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Math Help - Proof

  1. #1
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    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.
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  2. #2
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    Hello,
    Quote Originally Posted by muks View Post
    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.

    Proof-pn-ac-cb.jpg

    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
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