Circle A has a radius of $2$ $2$ and a center of $(2, 7)$ $(2 ,7 )$ . Circle B has a radius of $1$ $1$ and a center of $(3, 1)$ $(3 ,1 )$ . If circle B is translated by $< 1, 3 >$ $<1 ,3 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2016-12-06T20:14:14

$1- no overlap$ $"1- no overlap"$
$2-minimum distance is 0.12$ $"2-minimum distance is "0.12$

$1-draw circles given$ $"1-draw circles given"$

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$2-Translate circle B by (1,3)$ $"2-Translate circle B by (1,3)"$

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$the new position of circle B is shown in figure below$ $"the new position of circle B is shown in figure below"$

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$now find distance between O_{1} and O_{T}$ $"now find distance between "O_1" and "O_T$

$O_{1} O_{T} = \sqrt{{(2 - 1)}^{2} + {(7 - 3)}^{2}}$ $O_1O_T=sqrt((2-1)^2+(7-3)^2)$

$O_{1} O_{T} = \sqrt{1^{2} + 4^{2}} = \sqrt{1 + 16} = \sqrt{17} = 4.12$ $O_1O_T=sqrt(1^2+4^2)=sqrt(1+16)=sqrt(17)=4.12$

$radius r_{1} = 2$ $"radius "r_1=2$

$radius r_{2} = 1$ $"radius "r_2=1$

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

$O_{1} O_{T} > r_{1} + r_{2}; no overlap$ $O_1O_T>r_1+r_2" ; no overlap"$

$minimum distance is AB line segment(colored blue)$ $"minimum distance is AB line segment(colored blue)"$

$A B = O_{1} O_{T} - r_{1} + r_{2}$ $AB=O_1O_T-r_1+r_2$

$A B = 4.12 - 3$ $AB=4.12-3$

$A B = 0.12$ $AB=0.12$

Circle A has a radius of 2 22 and a center of (2 ,7 )(2,7)(2 ,7 ). Circle B has a radius of 1 11 and a center of (3 ,1 )(3,1)(3 ,1 ). If circle B is translated by <1 ,3 ><1,3><1 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?