-\pi r^2} \end{align}

and similarly for $y_c$

\begin{align} y_c&=\frac{A_1y_1+A_2y_2-A_3y_3}{A_1+A_2-A_3}\\ &=\frac{a^2\cdot\frac{2}{3}a+4a^2\cdot a-\pi r^2\cdot a}{a^2+4a^2-\pi r^2} \end{align}

given the centroid of the triangle $(2a/3, 2a/3)$ and the common centroid of the rectangle and of the circle $(2a,a)$.

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for case of $y_c$,if the same values would be,then what is difference? – dato datuashvili Dec 8 '13 at 13:50
@datodatuashvili: the difference is that you use $y_1, y_2, y_3$ – enzotib Dec 8 '13 at 13:53
and first one will be,i mean numerator,could you update please full answer – dato datuashvili Dec 8 '13 at 13:54
could you tell me please where does $2a/3$ and $a/3$ comes from? – dato datuashvili Dec 8 '13 at 13:56
@datodatuashvili: are the coordinates $x_1, y_1$ of the centroid of the triangle, given by a known formula as the arithmetic mean value of the coordinates of the vertices $(x_a+x_b+x_c)/3$, $(y_a+y_b+y_c)/3$ – enzotib Dec 8 '13 at 13:59