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

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Answer 1 · 2018-06-03T10:17:53

$no overlap \approx 2.82$ $"no overlap "~~2.82$

Explanation:

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

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

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

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

$under the translation < 2, 4 >$ $"under the translation "<2,4>$

$(3, 7) \to (3 + 2, 7 + 4) \to (5, 11) \leftarrow new centre of B$ $(3,7)to(3+2,7+4)to(5,11)larrcolor(red)"new centre of B"$

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

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

$let (x_{1}, y_{1}) = (1, 2) and (x_{2}, y_{2}) = (5, 11)$ $"let "(x_1,y_1)=(1,2)" and "(x_2,y_2)=(5,11)$

$d = \sqrt{{(5 - 1)}^{2} + {(11 - 2)}^{2}} = \sqrt{16 + 81} = \sqrt{117} \approx 10.82$ $d=sqrt((5-1)^2+(11-2)^2)=sqrt(16+81)=sqrt117~~10.82$

$sum of radii = 3 + 5 = 8$ $"sum of radii "=3+5=8$

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

$minimum distance = d - sum of radii$ $"minimum distance "=d-" sum of radii"$

$\times \times \times \times \times \times x = 10.82 - 8 = 2.82$ $color(white)(xxxxxxxxxxxxx)=10.82-8=2.82$
graph{((x-1)^2+(y-2)^2-9)((x-5)^2+(y-11)^2-25)=0 [-20, 20, -10, 10]}

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Circle A has a radius of $3$ $3$ and a center at $(1, 2)$ $(1 ,2 )$ . Circle B has a radius of $5$ $5$ and a center at $(3, 7)$ $(3 ,7 )$ . If circle B is translated by $< 2, 4 >$ $<2 ,4 >$ , 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 33 and a center at (1,2)(1 ,2 ). Circle B has a radius of 55 and a center at (3,7)(3 ,7 ). If circle B is translated by <2,4><2 ,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 $3$ $3$ and a center at $(1, 2)$ $(1 ,2 )$ . Circle B has a radius of $5$ $5$ and a center at $(3, 7)$ $(3 ,7 )$ . If circle B is translated by $< 2, 4 >$ $<2 ,4 >$ , does it overlap circle A? If not, what is the minimum distance between points on both circles?