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 = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ RL=√([0 - 2]^2 + [8 - 4]^2)\implies RL=√((-2)^2+4^2)\implies \boxed{RL=√(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 = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ RQ=√([6 - 2]^2 + [7 - 4]^2)\implies RQ=√(4^2+3^2)\implies RQ=√(25)`

needless to say RQ is larger than RL.