Question

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Answer 1

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

Let `bar"a", bar"b", bar"c", bar"d", bar"p", bar"q", bar"r",  "and"  bar"s"` be the position vectors of the points A, B, C, D, P, Q, R and S respectively.

Since P, Q, R and S are the midpoints of the sides AB, BC, CD and DA respectively,

`bar"p" = (bar"a" + bar"b")/2, bar"q" = (bar"b" + bar"c")/2, bar"r" = (bar"c" + bar"d")/2` and `bar"s" = (bar"d" + bar"a")/2`

∴ `bar"PQ" = bar"q" - bar"p"`

`= ((bar"b" + bar"c")/2) - ((bar"a" + bar"b")/2)`

`= 1/2 (bar"b" + bar"c" - bar"a" - bar"b") = 1/2(bar"c" - bar"a")`

`bar"SR" = bar"r" - bar"s"`

`= ((bar"c" + bar"d")/2) - ((bar"d" + bar"a")/2)`

`= 1/2(bar"c" + bar"d" - bar"d" - bar"a") = 1/2 (bar"c" - bar"a")`

∴ `bar"PQ" = bar"SR"`

∴ `bar"PQ" || bar"SR"`

Similarly `bar"QR" || bar"PS"`

∴ PQRS is a parallelogram.