# Circle A has a center at (7 ,5 ) and an area of 78 pi. Circle B has a center at (9 ,2 ) and an area of 54 pi. Do the circles overlap?

Jan 31, 2017

$\text{Circles do overlap.}$

#### Explanation:

$\text{Strategy:}$

$\text{1-Find radius of Circle A} : \textcolor{red}{{r}_{A}}$
$\text{2-Find radius of Circle B} : \textcolor{red}{{r}_{B}}$
$\text{3-Find distance from Center of Circle A to Center of Circle B} : \textcolor{red}{d}$
$4 - \text{Compare "color(red)(d)" and "color(red)( r_A+r_B)}$

$\text{1).........................................................}$
$A r e {a}_{A} = \pi \cdot {r}_{A}^{2} \text{ , "78 pi=pi*r_A^2" , } 78 \cancel{\pi} = \cancel{\pi} \cdot {r}_{A}^{2}$

${r}_{A}^{2} = 78 \text{ , "r_A=sqrt(78)" , "r_A=8.83" unit}$

$\text{2)..........................................................}$
$A r e {a}_{B} = \pi \cdot {r}_{B}^{2} \text{ , "54pi=pi*r_B^2" , } 54 \cancel{\pi} = \cancel{\pi} \cdot {r}_{B}^{2}$

${r}_{B}^{2} = 54 \text{ , "r_B=sqrt(54)" , "r_B=7.35" units}$

$\text{3)...........................................................}$
$d = \sqrt{{\left(9 - 7\right)}^{2} + {\left(2 - 5\right)}^{2}}$

$d = \sqrt{{2}^{2} + {\left(- 3\right)}^{2}}$

$d = \sqrt{4 + 9}$

$d = \sqrt{13}$

$d = 3.61 \text{ units}$

$\text{4).................................................................}$

$\text{if d>"r_A+r_B" than no overlap}$
$\text{if d<"r_A+r_B" than overlap}$

$d = 3.61 \text{units}$
${r}_{A} + {r}_{B} = 8.83 + 7.35 = 16.18$

$\text{Circles are overlap}$