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prove using vectors: The quadrilateral o...

prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides a rectangle is a rhombus.

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Let ABCD be a rhombus and P, Q, R, S be the midpoints of AB, BC, CD and DA respectively. Then, PQRS is a ||gm [prove it].
Diagonals of a rhombus bisect each other at right angles.
`therefore angle EOF= 90^(@).`
Now, RQ||DB `rArr` RE||FO, SR||AC `rArr` FR||OE.
`therefore angle EOF = 90^(@)`.
`therefore OERF " is a ||gm. So, " angle FRE = angle EOF=90^(@). `
Thus, PQRS is a ||gm. with `angle R =90^(@).`
Hence, PQRS is a rectangle.
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