Question

to verify that the quadrilateral obtained by joining the mid- point of a quadrilateral is a parallelogram​

Answer 1

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To verify that the quadrilateral obtained by joining the mid- point of a quadrilateral is a parallelogram

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• Let ABCD be any rhombus and P,Q,R,S be the mid points of AB,BC,CD and DA respectively.
• Then we know that by the property of rhombus that AC∥PQ.

Observing △ACB and △PQB

∠ABC=∠PBQ(same angles)

∠BAC=∠BPQ(corresponding angles)

∠BCA=∠BQP(corresponding angles)

△ABC is similer to △PBQ

⇒ AC/PQ = BC /BQ

⇒PQ= 1/2 AC

Similarly taking △BCD and △RCQ

RQ= 1/2 BD

Then again △ADC and △SDR

SSR1/2 AC

And at last

SP= 1/2BD

∴PQ=SRandRQ=SP

Opposite sides are equal.

Now,

∠APS+∠SPQ+∠BPQ=180

⇒∠ABD+∠SPQ+∠BAC=180

⇒ 1/2 (∠ABC)+∠SPQ+ 1/2(∠BAD)=180

o

⇒∠SPQ+1/2(∠ABC+∠BAD)=180

By properties of rhombus

⇒∠SPQ+1/2 (180 )=180

⟹∠SPQ=90

Similarly, ∠PQR=∠QRS=∠RSP=90

Thus, all angles measure 90 and opposite sides are equal

Hence, the figure obtained is a rectangle which is a parellogram

Answer 2

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