Question

ABCD is a quadrilateral and P, Q, R and S are the mia oints of the sides AB, BC, CD and DA
respectively. Show that the diagonals of PQRS bisect each other.​

Answer 1

ANSWER:-

Given:

ABCD is a quadrilateral & P,Q,R & S are the midpoints of the sides AB,BC,CD & DA respectively.

To prove:

Show that the diagonals of PQRS bisect each other.

Proof:

In ∆DAC,

S is the mid-point of DA & R is the mid points of DC.

SR||AC & SR = 1/2AC

[midpoints theorem]

Now,

In ∆BAC,

Therefore,

P is the mid-point of AB & Q is the mid-point of BC.

PQ||AC

PQ= 1/2AC...................(1)

[Mid points Theorem]

But from SR = 1/2AC........(2)

Therefore,

PQ = SR

So,

PQ||AC [from(2)]

SR ||AC [from(1)]

Therefore,

PQ||SR

Two lines parallel to the same line are parallel to each other.

&

PQ = SR [from(2)]

Therefore,

PQRS is a parallelogram.

⏺️A quadrilateral is a parallelogram if a pair of opposite sides are parallel & are of equal length.

Hope it helps ☺️