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ABCD is a quadrilateral in which P,Q,R,S,are the mid point of the side AB,BC,CD,DA.AC is the diagonal show that pq=sr

ABCD is a quadrilateral  in which P,Q,R,S,are the mid point of the side AB,BC,CD,DA.AC is the diagonal show that pq=sr
 

Grade:6

2 Answers

Arun
25750 Points
4 years ago
Given:In quadrilateral ABCD
P,Q,R,S are mid point of sides AB,BC,CD and DA respectively
T.P: 1
2
3
In triangle ADC,
SR is a line segment joining the mid point of DA and DC respectively
:. SR||AC ( mid point therom)
In triangle BAC
PQ is a line segment joining the mid point of BA and BC respectively.
PQ||AC (mid point therom)
pQ = half of AC
From 1 &2
SR=PQ
SR ||PQ
 
Vikas TU
14149 Points
4 years ago
Dear student 
In ΔABC, P and Q are the mid-points of sides AB and BC respectively.
∴ PQ || AC and PQ = 1/2 AC (Using mid-point theorem) ... (1)
In ΔADC,
R and S are the mid-points of CD and AD respectively.
∴ RS || AC and RS = 1/2 AC (Using mid-point theorem) ... (2)
From equations (1) and (2), we obtain
PQ || RS and PQ = RS
Since in quadrilateral PQRS, one pair of opposite sides is equal and parallel to
each other, it is a parallelogram.
Let the diagonals of rhombus ABCD intersect each other at point O.
In quadrilateral OMQN,
MQ || ON ( PQ || AC)
QN || OM ( QR || BD)
Therefore, OMQN is a parallelogram.
⇒ ∠MQN = ∠NOM
⇒ ∠PQR = ∠NOM
However, ∠NOM = 90° (Diagonals of a rhombus are perpendicular to each other)
∴ ∠PQR = 90°
Clearly, PQRS is a parallelogram having one of its interior angles as 90º.
Hence, PQRS is a rectangle.

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