Question

If a parallelogram has all its sides equal and one of its diagonal is equal to a side, show that its diagonals are in the ratio √3: 1. 

Answer 1

Given: ABCD is a parallelogram, where AC and BD are the diagonals meeting at O. AB = BC = AC.

To Prove: BD : AC :: √3 : 1

Proof : In △ABC, AB = BC = CA (given).

= a (say)

Hence ABC is an equilateral triangle . (definition of equilateral triangle)

AC and BD are the diagonals of parallelogram ABCD.

⇒ AC = BD (Diagonals of a parallelogram bisect each other)

or AO = OC.

i.e. BO is the median of the equilateral ABC.

Hence BO = √3/2a

∴ BD = √3a

⇒ BD : AC :: √3 a : a

⇒ BD : AC :: √3 : 1

Answer 2

Answer:

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