Question

Prove that quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral is a parallelogram

Answer 1

To prove :

PS || QR and SR || PQ. i.e. Quadrilateral PQRS is a parallelogram


Proof:

Draw diagonal BD.
As PS is the midsegment of ▲ ABD, we can say that PS || BD.

As QR is the midsegment of ▲ BCD, we can say that QR || BD.

∵ PS || BD and QR || BD by transitivity, we can say that PS || QR.

(Now draw diagonal AC.)

As SR is the midsegment of ▲ ACD, we can say that SR || AC.

As PQ is the midsegment of ▲ ABC, we can say that PQ || AC.

∵ SR || AC and PQ || AC by transitivity, we can say that SR || PQ.

∵ PS || QR and SR || PQ, ∴ quadrilateral PQRS is a parallelogram (by definition).

Answer 2

RS II PQ and SP II RQ

Step-by-step explanation:

Take  a quadrilateral ABCD.

step 2 - draw diagonals and let the intersecting point be names as O.

step 3 - Join the midpoints of adjacent sides of quadrilateral with each other forming a small quadrilateral inside the quadrilateral ABCD.

step 4 - name the points as shown in the figure attached below.

step 5 - in triangle DAC, using mid-point theorem RS II AC and 2RS = AC ----(1)

step 6 - similarly, in triangle ABC, using mid-point theorem PQ II AC and 2PQ=AC ----(2)

From, (1) and (2),

RS II PQ.

step 7 - In triangle ABD and triangle BCD, by using mid-point theorem, SP II RQ.

Since, RS II PQ and SP II RQ, opposite sides of quadrilateral are parallel. Therefore, it is a parallelogram.

Hence, verified.