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Question about II-A Geometry 2020 2.29

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Question about II-A Geometry 2020 2.29
by Bober Yang - Wednesday, June 30, 2021, 8:08 PM

Hello, I am reviewing my missed questions, and I don't get the explanation for this problem.

I am not sure how you can rotate triangle PDC around angle C 90 degrees and find angle PCQ is exactly half of that 90 degrees.

Can someone help me out? 

Here is the question:  Let ABCDABCD be a unit square and let P,QP,Q be on sides AD¯¯¯¯¯¯¯¯,AB¯¯¯¯¯¯¯¯AD¯,AB¯ respectively such that APQ△APQ has perimeter 22. Find the measure of PCQ∠PCQ.

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Re: Question about II-A Geometry 2020 2.29
by Areteem Professor - Monday, July 5, 2021, 12:47 PM

In the diagram below we can see the rotated triangle $\triangle BCP'$. 

The key is noticing that since $ABCD$ is a unit square, $\triangle APQ$ has perimeter $2$, and $\triangle PDC \cong \triangle P'BC$,  $\triangle PQC \cong \triangle P'QC$:

  • $PC = P'C$
  • $QC = QC$
  • $QP = 2 - AQ - AP = (1-AQ) + (1- AP) = QB + DP = QB + BP' = QP'$

In particular, $\angle P'CQ = \angle PCQ$. Since we rotated $90^\circ$, $\angle PCP' = 90^\circ$, and $\angle PCP' = \angle PCQ + \angle P'CQ$, so $\angle PCQ = 90^\circ \div 2 = 45^\circ$.