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.

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- Dec 6th 2008, 09:08 PMmuksProof
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. - Dec 7th 2008, 02:00 AMMoo
Hello,

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 $\displaystyle \angle PBM=90-\angle ABC=\angle CAB$.

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 ...... $\displaystyle \square$