Question

A circle is centered at R(2, 4). The point L(0,8) is on the circle.
Where does the point Q(6, 7) lie?

Answer 1

well, since we know the point L is on the circle, that means that the segment RL is really the length of the radius of that circle, hmmm let's see what that might be

$~~~~~~~~~~~~\textit{distance between 2 points} \\\\ R(\stackrel{x_1}{2}~,~\stackrel{y_1}{4})\qquad L(\stackrel{x_2}{0}~,~\stackrel{y_2}{8})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ RL=\sqrt{[0 - 2]^2 + [8 - 4]^2}\implies RL=\sqrt{(-2)^2+4^2}\implies \boxed{RL=\sqrt{20}}$

since RL is its radius, anything larger than that is outside the circle, and anything smaller than that is inside it.  Let's check for the distance RQ.

$~~~~~~~~~~~~\textit{distance between 2 points} \\\\ R(\stackrel{x_1}{2}~,~\stackrel{y_1}{4})\qquad Q(\stackrel{x_2}{6}~,~\stackrel{y_2}{7})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ RQ=\sqrt{[6 - 2]^2 + [7 - 4]^2}\implies RQ=\sqrt{4^2+3^2}\implies RQ=\sqrt{25}$

needless to say RQ is larger than RL.