1. ## Proof

1. In triangle ABC ,angle c=90 degree and ABPQ is a square on AB.If PN is perpendicular to AC from P, prove that PN=AC + CB.
Thank You.

2. Hello,
Originally Posted by muks
1. In triangle ABC ,angle c=90 degree and ABPQ is a square on AB.If PN is perpendicular to AC from P, prove that PN=AC + CB.
Thank You.
There are assumptions on the orientation of the figure, because there are situations where it doesn't work.

Okay, look at the sketch. M is the perpendicular to BC from P.

CMPN is a rectangle. So in order to prove that PN=AC+CB, it is sufficient and necessary to prove that BM=AC.

And this is easy if you consider triangles BMP and ABC.
• They both have right a right angle.
• Angle ABP is 90° since ABPQ is a square. Hence $\angle PBM=90-\angle ABC=\angle CAB$.

Therefore, the triangles BMP and ABC are similar.

But we know that measures BA and BP are equal since ABPQ is a square. Thus triangles BMP and ABC are congruent.

And we can conclude : BM=AC.

---------> PN=MC=BC+BM=BC+AC ...... $\square$