Question

class - 9 NCERT

ABCD is a quadrilateral in which pqr and s are midpoints of the sides ab BC CD and DA. AC is a diagonal . show that

SR II AC and SR= 1/2 AC

PQ=SR

pqrs is a parallelogram



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Answer 1

ANSWER:

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

(i) In △DAC , S is the mid point of DA and R is the mid point of DC.

Therefore, SR∥AC and SR= 1/2 AC

By mid-point theorem.

(ii) In △BAC , P is the mid point of AB and Q is the mid point of BC. Therefore,

PQ∥AC and PQ=1/2 AC

By mid-point theorem. But from

(i) SR= 1/2 AC therefore PQ=SR

(iii) PQ∥AC & SR∥AC therefore PQ∥SR and PQ=SR. Hence, a quadrilateral with opposite sides equal and paralle is a parallelogram. Therefore PQRS is a parallelogram.

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Answer 2

Answer:

(i) SR || AC and SR =

$\frac{1}{2} AC$

Considering ∆ACD

we observe that S and R are the mid points of side AD and DC respectively.

Hence,SR || AC and SR =

$\frac{1}{2}$

AC (As per the above theorem)...(1)

(ii) PQ = SR

Considering ∆ACB

We observe that P and Q are the mid points of side AB and BC respectively.

Hence, PQ || AC and PQ =

$\frac{1}{2}$

AC (As per above theorem…(2)

From (1) and (2) we can say,

PQ = SR

(iii) PQRS is a parallelogram

rom (i) and (ii) we can say that

PQ || AC and SR || AC

so, PQ || SR and PQ = SR

If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Hence, PQRS is a parallelogram.

Step-by-step explanation:

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