Question

ABCD is a quadrilateral in which P Q R and S are mid point of the side ab BC CD and BA AC is diagonal show that:

1)SR II AC and SR=1/2 AC
2)PQ =SR
3)PQRS is a parallelogram

Answer 1

sorry couldn't do the first one but if you somehow able to prove AC is parallel to SR then you can find you can prove that AC is equal to half of AC

Answer 2

Ello There!

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Answer

Given  

ABCD is a quadrilateral.

P, Q, R, S are midpoints of the sides AB, BC, CD and DA

AC is a Diagonal.

To Prove

i) SR II AC and SR=1/2 AC  

ii) PQ =SR  

iii) PQRS is a parallelogram

Proof

In Δ DAC

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

∴ By Midpoint theorem,

SR ║ AC                                (Let this be 1)

SR = $\frac{1}{2} AC$ (Let this be 2)

Hence i) is proved!

In Δ CBA

P and Q are the midpoints of AB and BC respectively.

∴ By Midpoint theorem,

PQ ║ AC                                (Let this be 3)

PQ = $\frac{1}{2} AC$ (Let this be 4)

From 1 and 3

SR║PQ                (Let this be 5)

ii) From 2 and 4

PQ = SR               (Let this be 6)

Hence ii) Is proved!

Now,

From 5 and 6

PQRS is a parallelogram. (one pair of opposite sides are equal and parallel) (Equal side is PQ and SR parallel sides are PQ and SR)

Hence iii) is proved!

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Hope It Helps!