A quadrilateral ABCD is drawn in which the mid- points of sides AB, BC, CD, and AD are P, Q, R and S respectively. If quadrilateral ABCD is a parallelogram, what can you say about quadrilateral PQRS?
A quadrilateral ABCD is drawn in which the mid- points of sides AB, BC, CD, and AD are P, Q, R and S respectively. If quadrilateral ABCD is a parallelogram, what can you say about quadrilateral PQRS?
The correct option is D Parallelogram
Join the diagonals BD and AC of quadrilateral ABCD.
In ΔABD , SP || BD and SP = 12 BD (mid point theorem) ------- I
In ΔBCD , QR || BD and QR = 12BD (mid point theorem) ------- II
In ΔACD , RS || AC and RS = 12 AC (mid point theorem) -------- III
In ΔABC , PQ || AC and PQ = 12 AC (mid point theorem) ---------IV
From I and II , SP || QR and SP = QR ---------------------------------------- V
From III and IV , RS || PQ and RS = PQ ---------------------------------------- VI
From V and VI , PQRS is a parallelogram since opposite sides are equal and parallel.
Parallelogram
Join the diagonals BD and AC of quadrilateral ABCD.
In ΔABD , SP || BD and SP = 12 BD (mid point theorem) ------- I
In ΔBCD , QR || BD and QR = 12BD (mid point theorem) ------- II
In ΔACD , RS || AC and RS = 12 AC (mid point theorem) -------- III
In ΔABC , PQ || AC and PQ = 12 AC (mid point theorem) ---------IV
From I and II , SP || QR and SP = QR ---------------------------------------- V
From III and IV , RS || PQ and RS = PQ ---------------------------------------- VI
From V and VI , PQRS is a parallelogram since opposite sides are equal and parallel.
