<s>[INST] Distance from point $D$ to $BI$ 
I found that : $BI = 2\sqrt 2, DI = \sqrt 5, DB = 2\sqrt 2$. It means $\triangle BDI$ is isosceles triangle. Lets say $B$ to $DI$ is $a = \sqrt {BI^2 - ( \frac 12 DI)^2} = \sqrt{\frac{27}{4}}$. Lets say $t$ is distance from D to BI.
$t. BI = a. DI$
$t = \frac{\sqrt{27}}{4}$.
But why my answer not in the options?
 [/INST] Note that $BI = \sqrt5$. So, △BDI is isosceles with DI = BI instead. Then, I to DB is 
$a = \sqrt {BI^2 - ( \frac 12 DB)^2} = \sqrt{3}$ and 
$$t = \frac{DB}{BI}a = \frac{2\sqrt2}{\sqrt5}\cdot\sqrt3=\frac{\sqrt{24}}5$$
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