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# Circle A has a radius of 2  and a center at (3 ,1 ). Circle B has a radius of 4  and a center at (8 ,3 ). If circle B is translated by <-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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#### Explanation:

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Jim G. Share
Mar 1, 2018

$\text{no overlap "~~0.71" units}$

#### Explanation:

$\text{What we have to do here is to "color(blue)"compare ""the}$
$\text{distance (d) between the centres with the "color(blue)"sum of radii}$

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

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

$\text{Before calculating d we require to find the centre of B}$
$\text{under the given translation}$

$\text{under the translation } < - 2 , 4 >$

$\left(8 , 3\right) \to \left(8 - 2 , 3 + 4\right) \to \left(6 , 7\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

$\text{to calculate d use the "color(blue)"distance formula}$

•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

$\text{let "(x_1,y_1)=(3,1)" and } \left({x}_{2} , {y}_{2}\right) = \left(6 , 7\right)$

$d = \sqrt{{\left(6 - 3\right)}^{2} + {\left(7 - 1\right)}^{2}} = \sqrt{9 + 36} = \sqrt{45} \approx 6.71$

$\text{sum of radii } = 2 + 4 = 6$

$\text{Since sum of radii"< d" then no overlap}$

$\text{min. distance "=d-" sum of radii}$

$\textcolor{w h i t e}{\text{min. distance }} = 6.71 - 6 = 0.71$
graph{((x-3)^2+(y-1)^2-4)((x-6)^2+(y-7)^2-16)=0 [-20, 20, -10, 10]}

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