19. A triangle $A B C$ is inscribed in a semicircle of radius 5 . If $A B=10$, find the maximum value of $s^{2}$ where $s=A C+B C$.
19. Ans: 200.
$A B C$ must be a right-angled triangle. Let $x=A C$ and $y=B C$, by Pythagoras theorem $x^{2}+y^{2}=10^{2}$.
$$
s^{2}=(x+y)^{2}=x^{2}+y^{2}+2 x y=100+2 \times \text { area of } A B C \text {. }
$$

Maximum area occurs when $x=y$, i.e. $\angle C A B=45^{\circ}$. So $x=y=\sqrt{50}$.