Prove that the diagonals of a rhombus are at right angles when O is origin c ( α ,β) B ( α + h ,β) A (h, 0)?

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Answer 1 · 2018-08-12T13:20:27

Please see the proof below

Let's work with vectors

#vec(OA)=((h),(0))#

#vec(CB)=((h),(0))#

#vec(OC)=((alpha),(beta))#

#vec(AB)=((alpha),(beta))#

As the figure is a rhombus

#||vec(OA)||=||vec(CB)||=||vec(OC)||=||vec(AB)||#

#h^2=alpha^2+beta^2#

The dot product of the diagonals is

#vec(OB). vec(AC)=((alpha+h),(beta)).((alpha-h),(beta))#

#=(alpha+h)(alpha-h)+beta^2#

#=alpha^2-h^2+beta^2#

#=0#

Therefore,

The vectors #vec(OB)# and #vec(AC)# are orthogonal, that is
the diagonals are perpendicular to one another.