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# geometry

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BC is tangent to both a circle with center at A and a circle with center at D. The area of the circle with center at A is 225π and the area of the circle with center at D is 36π.

If BC=16, find the distance between the centers of the two circles. May 27, 2020

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BC is tangent to both a circle with center at A and a circle with center at D.
The area of the circle with center at A is 225p and the area of the circle with center at D is 36p.
If BC=16, find the distance between the centers of the two circles. Formula area of the circle: $$A=\pi r^2$$

$$\begin{array}{|rcll|} \hline A=\pi r^2 \\ \hline 225\pi &=& \pi R^2 \\ 225 &=& R^2 \\ \mathbf{15} &=& \mathbf{R} \\\\ 36\pi &=& \pi r^2 \\ 36 &=& r^2 \\ \mathbf{6} &=& \mathbf{r} \\ \hline \end{array}$$

Pythagoras' theorem:

$$\begin{array}{|rcll|} \hline AD^2 &=& 16^2+(R+r)^2 \\ AD^2 &=& 16^2+(15+6)^2 \\ AD^2 &=& 16^2+21^2 \\ AD^2 &=& 256+441 \\ AD^2 &=& 697 \\ AD &=& 26.4007575649 \\ \mathbf{AD} &\approx& \mathbf{26.4} \\ \hline \end{array}$$

The distance between the centers of the two circles is $$\approx \mathbf{26.4}$$ May 27, 2020