# Circle A has a center at (5 ,3 ) and an area of 4 pi. Circle B has a center at (1 ,2 ) and an area of 16 pi. Do the circles overlap? If not, what is the shortest distance between them?

Jul 8, 2016

The circles overlap.

#### Explanation:

Since $\text{Area"_"circle} = \pi {r}^{2}$
$\textcolor{w h i t e}{\text{XXX}}$Circle A (with area $4 \pi$) has a radius ${r}_{A} = 2$
and
$\textcolor{w h i t e}{\text{XXX}}$Circle B (with area $16 \pi$) has a radius of ${r}_{B} = 4$

The length of the line segment from the center of A at $\left(5 , 3\right)$ to the center of B at $\left(1 , 2\right)$ can be calculated using the Pythagorean Theorem as
color(white)("XXX")d_"AB" = sqrt( (5-1)^2+(3-2)^2)=sqrt(17) (approximately $4.123$)

Circle A covers $2$ units along this line segment and
Circle B covers $4$ units along this line segment.

Since $2 + 4 > {d}_{\text{AB}}$
$\textcolor{w h i t e}{\text{XXX}}$the circles overlap (by $\left(2 + 4\right) - 4.123 = 1.877$ units)