Question

ABCD is a quadrilateral in which PQRS are mid points of the sides AB, BC, CD and DA.

AC is the diagonal. Show that: -

(i) SR ∥ AC and SR = 1/2AC

(ii) PQ = SR

(iii) PQRS is a parallelogram​

Answer 1

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

(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.