Triangle $ABC$ is right angled at $C$. $AD$ splits $\angle BAC$ in the ratio $2:1,$ where $\angle CAD$ is smaller. If $AC=2$ and $CD=1$, find the length of $BD$.
B
D
C A
tan DAC = 1/2
BAC = 3* DAC
Using a special trig formula
tan (3* DAC) = 3tan ( DAC) - tan^3(DAC)
________________________ =
1 - 3tan^2 (DAC)
[ 3 (1/2) - (1/2)^3 ] / [ 1 - 3/4 ] = [ 3/2 - 3/8] / (1/4) = (11/8)/(1/4) = 44/8 = 11/2
So tan BAC = 11/ 2 = BC / CA
So BC = 11
And BD = BC - AC = 11 - 1 = 10