#### Answer

$AD=\sqrt 89$

#### Work Step by Step

Using Pythagorean's Theorem (for the small triangle in the diagram...
$a^{2}+b^{2}=c^{2}$
Substitute in known lengths...
$4^{2}+3^{2}=c^{2}$
Simplify...
$16+9=c^{2}$
$25=c^{2}$
Square root both sides...
$c=5$ or $AC=5$
Using Pythagorean's Theorem for the larger triangle...
$a^{2}+b^{2}=c^{2}$
Substitute in known lengths...
$5^{2}+8^{2}=c^{2}$
Simplify...
$25+64=c^{2}$
$89=c^{2}$
Square root both sides...
$c=\sqrt 89$ or $AD=\sqrt 89$