Q. 234.0( 5 Votes )

P is the mi

Answer :

To Prove: DA = AR and CQ = QR

Given: P is the mid-point of the CD. A line through C parallel to PA intersects AB at Q and DA produced to R.


Concept Used:


Opposite sides of a parallelogram are parallel and equal.


ASA Theorem: If two angles and one side of a triangle is equal to two angles and one side of another triangle, then the triangles are congruent.


Diagram:



Explanation:


Now,


BC = AD [Opposite sides of a parallelogram]


BC || AD [Opposite sides of a parallelogram]


DC = AB and DC || AB [ Opposite sides of a parallelogram]


Now, it is given that P is mid-point of the CD.


DP = PC = 1/2 DC


Now,


QC || AP and PC || AQ,


As the opposite sides are equal and parallel.


APCQ is a parallelogram.


AQ = PC = 1/2 DC = 1/2 AB = BQ


Now, in ΔAQR and BQC,


AQ = BQ


AQR = BQC [Vertically opposite angles]


ARQ = BCQ [Alternate angles]


Therefore, ΔAQR and BQC are congruent by ASA theorem.


AR = BC [By C.P.C.T]


BC = DA


AR = DA


And, CQ = QR


Hence, Proved.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

The figure formedRS Aggarwal & V Aggarwal - Mathematics

A <imRS Aggarwal & V Aggarwal - Mathematics

In a parallelpgraRD Sharma - Mathematics