Dear teacher,
Please help me in solving the below questions . l am facing difficulty as l am stucking in the halfway of solution .
I would be highly thankful to you

Dear student,
Given: In a quadrilateral ABCD, P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively.

To prove: PQRS is a parallelogram.

Construction: Join AC.


Proof:

In $∆$ACD,

As, R and S are mid-points of DC and DA, respectively.

So, by using the Mid-point theorem − The segment connecting the mid-points of two sides of a triangle is parallel to the third side and is half the length of the third side, we get

RS$\parallel$AC and RS = $\frac{1}{2}$AC           .....(1)

Similarly, in $∆$ABC, by using the mid-point theorem, we get

PQ$\parallel$AC and PQ = $\frac{1}{2}$AC           .....(2)

Therefore, from (1) and (2), we get

PQ$\parallel$RS and PQ = RS

But this is a pair of opposite sides of the quadrilateral PQRS.

Therefore, PQRS is a parallelogram.
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