Question

in quadrilateral PQRS the points A,B,C and D are the midpoints of side PQ,side QR,side RS and side SP respectively. Prove that quadrilateral ABCD is a parallelogram. Take a look at the diagram!​

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.

Answer 2

Step-by-step explanation:

Hint: draw the diagonals of the quadrilateral PR.

In triangle QPR,

using midpoint theorem for sides PQ with A as midpoint and QR with B as midpoint,

AB|| PR and AB=1/2(PR)

Similarly for triangle PSR

CD||PR||AB and CD= 1/2(PR)=AB

Since, 2 sides of ABCD are parallel and equal, Hence ABCD is a parallelogram.