Circle A has a radius of $4$ $4$ and a center at $(7, 2)$ $(7 ,2 )$ . Circle B has a radius of $3$ $3$ and a center at $(4, 5)$ $(4 ,5 )$ . If circle B is translated by $< - 3, 4 >$ $<-3 ,4 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Circle A has a radius of $4$ $4$ and a center at $(7, 2)$ $(7 ,2 )$ . Circle B has a radius of $3$ $3$ and a center at $(4, 5)$ $(4 ,5 )$ . If circle B is translated by $< - 3, 4 >$ $<-3 ,4 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2018-02-17T16:42:08

$no overlap, \approx 2.22$ $"no overlap ",~~2.22$

Explanation:

$what we have to do here is compare the distance (d)$ $"what we have to do here is "color(blue)"compare ""the distance (d)"$
$to the sum of the radii$ $"to the "color(blue)"sum of the radii"$

$∙ if sum of radii > d then circles overlap$ $• " if sum of radii">d" then circles overlap"$

$∙ if sum of radii < d then no overlap$ $• " if sum of radii "< d" then no overlap"$

$before calculating d we require to find the centre of B$ $"before calculating d we require to find the centre of B"$
$under the given translation$ $"under the given translation"$

$under a translation < - 3, 4 >$ $"under a translation "<-3,4>$

$(4, 5) \to (4 - 3, 5 + 4) \to (1, 9) \leftarrow new centre of B$ $(4,5)to(4-3,5+4)to(1,9)larrcolor(red)"new centre of B"$

$to calculate d use the distance formula$ $"to calculate d use the "color(blue)"distance formula"$

$∙ x d = \sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(y_{2} - y_{1})}_{2}^{2}}$ $•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$

$let (x_{1}, y_{1}) = (7, 2) and (x_{2}, y_{2}) = (1, 9)$ $"let "(x_1,y_1)=(7,2)" and "(x_2,y_2)=(1,9)$

$d = \sqrt{{(1 - 7)}^{2} + {(9 - 2)}^{2}} = \sqrt{36 + 49} \approx 9.22$ $d=sqrt((1-7)^2+(9-2)^2)=sqrt(36+49)~~9.22$

$sum of radii = 4 + 3 = 7$ $"sum of radii "=4+3=7$

$since sum of radii < d then no overlap$ $"since sum of radii"< d" then no overlap"$

$min. distance = d - sum of radii$ $"min. distance "=d-"sum of radii"$

$min. distance = 9.22 - 7 = 2.22$ $color(white)("min. distance ")=9.22-7=2.22$
graph{((x-7)^2+(y-2)^2-16)((x-1)^2+(y-9)^2-9)=0 [-40, 40, -20, 20]}

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Circle A has a radius of $4$ $4$ and a center at $(7, 2)$ $(7 ,2 )$ . Circle B has a radius of $3$ $3$ and a center at $(4, 5)$ $(4 ,5 )$ . If circle B is translated by $< - 3, 4 >$ $<-3 ,4 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Circle A has a radius of 44 and a center at (7,2)(7 ,2 ). Circle B has a radius of 33 and a center at (4,5)(4 ,5 ). If circle B is translated by <−3,4><-3 ,4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Explanation:

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Circle A has a radius of $4$ $4$ and a center at $(7, 2)$ $(7 ,2 )$ . Circle B has a radius of $3$ $3$ and a center at $(4, 5)$ $(4 ,5 )$ . If circle B is translated by $< - 3, 4 >$ $<-3 ,4 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?