ABCD is a rhombus in which `angleC= 60^(@)`then AC:BD=.

Options

A. `sqrt3 : 1`
B. `sqrt3 : sqrt2`
C. `3 : 1`
D. `3 : 2`

Answer

A

Explanation

ABCD is a rhombus. So its all sides are equal.
Now, `BC = DC rArr angleBDC = angleDBC =x^(@)` (say).
Also, `angleBCD = 60^(@)` (given).
`therefore x^(@)+x^(@)+60^(@)=180^(@)rArr2x=120rArrx = 60`.
`therefore angle BDC=angleDBC=angleBCD=60^(@).` So, `triangleBCD` is an equilateral triangle.
`therefore BD = BC =a`(say).
`AB^(2)=OA^(2)+OB^(2)rArrOA^(2)=AB^(2)-OB^(2)=a^(2)-((a)/(2))^(2)=a^(2)-(a^(2))/4=(3a^(2))/(4)`
`therefore AC : BD =sqrt3a : a=sqrt3 : 1`.