Circle A has a center at #(3 ,4 )# and an area of #16 pi#. Circle B has a center at #(8 ,1 )# and an area of #40 pi#. Do the circles overlap?

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Answer 1 · 2016-03-21T23:06:44

Yes. In fact the centre of circle A is contained in circle B.

The area of a circle is #pi r^2# where #r# is the radius.

So the radius of circle A is #sqrt(16) = 4# and the radius of circle B is #sqrt(40) = 2 sqrt(10) ~~ 6.325#

The distance between #(3, 4)# and #(8, 1)# is given by the distance formula as:

#sqrt((8-3)^2+(1-4)^2) = sqrt(5^2+3^2) = sqrt(25+9) = sqrt(34) ~~ 5.831#

So the point #(3, 4)# is actually contained in the circle B, being closer to #(8, 1)# than the radius of circle B.

graph{((x-3)^2+(y-4)^2-16)((x-3)^2+(y-4)^2-0.02)((x-8)^2+(y-1)^2-40)((x-8)^2+(y-1)^2-0.02) = 0 [-14.58, 25.42, -8.4, 11.6]}