<s>[INST] In right triangle $ABC$ ( $BC$ is hypotenuse),$D$ is a point on $BC$.Calculate $AB$ given that... 
In right triangle $ABC$ ( $BC$ is hypotenuse),$D$ is a point on $BC$.Calculate $AB$ given that: $AD$=5,$BD=3$,$CD=6$.

 [/INST] Let $\angle{ADC}$ be $\alpha$. Then by the cosine theorem
$b^2=25+36-60\cos\alpha$,
$c^2=25+9-30\cos(180^{\circ}-\alpha)=34+30\cos\alpha$.
This gives
$b^2+2c^2=129$.
Since $b^2+c^2=9^2=81$, it gives $c^2=48.$
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