Question

prove that the lines joining the mid points of the opposite sides of a quadrilateral bisect each other​

Answer 1

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. 

Answer 2

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.