In right triangle $ABC$ ( $BC$ is hypotenuse),$D$ is a point on $BC$.Calculate $AB$ given that...

Question

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

Answer 1

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 $c^2=48.$

Answer 2

You also can use Stewart's Theorem:

$$\frac{AC^2}{CD\cdot BC}-\frac{AD^2}{CD\cdot BD}+\frac{AB^2}{BD\cdot BC}=1\\ \frac{81-AB^2}{6\cdot 9}-\frac{5^2}{6\cdot 3}+\frac{AB^2}{3\cdot 9}=1$$