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

Apr 24, 2016

cicles overlap

Explanation:

What we have to do here is to calculate the distance (d) between the centres of the circles and compare this with the sum of the radii.

• If sum of radii > d , then circles overlap.

• If sum of radii < d , then no overlap.

Under a translation of $\left(\begin{matrix}2 \\ - 3\end{matrix}\right)$

centre of B (4 ,7) → (4+2 , 7-3) → (6 , 4)

To calculate the distance between centres use the $\textcolor{b l u e}{\text{ distance formula }}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points }$

let $\left({x}_{1} , {y}_{1}\right) = \left(3 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(6 , 4\right)$

 d=sqrt((6-3)^2+(4-2)^2)=sqrt(9+4)=sqrt13 ≈ 3.61

radius of A + radius of B = 3 + 1 = 4

Since sum of radii > d , then circles overlap