Question

PLEASE HELP ASAP!!!

Prove that the median to the hypotenuse of a right triangle is half the hypotenuse.

Plan: Since midpoints will be involved, use multiples of ______ to name coordinates for M and N.

Answer 1

Answer:

The proof is explained below

Step-by-step explanation:

Given right angled triangle and also the median we have to prove that  the median to the hypotenuse of a right triangle is half the hypotenuse i.e $OP=\frac{1}{2}MN.$

In ΔORN, by Pythagoras theorem

$ON^2=OR^2+NR^2$

In ΔORM, by Pythagoras theorem

$OM^2=OR^2+MR^2$

Adding above two equations, we get

$ON^2+OM^2=2OR^2+NR^2+MR^2$

⇒ $MN^2=2(OP^2-PR^2)+(NP-PR)^2+(MP+PR)^2$

⇒ $MN^2=2OP^2-2PR^2+NP^2+PR^2-2(NP)(PR)+MP^2+PR^2+2(MP)(PR)$

Since P is the mid-point i.e MP=PN

  $MN^2=2OP^2-2PR^2+\frac{MN^2}{4}+PR^2-\frac{MN^2}{4}+PR^2+2PR^2$

             =$2OP^2+\frac{MN^2}{2}$

⇒ $\frac{MN^2}{2}=2OP^2$

⇒ $MN^2=4OP^2$

⇒ MN=2OP ⇒ $OP=\frac{1}{2}MN$

Hence Proved.