IN a triangle LMN,the angle at L is a right-angle. p is the foot of the perpendicular from L to MN.proof that |MP|^2 - |PN|^2 = |MN|^2 - 2|LN|^2

Question

Its in new general mathematics textbook for senior secondary 1 for excercise 11c question 13c

Answer 1

In triangle LMP, LP^2=LM^2-MP^2 and in triangle LNP, LP^2=LN^2-NP^2,

so (1) LM^2-MP^2=LN^2-NP^2. Therefore, MP^2-NP^2=LM^2-LN^2.

Also, in triangle LMN, MN^2=LM^2+LN^2, and (2) LM^2=MN^2-LN^2.

Substituting for LM in (1): MN^2-LN^2-MP^2=LN^2-NP^2. And MN^2-2LN^2=MP^2-NP^2. This is the same as MP^2-PN^2=MN^2-2LN^2. QED.