Circle A has a center at $(5, 4)$ $(5 ,4 )$ and a radius of $3$ $3$ . Circle B has a center at $(6, - 8)$ $(6 ,-8 )$ and a radius of $2$ $2$ . Do the circles overlap? If not, what is the smallest distance between them?

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Circle A has a center at $(5, 4)$ $(5 ,4 )$ and a radius of $3$ $3$ . Circle B has a center at $(6, - 8)$ $(6 ,-8 )$ and a radius of $2$ $2$ . Do the circles overlap? If not, what is the smallest distance between them?

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Answer 1 · 2016-02-09T20:10:11

They don't overlap. The smallest distance between them is $\sqrt{145} - 5 ≅ 7.042$ $sqrt(145)-5~=7.042$ .

Explanation:

The distance between centers is:
$d = \sqrt{{(6 - 5)}^{2} + {(- 8 - 4)}^{2}} = \sqrt{1 + 144} = \sqrt{145} ≅ 12.042$ $d=sqrt((6-5)^2+(-8-4)^2)=sqrt(1+144)=sqrt(145)~=12.042$

$r_{1} + r_{2} = 3 + 2 = 5$ $r_1+r_2=3+2=5$

$d > r_{1} + r_{2}$ $d>r_1+r_2$

Since the distance between centers is longer than the sum of the radii, they don't overlap and the smallest distance between these circles is
$= d - r_{1} - r_{2} = \sqrt{145} - 5 ≅ 7.042$ $=d-r_1-r_2=sqrt(145)-5~=7.042$

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Circle A has a center at $(5, 4)$ $(5 ,4 )$ and a radius of $3$ $3$ . Circle B has a center at $(6, - 8)$ $(6 ,-8 )$ and a radius of $2$ $2$ . Do the circles overlap? If not, what is the smallest distance between them?

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Explanation:

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