Question

The point (8,-4) lies on a circle. What is the length of the radius of this circle if the center is located at (5,-7)

Answer 1

Notice the picture, the radius is just the distance from the
center to a point on the circle, thus

$\bf \textit{distance between 2 points}\\ \quad \\\\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{5}}\quad ,&{{ -7}})\quad % (c,d) &({{ 8}}\quad ,&{{ -4}}) \end{array}\qquad % distance value \\\\\\ \begin{array}{llll} d = &\sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2}\\ &\qquad \uparrow \\ &radius \end{array}$

$\bf \begin{array}{llll} d = &\sqrt{({{ 8}}-{{ 5}})^2 + ({{ -4}}-{{(-7)}})^2}\\ &\qquad \uparrow \\ &radius \end{array} \\\\\\ d=\sqrt{(3)^2+(-4+7)^2}\implies d=\sqrt{3^2+3^2}\implies d=\sqrt{18} \\\\\\ d=\sqrt{9\cdot 2}\implies d=\sqrt{3^2\cdot 2}\implies d=3\sqrt{2}$