Question

A line segment has endpoints (-2,-3) and (4,4). Find the length of the segment and find the midpoint.

Answer 1

$\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ -2 &,& -3~) % (c,d) &&(~ 4 &,& 4~) \end{array} \\\\\\ d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{[4-(-2)]^2+[4-(-3)]^2}\implies d=\sqrt{(4+2)^2+(4+3)^2} \\\\\\ d=\sqrt{36+49}\implies \boxed{d=\sqrt{85}}\\\\ -------------------------------$

$\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ -2 &,& -3~) % (c,d) &&(~ 4 &,& 4~) \end{array}\qquad % coordinates of midpoint \left(\cfrac{ x_2 + x_1}{2}\quad ,\quad \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{4-2}{2}~~,~~\cfrac{4-3}{2} \right)\implies \left(\cfrac{2}{2}~~,~~\cfrac{1}{2} \right)\implies \boxed{\left(1~,~\frac{1}{2} \right)}$