All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
Quadrilateral Side Query (Posted on 2015-10-31) Difficulty: 3 of 5
PQRS is a quadrilateral such that :
∠SPQ = ∠PSR and ∠PQS = ∠QRS.

Find each of RS and SP in terms of PQ, QR, and QS.

.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution Comment 1 of 1

From triangle QRS let
   a = /RSQ, b = /QRS, c = /SQR
From triangle PQS let
   d = /PQS, e = /SPQ, f = /QSP
   a + b + c = 180 = d +    e    + f
                    = b + (a + f) + f
       ==> c = 2*f
Let T be the point on side RS such that QT bisects /SQR.
                  QR*RS
Therefore, RT = ---------  and 
                 QR + QS
           QT^2 = QR*QS*sqrt[1 - (RS/[QR + QS])^2]           (1)
Checking the angles we see that triangles RTQ and QPS
are similar. Therefore,
    RT     QT     QR
   ---- = ---- = ----                                      (2)
    QP     SP     QS
Combining (1) and (2) gives
   RS = QP*(QR + QS)/QS  and  
   SP = sqrt[ QS*(QS^2 - QP^2)/QR ]
QED


  Posted by Bractals on 2015-11-01 09:43:08
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information