Question

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?

Answer 1

`"Circles do overlap."`

Explanation:

`"Strategy:"`

`"1-Find radius of Circle A":color(red)(r_A)`
`"2-Find radius of Circle B":color(red)(r_B)`
`"3-Find distance from Center of Circle A to Center of Circle B":color(red)(d)`
`4-"Compare "color(red)(d)" and "color(red)( r_A+r_B)"`

`"1)........................................................."`
`Area_A=pi*r_A^2" , "78 pi=pi*r_A^2" , "78cancel( pi)=cancel(pi)*r_A^2`

`r_A^2=78" , "r_A=sqrt(78)" , "r_A=8.83" unit"`

`"2).........................................................."`
`Area_B=pi*r_B^2" , "54pi=pi*r_B^2" , "54cancel(pi)=cancel(pi)*r_B^2`

`r_B^2=54" , "r_B=sqrt(54)" , "r_B=7.35" units"`

`"3)..........................................................."`
`d=sqrt((9-7)^2+(2-5)^2)`

`d=sqrt(2^2+(-3)^2)`

`d=sqrt(4+9)`

`d=sqrt(13)`

`d=3.61" units"`

`"4)................................................................."`

`"if d>"r_A+r_B" than no overlap"`
`"if d<"r_A+r_B" than overlap"`

`d=3.61 "units"`
`r_A+r_B=8.83+7.35=16.18`

`"Circles are overlap"`