Q. 18

ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.

Answer :

Given: AD = BC


To prove: PQRS is a rhombus


Theorem Used:


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.


Proof:



In ΔBAD,


P and S are the mid points of sides AB and BD,


So, by midpoint theorem,


PS||AD and PS = 1/2 AD … (i)


In ΔCAD,


R and Q are the mid points of CD and AC,


So, by midpoint theorem


QR||AD and QR = 1/2 AD … (ii)


Compare (i) and (ii)


PS||QR and PS = QR


Since one pair of opposite sides is equal as well as parallel then


PQRS is a parallelogram … (iii)


Now, In ΔABC, by midpoint theorem


PQ||BC and PQ = 1/2 BC … (iv)


And Ad = BC … (v)


Compare equations (i) (iv) and (v)


PS = PQ … (vi)


From (iii) and (vi)


Since PQRS is a parallelogram with PS = PQ then 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
Basic Proportionality TheoremBasic Proportionality TheoremBasic Proportionality Theorem42 mins
A Peep into Pythagoras TheoremA Peep into Pythagoras TheoremA Peep into Pythagoras Theorem43 mins
R.D Sharma | Solve Exercise -4.2 and 4.3R.D Sharma | Solve Exercise -4.2 and 4.3R.D Sharma | Solve Exercise -4.2 and 4.3FREE Class
NCERT | Strong Your Basics of TrianglesNCERT | Strong Your Basics of TrianglesNCERT | Strong Your Basics of Triangles39 mins
RD Sharma | Imp. Qs  From TrianglesRD Sharma | Imp. Qs  From TrianglesRD Sharma | Imp. Qs From Triangles41 mins
Champ Quiz | Thales TheoremChamp Quiz | Thales TheoremChamp Quiz | Thales Theorem49 mins
NCERT | Concept Based Questions on TrianglesNCERT | Concept Based Questions on TrianglesNCERT | Concept Based Questions on Triangles31 mins
Quiz | Criterion of Similarity of TriangleQuiz | Criterion of Similarity of TriangleQuiz | Criterion of Similarity of Triangle45 mins
How to Ace Maths in NTSE 2020?How to Ace Maths in NTSE 2020?How to Ace Maths in NTSE 2020?36 mins
Know About Important Proofs in TrianglesKnow About Important Proofs in TrianglesKnow About Important Proofs in Triangles33 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