# 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 9 pi. Do the circles overlap? If not, what is the shortest distance between them?

May 3, 2018

they don't over lap
the shortest distance between them is $\sqrt{65} - 7 \approx$ 1.062257748

#### Explanation:

area of Circle A = $\pi \cdot$ ${\left(R a\right)}^{2}$
16$\pi$ = $\pi \cdot R {a}^{2}$
$R {a}^{2}$ = 16
Ra = 4

area of Circle B = $\pi \cdot R {b}^{2}$
$9 \pi$ = $\pi \cdot R {b}^{2}$
9 = $R {b}^{2}$
Rb = 3

the distance between the center points is:
$\sqrt{{\left(\delta x\right)}^{2} + {\left(\delta y\right)}^{2}}$
$\delta x$ = Xa - Xb = 5 - 12 = -7
$\delta y$ = Ya - Yb = 4 - 8 = -4
$\sqrt{{\left(- 7\right)}^{2} + {\left(- 4\right)}^{2}}$ = $\sqrt{65}$ $\approx$ 8.062257748

$\because$ Ra + Rb < $\sqrt{{\left(\delta x\right)}^{2} + {\left(\delta y\right)}^{2}}$
$\therefore$ they didn't overlap.

the shortest distance between them is:
Gap = Distance between 2 center points - Ra - Rb
= $\sqrt{65}$ - 4 -3
$\approx$ 1.062257748

keep on learning~