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Suppose that a quadrilateral has all sides of equal length and opposite sides are...

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haqc | eNoter

Posted October 2, 2013 at 4:57 AM via web

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Suppose that a quadrilateral has all sides of equal length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular. 

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aruv | High School Teacher | Valedictorian

Posted October 2, 2013 at 12:18 PM (Answer #1)

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Let `veca and vecb ` represented by sides of quadrilateral of equal length. i.e.

`|veca|=|vecb|=>|a|^2=|b|^2`

Also let define

`veca.veca=|a|^2` and `vecb.vecb=|b|^2`

The diagonals of the quadrilateral will be represented by `veca+vecb `

`and vecb-veca.`

Two vectors are perpendicular if their dot product is zero. Thus

`(veca+vecb).(vecb-veca)=veca.vecb+vecb.vecb-veca.veca-vecb.veca`

`since `

`veca.vecb=vecb.veca`

`therefore`

`(veca+vecb).(vecb-veca)=veca.vecb+|b|^2-|a|^2-veca.vecb`

`=0`

This implies diagonals are perpendicular.

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