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NotTheSmartest  Mar 7, 2017
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We can prove this as follows :

 

Connect all four midpoints

 

Distance from    (-a, b) to (a,b)   =  2a  

Distance from    (-a, -b) to (a, - b) =  2a ....so  these  opposite sides are congruent

 

Distance from (-a, b) to (-a, -b)  = 2b    

Distance from (a,b) to ( a, -b) = 2b......so these opposite sides are congruent

 

And the slope between (-a,b) and (a,b)  =   [b - b] / [-a - a]  =  0

And the slope between (-a,-b) and (a, -b)  = [ - b - -b] / [ -a -a] =  0.....so.....these opposite sides are parallel

 

And the slope between (-a,b) and (-a, -b)  = [-b - b] / [ -a - - a] =  is undefined

And the slope between (a,b) and (a,-b) = [-b - b] / [a - a]  is undefined....so these opposite sides are parallel

 

And the diagonals drawn from (-a,b) to (a, -b)  and from (-a,-b) to (a,b)  meet at the origin and they bisect each other since the distance from the origin to each of these points = sqrt(a^2 + b^2)

 

So...the opposite sides are congruent and parallel  and the diagonals bisect each other.....thus.....this is a rectangle

 

 

cool cool cool

CPhill  Mar 7, 2017

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