The acute angle of a rhombus whose side is a mean proportional between its diagonals is

Options

A. `15^(@)`
B. `20^(@)`
C. `30^(@)`
D. `80^(@)`

Answer

C

Explanation

Let the required angle be `'theta'`
`cos(pi-theta)=(a^(2)+a^(2)-d_(1)^(2))/(2a^(2))`
`rArr 2a^(2)-d_(1)^(2)=-2a^(2)cos theta`
`rArr d_(1)^(2)=4a^(2)"cos"^(2)(theta)/(2)`
`rArr d_(1)=2a cos.(theta)/(2)`
Similarly, `d_(2)^(2)=2a^(2)(1-cos theta)`
`rArr d_(2)=2a sin.(theta)/(2)`
Given , `a^(2)=d_(1)d_(2)`
`rArr a^(2)=4a^(2)sin.(theta)/(2)cos.(theta)/(2)`
`rArr theta = 30^(@)`