Question

HELP!!!PLEASEEEE 20 POINTS!!!

Answer 1

The fomula pf a midpoint:

$\left(\dfrac{x_1+x_2}{2};\ \dfrac{y_1+y_2}{2}\right)$

N(2a, 0), M(0, 2b). Substitute:

$x=\dfrac{2a+0}{2}=\dfrac{2a}{2}=a\\\\y=\dfrac{0+2b}{2}=\dfrac{2b}{2}=b$

$\boxed{P(a,\ b)}$

The formula of a distance between two points:

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

Calculate the distance $MP=\dfrac{1}{2}MN$:

$|MP|=\sqrt{(a-2a)^2+(b-0)^2}=\sqrt{(-a)^2+b^2}=\sqrt{a^2+b^2}$

Calculate the distance OP  [O(0, 0)]:

$\|OP|=\sqrt{a-0)^2+(b-0)^2}=\sqrt{a^2+b^2}$

$\boxed{OP=MP\to OP=\dfrac{1}{2}MN}$