problem based on pythagoras theorem

Question

In triangle PQR angle P= 90, PQ=PR. PM is an altitude from P, S is any point on QR such that M is between Q and S,  S is betweenM and R.  prove that : QS square + SR square = 2PS square.

Answer 1

This one is about setting it up and then doing a bunch of algebra. I'll do my best to get you started and leave the rest to you.

 

What we know:

 

QM=MR=PM and some perpendicular stuff

 

What we want to show:

2PS² = QS²+SR²

We will leave the left side alone for now. When doing a proof like this it is usually easier to simplify the complex side than it is to expand the simple side. To Continue...

        = (QM + MS)² + (RM - MS)²

Note that QM=RM so this can be written as

        = (QM + MS)² + (QM - MS)²

From here it is just an exercise of expanding and cancelling that I will leave up to you.

 

Good luck!