Question

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

Answer 1

Given:-

ABCD is a quadrilateral P,Q,R & S are the midpoints of the respective sides.

To Prove:-

PR and QS bisect each other

Proof:-

By midpoint Theorem:-

 Join PQ,QR,RS,PS

Join diagonals AC and BD

• In ΔABC,

→P and Q r the midpoints of AB and BC respectively

→Therefore by midpoint theorem, PQ is parallel to AC and PQ=1/2AC

→In the same way prove that SR is parallel to AC and SR=1/2AC

→Therefore, since the opposite sides are equal and parallel PQRS is a parallelogram

→In a parallelogram diagonals bisect each other

[Hence Proved!!]

Answer 2

Answer:

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.

Hence proved.

Step-by-step explanation:

I HOPE THAT IT'S HELPFUL FOR YOU.