Geometry

I solved this question few days ago and found interesting. In a convex quadrilateral PQRSPQRS, PQ=RS,(3+1)QR=SPPQ=RS,(\sqrt {3}+1)QR=SP and ∠RSP−∠SPQ=30∘.\angle RSP- \angle SPQ=30^{\circ}.Find ∠PQR−∠QRS.\angle PQR-\angle QRS.This is also a quite easy problem, but let's see who give the shortest solution.Then I will give my solution.

I am observing since few weeks that the geometry problems in the combinatorics and geometry section are very low.This time only 2.They are also quite easy,.What do u think.?

Note by Kishan K
7 years, 10 months ago

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1 vote

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Comments

let PQ=a;QR=b;SR=c and PS=d then a=c and d=(3+1)bd=(\sqrt{3}+1)b we have PR2=c2+d2−2cdcos∠S=a2+b2−2abcos∠QPR^{2}=c^{2}+d^{2}-2cdcos\angle S=a^{2}+b^{2}-2abcos\angle Q FROM THIS b2−d2=2abcos∠Q−2cdcos∠Sb^{2}-d^{2}=2abcos\angle Q-2cdcos\angle S similarly b2−d2=2bccos∠R−2adcos∠Pb^{2}-d^{2}=2bccos\angle R-2adcos\angle P comparing we get 2abcos∠Q−2cdcos∠S=2bccos∠R−2adcos∠P2abcos\angle Q-2cdcos\angle S=2bccos\angle R-2adcos\angle P or simflifying it we get [putting the values a=c and d=(3+1)bd=(\sqrt{3}+1)b]=== cosQ−(3+1)cosS=cosR−(3+1)cosPcosQ-(\sqrt{3}+1)cosS=cosR-(\sqrt{3}+1)cosP OR cosQ−cosR=(3+1)(cosS−cosP)cosQ-cosR=(\sqrt{3}+1)(cosS-cosP) OR sinQ+R2sinQ−R2=(3+1)sinS+P2sinS−P2sin\frac{Q+R}{2}sin\frac{Q-R}{2}=(\sqrt{3}+1)sin\frac{S+P}{2}sin\frac{S-P}{2} we see sinQ+R2=sinS+P2sin\frac{Q+R}{2}=sin\frac{S+P}{2} putting this value in the above equation we get sinQ−R2=(3+1)sin302sin\frac{Q-R}{2}=(\sqrt{3}+1)sin\frac{30}{2}[as S-P=30]...... putting the value of sin15 we get sinQ−R2=12sin\frac{Q-R}{2}=\frac{1}{\sqrt{2}} so Q−R2=45\frac{Q-R}{2}=45 or Q-R=90. so we have done now. now you give me your solutin;kishan. i know this is not the shortest solution;nor the best!!!

krishan Chakraborty - 7 years, 6 months ago

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My solution is slightly shorter than yours.I would like you to do it.I would just give you a path to my solution.Try to get the area of the quadrilateral in 2 different ways.[PQS]+[QRS]=[PQR]+[PSR][PQS]+[QRS]=[PQR]+[PSR] and try to use the sine rule area formula.After few simplifications,you will get your answer.

Kishan k - 7 years, 6 months ago

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thanks!! i will be waiting for your next beautiful problems;kishan

krishan Chakraborty - 7 years, 6 months ago
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