Question

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

Answer 1

ABCD is a quadrilateral with `vec"OA" = vec"a", vec"OB" = vec"b", vec"OC" = vec"c"` and `vec"OD" = vec"d"`

P = midpoint of AB ⇒ `vec"OP" = (vec"a" + vec"b")/2`

Q = midpoint of BC ⇒ `vec"OQ" = (vec"b" + vec"c")/2`

R = midpoint of CD ⇒ `vec"OR" = (vec"c" + vec"d")/2`

S = midpoint of DA ⇒ `vec"OS" = (vec"d" + vec"a")/2`

To prove: P Q R S is a parallelogram.

`vec"PQ" = vec"OQ" - vec"OP"`

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

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

= `(vec"c" - vec"a")/2`

⇒ `vec"PQ" = vec"SR"`   ........(1)

`vec"PS" = vec"OS" - vec"OP"`

= `(vec"d" + vec"a")/2 - (vec"a" + vec"b")/2`

= `(vec"d" - vec"b")/2`

`vec"QR" = vec"OR" - vec"OQ"`

= `(vec"c" + vec"d")/2 - (vec"c" + vec"b")/2`

= `(vec"d" - vec"b")/2`

⇒ `vec"PS" = vec"QR"`  ........(2)

In a quadrilateral when opposite sides are equal and parallel it is a parallelogram.

So, PQRS is a parallelogram, from (1) and (2).