Circle A has a center at (2 ,2 ) and a radius of 2 . Circle B has a center at (5 ,9 ) and a radius of 3 . Do the circles overlap? If not what is the smallest distance between them?

Apr 13, 2018

color(crimson)("Since " bar(AB) > r + r', " two circles do not overlap"

color(blue)("Smallest distance between the circles " = bar(AB) - r - r'

color(brown)(=> 7.62 - 2 - 3 = 2.62

Explanation:

From the figure above,

color(purple)("The two circles A & B overlap each other if "

color(purple)(bar(AB) < r + r'

color(purple)("Touch each other if " bar(AB) = r + r'

color(purple)("Do not intersect if " bar(AB) > r + r'

$\text{Given : Circle A } \left(g , f\right) = \left(2 , 2\right) , r = 2$

$\text{Given : Circle B } \left(g ' , f '\right) = \left(5 , 9\right) , r ' = 3$

$\overline{A B} = \sqrt{{\left(g - g '\right)}^{2} + {\left(f - f '\right)}^{2}}$

$\overline{A B} = \sqrt{{\left(2 - 5\right)}^{2} + {\left(2 - 9\right)}^{2}} \approx 7.62$

$r + r ' = 2 + 3 = 5$

color(crimson)("Since " bar(AB) > r + r', " two circles do not overlap"

color(blue)("Smallest distance between the circles " = bar(AB) - r - r'

color(brown)(=> 7.62 - 2 - 3 = 2.62