the circle shown below is centered at the origin and contains the point (-4,-2). Which of the following is closest to the length of the diameter of the circle?

we know the circle has its center at the origin, 0,0, and the point -4,-2 is on the circle, is just the distance from the center to a point on it, thus

[tex]\bf \textit{distance between 2 points}\\ \quad \\ \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 0}}\quad ,&{{ 0}})\quad % (c,d) &({{ -4}}\quad ,&{{ -2}}) \end{array}\qquad % distance value d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2} \\\\\\ r=\sqrt{(-4-0)^2+(-2-0)^2}\implies r=\sqrt{(-4)^2+(-2)^2} \\\\\\ r=\sqrt{16+4}\implies r=\sqrt{20}\implies r=\sqrt{4\cdot 5}\implies r=\sqrt{2^2\cdot 5} \\\\\\ r=2\sqrt{5}[/tex]