Question

The midpoints of an irregular quadrilateral ABCD are connected to form another quadrilateral inside ABCD. Complete the explanation of why the quadrilateral is a parallelogram.​

Answer 1

Suppose: M, N, P, Q are the midpoints of AB, BC, CD, AD respectively

=> MNPQ is the quadrilateral inside ABCD

connect B to D, ΔABD has : M is the midpoint of AB

Q is the midpoint of AD

=> MQ is the midpoint polygon of ΔABD

=> MQ // BD and MQ = 1/2.BD (1)

ΔBCD has: N is the midpoint of BC

P is the midpoint of DC

=> NP is the midpoint polygon of ΔBCD

=> NP // BD and NP = 1/2.BD (2)

from (1) and (2) => MQ // NP ( //BD)

MQ = NP (=1/2.BD)

=> MNPQ is a parallelogram.​

=> the quadrilateral inside ABCD is a parallelogram.​

Explanation: