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?

1 Answer
Jan 31, 2017

#"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"#

enter image source here

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

#"Circles are overlap"#