How Do You Prove This?

Question

http://i107.photobcket.com/albums/m295/halfxhearts/math.png

It's all in the picture. You don't have to write out all the steps, just tell me how to find if a line is a perpendicular bisector.

***WHOOPS, THIS IS THE CORRECT LINK http://i107.photobucket.com/albums/m295/halfxhearts/math.png

Answer 1

To avoid any confusion I have called the point of intersection of MO and NP, Q

ΔMNP is isosceles (MN = MP)

In Δ's MNQ, MPQ
MN = MP (Data)
MQ is common
Therefore ΔMNQ ≡ ΔMPQ (SAS Property)

Therefore NQ = PQ (Corresponding sides of congruent triangles are equal)

And BUT Therefore
So MO is the perpendicular bisector of PN

Answer 2

By SAS, we have that triangles MNQ and MPQ are congruent. This gives NQ=PQ, therefore MO bisects NP. This also gives that the angles MQP and MQN are the same. Since their sum is 180º. these two facts give that each angle is 90º, and in particular, MO is perpendicular to NP.