Question

Show that the line segment joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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Answer 1

Given : line segment joining the mid-points of the opposite sides of a quadrilateral bisect each other.

To Find :  Prove

Solution:

Let say a Quadrilateral  ABCD

P , Q , R and S are mid points of   AB , CD , CD and AD respectively

PR and QS are joined

and to be Proved that they bisect each other

PQRS is a quadrilateral

PR and QS are diagonals

They will bisect if PQRS is a parallelogram

line joining the mid-point of two sides of a triangle is equal to half the length of the third side and parallel to 3rd side

Join AC and BC

in ΔABC   P and Q are mid points of AB and BC

Hence PQ = AC/2  , PQ || AC

in ΔADC   R and S  are mid points of AB and AD

Hence RS = AC/2  , RS || AC

PQ = RS  = AC/2    and PQ || RS

Similarly

QR = PS = BD/2 and QR || PS

Hence PQRS is a parallelogram and PR and QS will bisect

Hence proved line segment joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Learn More:

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Answer 2

Answer:

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Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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Solution

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In △ADC,S is the mid-point of AD and R is the mid-point of CD

In △ABC,P is the mid-point of AB and Q is the mid-point of BC

Line segments joining the mid-points of two sides of a triangle is parallel to the third side and is half of of it.

∴SR∥AC and SR=

2

1

AC ....(1)

∴PQ∥AC and PQ=

2

1

AC ....(2)

From (1) and (2)

⇒PQ=SR and PQ∥SR

So,In PQRS,

one pair of opposite sides is parallel and equal.

Hence, PQRS is a parallelogram.

PR and SQ are diagonals of parallelogram PQRS

So,OP=OR and OQ=OS since diagonals of a parallelogram bisect each other