ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal. Show that : (i) `S R"\ "||"\ "A C` and `S R=1/2A C` (ii) `P Q"\ "="\ "S R` (iii) PQRS is a parallelogram

Explanation

The midpoint theorem states that the segment joining sides of a triangle at the midpoint of those sides is parallel to the third side and is half the length of the third side.

In` /_\DAC, S is the mid point of DA and R is the mid point of DC.` Therefore, SR // and SR = 1/2AC.By mid-point theorem.`

In` /_\ BAC,` P is the mid point of AB and Q is the mid point of BC. Therefore, `PQ // AC` and `PQ // AC` and `PQ = 1/2AC`. By mid point theorem. But from (1)` SR = 1/2AC` therefore` PQ = SR`

`PQ // AC & SR // AC` therefore `PQ // SR and PQ = SR`. Hence, a quadrilateral with opposite sides equal and paralle is a parallelogram. Atherefore PQRS is a parallelogram.