Question

find the midpoint and distance given the two endpoints 13. (12,15) and (-8, -22) 14. (-3,5) and (14, 28)

Explanation:

Step1: Midpoint formula: $M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

For (12,15) & (-8,-22): $x_m=\frac{12+(-8)}{2}=2$, $y_m=\frac{15+(-22)}{2}=-\frac{7}{2}$

Step2: Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$d=\sqrt{(-8-12)^2+(-22-15)^2}=\sqrt{(-20)^2+(-37)^2}=\sqrt{400+1369}=\sqrt{1769}$

Step3: Midpoint for (-3,5) & (14,28): $x_m=\frac{-3+14}{2}=\frac{11}{2}$, $y_m=\frac{5+28}{2}=\frac{33}{2}$

Step4: Distance for (-3,5) & (14,28): $d=\sqrt{(14-(-3))^2+(28-5)^2}=\sqrt{17^2+23^2}=\sqrt{289+529}=\sqrt{818}$

Answer:

1. Midpoint: $(2, -\frac{7}{2})$, Distance: $\sqrt{1769}$
2. Midpoint: $(\frac{11}{2}, \frac{33}{2})$, Distance: $\sqrt{818}$