Circle A has a center at (5 ,4 ) and an area of 16 pi. Circle B has a center at (12 ,8 ) and an area of 25 pi. Do the circles overlap? If not, what is the shortest distance between them?

Oct 14, 2017

The given circles intersect

Explanation:

Since the area of a circle is $\pi \cdot {r}^{2}$

we get

$16 \cancel{\pi} = \cancel{\pi} \cdot {r}^{2}$ (circle A)

then ${r}_{A} = 4$

and

$25 \cancel{\pi} = \cancel{\pi} \cdot {r}^{2}$ (circle B)

then

${r}_{B} = 5$

The distance between the centers is:

${C}_{A} {C}_{B} = \sqrt{{\left({x}_{B} - {x}_{A}\right)}^{2} + {\left({y}_{B} - {y}_{A}\right)}^{2}}$

$= \sqrt{{\left(12 - 5\right)}^{2} + {\left(8 - 4\right)}^{2}} = \sqrt{49 + 16} = \sqrt{65}$

that's less than the sum of the radiuses

${C}_{A} {C}_{B} = \sqrt{65} < 4 + 5$

We conclude that the circles intersect: