Sunday, December 27, 2015

Isosceles Trapeziums are Con-Cyclic

Problem : We need to prove that any isosceles trapezium is con-cyclic.






Solution :Here’s an isosceles trapezium with sides PQ and RS are parallel and PS = QR. A quadrilateral is cyclic if it’s opposite angles are supplementary (i.e. they add up to 180˚). So , we need to prove that QPS + QRS = 180˚ and PSR + PQR = 180˚.

Let us construct two perpendiculars, PU and QT, from point P and Q to segment RS.

Now, in ΔPSU and ΔQRT
PUS = QTR        by construction - both are right angles
PS = QR           Given trapezium is isosceles
PU = QT           perpendicular distance between two parallel lines
Thus, ΔPSU and ΔQRT are congruent by Right angle-Hypotenuse-Side (RHS) congruency rule
Now angles , PSU = QRT        Corresponding Parts of Congruent Triangles (CPCT)
Therefore ,angle PSR and angle QRS are equal. -------- (1)
Also, angle RQT is equal to angle SPU by CPCT. Adding right angles UPQ and TQP to the above angles, we get
RQT + TQP = SPU + UPQ
Thus, angle RQP is equal to SPQ -------- (2)
Adding equations 1 and 2 we get the following relations for angles
PSR + RQP = QRS + SPQ --------(3)
Since the sum of all the angles in a quadrilateral is 360˚,from equation (3)
PSR + RQP + QRS + SPQ = 360˚
2 (PSR + RQP) = 2 (QRS + QPS) = 360˚
PSR + RQP =QRS + QPS = 180˚
Since the opposite angles are supplementary, it can be concluded that an isosceles trapezium is a cyclic quadrilateral.

6 comments:

  1. Its really helpfull.., thanks for sharing this.

    Papa jobs Providing latest information about Current affairs, Bank Recruitment, Government jobs, Bank jobs, IT jobs., check this link Job Notifications i hope this is very use full to you and all the very best Guys.

    ReplyDelete
  2. Its really helpfull.., thanks for sharing this.

    Papa jobs Providing latest information about Current affairs, Bank Recruitment, Government jobs, Bank jobs, IT jobs., check this link Job Notifications i hope this is very use full to you and all the very best Guys.

    ReplyDelete
  3. Nice blog and your all presenting information are very great and it's really good well done.

    Dissertation Writing Service UK

    ReplyDelete
  4. Hey keep posting such good and meaningful articles.

    ReplyDelete
  5. This is the precise weblog for anybody who needs to seek out out about this topic. You notice so much its almost arduous to argue with you. You positively put a brand new spin on a subject that's been written about for years. Nice stuff, simply nice!

    ReplyDelete