Question

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.​

Answer 1

Answer:

Let us consider the figure:



The figure ABCDABCD is a quadrilateral and it has midpoints J,K,L,MJ,K,L,M of sides AB,BC,CD,DAAB,BC,CD,DA respectively.

By using the midpoint theorem, in the triangle ABCABC, we can identify the midpoints JKJK is parallel to the third side ACAC.

So, JK=12ACJK=12AC.

Like the same, in the triangle CDACDA, the midpoints LMLM is parallel to the third side ACAC.

So, LM=12ACLM=12AC.

Therefore, the midpoints are parallel JK∥LMJK∥LM and equal in length JK=LMJK=LM.

Hence, it is proved that quadrilateral JKLMJKLM is a parallelogram.

Answer 2

➝ Given :-

• A quadrilateral ABCD in which P, Q, R, S are the midpoints of AB, BC, CD and DA respectively

➝ To prove :-

• PQRS is a parallelogram.

➝ Construction :-

• Join AC.

➝ Proof :-

→ In ∆ABC,

P and Q are the midpoints of AB and BC

respectively

PQ || AC. ....(i) [by midpoint theorem]

→ In ∆DAC,

S and R are the midpoints of AD and CD respectively.

SR || AC .... (ii) [by midpoint theorem]

From (i) and (ii), we get PQ || SR.

Similarly, by joining BD, we can prove that

PS || QR.

Hence, PQRS is a parallelogram.

Note :-

• refer the above attachment....