Question

HELP!!!PLEASEEEE 20 POINTS!!!

Answer 1

The fomula pf a midpoint:

`\left((x_1+x_2)/(2);\ (y_1+y_2)/(2)\right)`

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

`x=(2a+0)/(2)=(2a)/(2)=a\\\\y=(0+2b)/(2)=(2b)/(2)=b`

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

The formula of a distance between two points:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

Calculate the distance
`MP=(1)/(2)MN`:

`|MP|=√((a-2a)^2+(b-0)^2)=√((-a)^2+b^2)=√(a^2+b^2)`

Calculate the distance OP [O(0, 0)]:

`\|OP|=√(a-0)^2+(b-0)^2)=√(a^2+b^2)`

`\boxed{OP=MP\to OP=(1)/(2)MN}`