Circle A has a radius of 1  and a center at (2 ,4 ). Circle B has a radius of 3  and a center at (6 ,5 ). If circle B is translated by <-3 ,4 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Apr 12, 2016

no overlap , d ≈ 1.099

Explanation:

A translation does not change the shape of a figure, only it's position.

Under a translation of $\left(\begin{matrix}- 3 \\ 4\end{matrix}\right)$

centre of B(6 , 5) → (6 -3 ,5 + 4) → (3 , 9)

Now, require to calculate the distance between the centres of A and B using 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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points }$

let $\left({x}_{1} , {y}_{1}\right) = \left(2 , 4\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 9\right)$

d= sqrt((3-2)^2 + (9-4)^2) = sqrt(1+25) = sqrt26 ≈ 5.099

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

Since sum of radii < distance between centres , no overlap.

and distance (d) between circles = 5.099 - 4 = 1.099