Q. 34.0( 330 Votes )

ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Answer :

Let us join AC and BD.

In ΔABC,

P and Q are the mid-points of AB and BC respectively


To Prove: PQRS is a rhombus.
Proof:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

PQ || AC and PQ = AC (Mid-point theorem) (1)


Similarly in ΔADC,

SR || AC and SR = AC (Mid-point theorem) (2)


Clearly,


PQ || SR and PQ = SR


Since in quadrilateral PQRS, one pair of opposite sides is equal and parallel to each other, it is a parallelogram


PS || QR and PS = QR (Opposite sides of parallelogram) (3)


In ΔBCD, Q and R are the mid-points of side BC and CD respectively.


QR || BD and QR = BD (Mid-point theorem) (4)


However, the diagonals of a rectangle are equal.


AC = BD (5)


By using equation (1), (2), (3), (4), and (5), we obtain


PQ = QR = SR = PS

Now, as all sides of the rhombus are equal.

Hence, PQRS is a rhombus

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Critical Thinking Problems on QuadrilateralsCritical Thinking Problems on QuadrilateralsCritical Thinking Problems on Quadrilaterals44 mins
Quiz | Properties of ParallelogramQuiz | Properties of ParallelogramQuiz | Properties of Parallelogram31 mins
Extras on QuadrilateralsExtras on QuadrilateralsExtras on Quadrilaterals40 mins
Smart Revision | QuadrilateralsSmart Revision | QuadrilateralsSmart Revision | Quadrilaterals43 mins
Quiz | Basics of QuadrilateralsQuiz | Basics of QuadrilateralsQuiz | Basics of Quadrilaterals36 mins
RD Sharma |  Extra Qs. of Cyclic QuadrilateralsRD Sharma |  Extra Qs. of Cyclic QuadrilateralsRD Sharma | Extra Qs. of Cyclic Quadrilaterals31 mins
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 :