Circle A has a radius of 4  and a center of (8 ,5 ). Circle B has a radius of 3  and a center of (6 ,7 ). If circle B is translated by <3 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Apr 16, 2016

circles overlap

Explanation:

To determine wether the circles overlap or not , requires calculating the distance (d) between the centres and comparing 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 $\left(\begin{matrix}3 \\ 1\end{matrix}\right)$

centre of B(6 , 7) → (6 + 3 , 7 + 1) → (9 , 8)

To calculate the distance (d) 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$\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points }$

let $\left({x}_{1} , {y}_{1}\right) = \left(8 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(9 , 8\right)$

rArr d =sqrt((9-8)^2+(8-5)^2)=sqrt(1+9)=sqrt10 ≈ 3.16

now radius of A + radius of B = 4 + 3 = 7

Since sum of radii > d , then circles overlap.