Mathematics
In the given figure, A, B, C and D are mid-points of PQ, QR, RS and PS respectively. E, F, G and H are mid-points of AB, BC, CD and AD respectively. Which type of quadrilaterals are ABCD and EFGH?
Mid-point Theorem
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Answer
Join QS, PR, AC and BD.
By mid-point theorem,
The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it.
In △QRP,
A and B are midpoints of PQ and QR respectively.
∴ AB || PR and AB = PR (By midpoint theorem) …..(1)
Similarly in △PRS,
D and C are midpoints of PS and RS respectively.
∴ DC || PR and DC = PR (By midpoint theorem) …..(2)
In △PQS,
D and A are midpoints of PS and PQ respectively.
∴ DA || QS and DA = QS (By midpoint theorem) …..(3)
Similarly in △QRS,
B and C are midpoints of QR and SR respectively.
∴ BC || QS and BC = QS (By midpoint theorem) …..(4)
From (1) and (2) we get,
AB = DC and AB || DC
From (3) and (4) we get,
DA = BC and DA || BC
Since, opposite sides are parallel and equal.
Thus, ABCD is a parallelogram.
In △ABC,
E and F are midpoints of AB and BC respectively.
∴ EF || AC and EF = AC (By midpoint theorem) …..(5)
Similarly in △ADC,
H and G are midpoints of AD and CD respectively.
∴ GH || AC and GH = AC (By midpoint theorem) …..(6)
In △ABD,
H and E are midpoints of AD and AB respectively.
∴ EH || BD and EH = BD (By midpoint theorem) …..(7)
Similarly in △BCD,
F and G are midpoints of BC and CD respectively.
∴ FG || BD and FG = BD (By midpoint theorem) …..(8)
From (5) and (6) we get,
EF = GH AND EF || GH
From (7) and (8) we get,
EH = FG and EH || FG
Since, opposite sides are parallel and equal.
Thus, EFGH is a parallelogram.
Hence, ABCD and EFGH are parallelograms.
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