complex analysis

Question

prove that diagonal of rhombus are orthogonal.    

 

this question belongs to complex analysis,

 

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Answer 1

starting at the top left draw rhombus ABCD

draw the diagonals AC and BD intersecting at point E

 

proof:

 

the diagonals of a parallelogram bisect each other   Theorem

a rhombus is a parallelogram   definition of a rhombus

the diagonals of rhombus ABCD bisect each other at point E   Theorem

mAE=mEC  definition of bisect  definition of bisect

mAB= mBC   all sides of a rhombus have the same measure

mBE=mBE  reflexive property

triangle ABE≅triangle BEC    Theorem SSS(side-side-side)

<AEB≅<BEC   corresponding angles of congruent triangles are congruent

<AEB and <BEC are supplementary angles    <AEB and <BEC together form a straight angle (180º)

m<AEB+m<BEC=180º   definition of supplementary angles

m<AEB=m<BEC=90º  two angles that are equal and supplementary must each be 90º angles

therefore BD ⊥ AC  perpendicular segments form 90º angles (definition)

the diagonals are orthogonal (perpendicular)

 

Answer 2

Draw a diagram.

The diagonals of a rhombus bisect the angles.

This creates four congruent triangles.

By corresponding parts, the angles at the intersection of the diagonals are congruent.

Therefore, each of them is 90°

The diagonals are orthogonal (fancy for perpendicular).