Circle A has a radius of $5$ $5$ and a center of $(2, 7)$ $(2 ,7 )$ . Circle B has a radius of $4$ $4$ and a center of $(7, 3)$ $(7 ,3 )$ . If circle B is translated by $< - 1, 2 >$ $<-1 ,2 >$ , 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 $5$ $5$ and a center of $(2, 7)$ $(2 ,7 )$ . Circle B has a radius of $4$ $4$ and a center of $(7, 3)$ $(7 ,3 )$ . If circle B is translated by $< - 1, 2 >$ $<-1 ,2 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2018-01-21T09:55:01

$circles overlap$ $"circles overlap"$

Explanation:

What we have to do here is $compare$ $color(blue)"compare"$ the distance (d) between the centres of the circles to the $sum of the radii$ $color(blue)"sum of the radii"$

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

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

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

$under a translation < - 1, 2 >$ $"under a translation "<-1,2>$

$(7, 3) \to (7 - 1, 3 + 2) \to (6, 5) \leftarrow new centre of B$ $(7,3)to(7-1,3+2)to(6,5)larrcolor(red)"new centre of B"$

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

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

$let (x_{1}, y_{1}) = (2, 7) and (x_{2}, y_{2}) = (6, 5)$ $"let "(x_1,y_1)=(2,7)" and "(x_2,y_2)=(6,5)$

$d = \sqrt{{(6 - 2)}^{2} + {(5 - 7)}^{2}} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47$ $d=sqrt((6-2)^2+(5-7)^2)=sqrt(16+4)=sqrt20~~4.47$

$sum of radii = 5 + 4 = 9$ $"sum of radii "=5+4=9$

$since sum of radii > d then circles overlap$ $"since sum of radii">d" then circles overlap"$
graph{((x-2)^2+(y-7)^2-25)((x-6)^2+(y-5)^2-16)=0 [-20, 20, -10, 10]}

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Circle A has a radius of $5$ $5$ and a center of $(2, 7)$ $(2 ,7 )$ . Circle B has a radius of $4$ $4$ and a center of $(7, 3)$ $(7 ,3 )$ . If circle B is translated by $< - 1, 2 >$ $<-1 ,2 >$ , 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 5 55 and a center of (2 ,7 )(2,7)(2 ,7 ). Circle B has a radius of 4 44 and a center of (7 ,3 )(7,3)(7 ,3 ). If circle B is translated by <-1 ,2 ><−1,2><-1 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer

Explanation:

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Circle A has a radius of $5$ $5$ and a center of $(2, 7)$ $(2 ,7 )$ . Circle B has a radius of $4$ $4$ and a center of $(7, 3)$ $(7 ,3 )$ . If circle B is translated by $< - 1, 2 >$ $<-1 ,2 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?