In right-angled triangle ABC, `angleB = 90^(@)`, D is such a point on AB that AB : BC : BD `= sqrt(3)` : 1 : 1. Find the value of `angleACD`.

Explanation

Let `angleBCD = theta`.
Given that AB : BC : BD =`sqrt(3)` : 1 : 1,
i.e.,BC : BD = 1 : 1
`rArr " " (BC)/(BD) = 1 " " rArr " " (BD)/(BC) = 1` ………..(1)
Now, in right-angled triangle BCD, we get,
`tan theta = (BD)/(BC)` [ by definition ]
`rArr " " tan theta = 1` [from (1) ]
`rArr tan theta = tan 45^(@)`
`rArr theta = 45^(@)`
Again, given that AB : BC `= sqrt(3)` : 1
Now, from the right-angled triangle ABC we get,
`tan angleACB = (AB)/(BC)`
`rArr " " tan angleACB = sqrt(3) = tan 60^(@)`,
`rArr " " angleACB = 60^(@)`.
`therefore angleACD = angleACB - angleBCD` [by the figure]
` = 60^(@) - 45^(@)`
` = 15^(@)`.
Hence the required value of `angleACD = 15^(@)`.