# Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram. - Mathematics and Statistics

Sum

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

#### Solution

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.

Concept: Section Formula
Is there an error in this question or solution?