17.In a parallelogram ABCD, E and F are the mid points of sides AB and CD
respectively. Show that the line segments AF and EC trisect the diagonal BD.
Answers
Parallelogram :
A quadrilateral in
which both pairs of opposite sides are parallel is called a parallelogram.
A
quadrilateral is a parallelogram if
i)Its
opposite sides are equal
ii)
its opposite angles are equal
iii)
diagonals bisect each other
iv)
a pair of opposite sides is equal and parallel.
Converse
of mid point theorem:
The
line drawn through the midpoint of one side of a triangle, parallel to another
side bisect the third side.
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Given,
ABCD is a parallelogram. E and F are the mid-points of sides AB and CD
respectively.
To show: line segments AF and EC trisect the diagonal BD.
Proof,
ABCD is a parallelogram
Therefore, AB || CD
also, AE || FC
Now,
AB = CD
(Opposite sides of parallelogram ABCD)
1/2 AB = 1/2 CD
AE = FC
(E and F are
midpoints of side AB and CD)
Since a pair of opposite sides of a
quadrilateral AECF is equal and parallel.
so,AECF is a
parallelogram
Then, AF||EC,
AP||EQ & FP||CQ
(Since opposite sides of a
parallelogram are parallel)
Now,
In ΔDQC,
F is mid point of side DC & FP || CQ
(as AF || EC).
So,P is the
mid-point of DQ
(by Converse of mid-point theorem)
DP = PQ — (i)
Similarly,
In APB,
E is mid point of side AB and EQ || AP
(as AF || EC).
So,Qis the mid-point of PB
(by Converse of mid-point theorem)
PQ = QB —
(ii)
From equations (i) and (ii),
DP = PQ = BQ
Hence, the line segments AF and EC trisect the diagonal BD.
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Hope this will help you...
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