Not to Scale

By Pythagorean theorem , we have:

$\begin{cases} \overline{AD}^2 + \overline{AB}^2 = \overline{BD}^2 & ...(1) \\ \overline{BC}^2 + \overline{CD}^2 = \overline{BD}^2 & ...(2) \end{cases}$

$\begin{aligned} \implies \overline{AD}^2 + \overline{AB}^2 & = \overline{BC}^2 + \overline{CD}^2 \\ \implies \overline{AD}^2 & = \overline{BC}^2 + \overline{CD}^2 - \overline{AB}^2 = 5^2 + 4^2 - (\sqrt 5)^2 = 36 \\ \implies \overline{AD} & = \sqrt{36} = \boxed 6 \end{aligned}$

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Thanks, @Kaizen Cyrus for the image.

Scaled diagram.