Pqrs is a parallelogram in which a, b, c and d are mid points of its sides respectively. prove that abcd is a parallelogram?

Step-by-step explanation:

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=  

2

1

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=  

2

1

AC.By mid-point theorem. But from (i) SR=  

2

1

AC therefore PQ=SR

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=  

2

1

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=  

2

1

AC.By mid-point theorem. But from (i) SR=  

2

1

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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