Show that the line joining the middle points of the consecutive sides of a quadrilateral is a parallelogram.

Solution

Let P, Q, R, S be the mid-points of the sides AB, BC, CD, DA respectively of quadrilateral ABCD.

therefore space space space PQ with rightwards arrow on top space equals space PB with rightwards arrow on top space plus space BQ with rightwards arrow on top space equals space 1 half space AB with rightwards arrow on top space plus space 1 half BC with rightwards arrow on top space equals space 1 half open parentheses AB with rightwards arrow on top plus BC with rightwards arrow on top close parentheses

therefore space space space space PQ with rightwards arrow on top space equals space 1 half AC with rightwards arrow on top

...(1)
Again,

SR with rightwards arrow on top space equals SD with rightwards arrow on top space plus space DR with rightwards arrow on top space equals space 1 half AD with rightwards arrow on top space plus space 1 half DC with rightwards arrow on top space equals space 1 half open parentheses AD with rightwards arrow on top space plus space DC with rightwards arrow on top close parentheses

therefore space space space space space space space space space SR with rightwards arrow on top space equals space 1 half space AC with rightwards arrow on top

...(2)
From (1) and (2), we get

PQ with rightwards arrow on top space equals space SR with rightwards arrow on top

∴ sides PQ and SR are equal and parallel.
Similarly sides PS and QR are equal and parallel.
∴ PQRS is a parallelogram.

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