Circle A has a radius of $5$ $5$ and a center of $(5, 2)$ $(5 ,2 )$ . Circle B has a radius of $2$ $2$ and a center of $(4, 5)$ $(4 ,5 )$ . 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 $5$ $5$ and a center of $(5, 2)$ $(5 ,2 )$ . Circle B has a radius of $2$ $2$ and a center of $(4, 5)$ $(4 ,5 )$ . 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 · 2017-10-17T03:03:34

They do not overlap, minimum distance is $\sqrt{65} - 7 \approx 1.06$ $sqrt(65) - 7 ~~ 1.06$ .

Explanation:

First let's move the center of circle B.
Translation simply moves the circle's center, so we can just add the translation amount to the x and y values of the center.

$(4, 5) + (- 3, 4)$ $(4,5) + (-3,4)$
$(4 - 3, 5 + 4)$ $(4-3,5+4)$
$(1, 9)$ $(1,9)$

So now we just need to figure out if A and B overlap.
B has a radius of 2, and A has a radius 5.
$5 + 2 = 7$ $5+2 = 7$
So, they will overlap if A's center and B's center are less than 7 away.

Distance from A's center to B's center can be calculated using distance formula.

$d = \sqrt{{(5 - 1)}^{2} + {(2 - 9)}^{2}}$ $d = sqrt((5-1)^2 + (2-9)^2)$

$d = \sqrt{{(4)}^{2} + {(- 7)}^{2}}$ $d = sqrt((4)^2 + (-7)^2)$

$d = \sqrt{16 + 49}$ $d = sqrt(16 + 49)$

$d = \sqrt{65} \approx 8.06$ $d = sqrt(65) ~~8.06$

The distance is greater than $7$ $7$ , so they don't intersect.
If you do not have a calculator to calculate the square root, we know that $\sqrt{49} = 7$ $sqrt(49) = 7$ , so $\sqrt{65}$ $sqrt(65)$ must be greater than $7$ $7$ .

The minimum distance between the circles can be calculated by taking the difference of the distance and 7.
This is because a line drawn from one center to the other will be the length of both radii plus the minimum distance between the circles.

$\sqrt{65} - 7 \approx 1.06$ $sqrt(65) - 7 ~~ 1.06$

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Circle A has a radius of $5$ $5$ and a center of $(5, 2)$ $(5 ,2 )$ . Circle B has a radius of $2$ $2$ and a center of $(4, 5)$ $(4 ,5 )$ . 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 55 and a center of (5,2)(5 ,2 ). Circle B has a radius of 22 and a center of (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?

1 Answer

Explanation:

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Circle A has a radius of $5$ $5$ and a center of $(5, 2)$ $(5 ,2 )$ . Circle B has a radius of $2$ $2$ and a center of $(4, 5)$ $(4 ,5 )$ . 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?