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Geometry Level pending

Given that the side of the square ABCD is 1, points P and Q are on AB and AD respectively, such that the perimeter of △ A P Q \bigtriangleup APQ is 2. Find ∠ P C Q \angle PCQ in degrees.


The answer is 45.

Vilakshan Gupta by Vilakshan Gupta , 19, India

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2 solutions

Vilakshan Gupta
Mar 19, 2017

Let a=AP , b=AQ , Extend AD to P' such that DP'=PB, then R t △ P B C ≅ R t △ P ′ D C Rt \bigtriangleup PBC \cong Rt \bigtriangleup P'DC by (SSS) , so CP'=PC and QP' = (1-b) + (1-a)=2-(a+b) = PQ , ∴ △ C Q P ≅ △ C Q P ′ \therefore \bigtriangleup CQP \cong \bigtriangleup CQP' , ∴ ∠ P C Q = ∠ P ′ C Q \therefore \angle PCQ = \angle P'CQ , ∴ ∠ P C B = ∠ P ′ C D \therefore \angle PCB = \angle P'CD , ∴ ∠ P C P ′ = ∠ D C B = 9 0 ∘ \therefore \angle PCP' = \angle DCB =90^{\circ} ∴ ∠ P C Q = 1 2 . 90 = 4 5 ∘ \therefore \angle PCQ = \frac{1}{2}.90 = 45^{\circ}

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Ahmad Saad
Mar 19, 2017

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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