Circle A has a center at #(5 ,3 )# and an area of #4 pi#. Circle B has a center at #(1 ,2 )# and an area of #16 pi#. Do the circles overlap? If not, what is the shortest distance between them?

1 Answer
Jul 8, 2016

The circles overlap.

Explanation:

Since #"Area"_"circle"=pir^2#
#color(white)("XXX")#Circle A (with area #4pi#) has a radius #r_A=2#
and
#color(white)("XXX")#Circle B (with area #16pi#) has a radius of #r_B=4#

The length of the line segment from the center of A at #(5,3)# to the center of B at #(1,2)# can be calculated using the Pythagorean Theorem as
#color(white)("XXX")d_"AB" = sqrt( (5-1)^2+(3-2)^2)=sqrt(17)# (approximately #4.123#)

Circle A covers #2# units along this line segment and
Circle B covers #4# units along this line segment.

Since #2+4 > d_"AB"#
#color(white)("XXX")#the circles overlap (by #(2+4)-4.123 = 1.877# units)