Question

prove that figure formed by joining the midpoint of the pairs of consecutive side of a quadrilateral is parallelogram​

Answer 1

The midpoint theorem states that “The line segment in a triangle joining the midpoints of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

Let the quadrilateral be ABCD and P,Q,R,S be the midpoints of sides AB, BC, CD, and AD respectively

Join A and C

In triangle ABC, P and Q are the midpoints of sides AB and BC respectively.

By midpoint theorem we get PQ is parallel to AC and PQ= 1/2AC ………….(1)

similarly in triangle ADC we can prove RS is parallel to AC and

RS = 1/2AC …………………..(2)

From (1) and (2) we get , in quadrilateral PQRS

PQ ll RS and PQ = RS

We have a theorem related to parallelogram that “ If two opposite sides of a quadrilateral are parallel and congruent then it is a parallelogram”

Hence we can prove quadrilateral PQRS is a parallelogram.