Group Event 6
As shown in the figure, $\triangle A B C$ and $\triangle X Y Z$ are equilateral triangles and are ends of a right prism. $P$ is the mid-point of $B Y$ and $B P=3 \mathrm{~cm}$, $X Y=4 \mathrm{~cm}$.
G6.1 If $a=\frac{C P}{P X}$, find $a$.
G6.2 If $C X=\sqrt{b} \mathrm{~cm}$, find $b$.
G6.3 If $\cos \angle P C X=\frac{\sqrt{c}}{5}$, find $c$.
G6.4 If $\sin \angle P C X=\frac{2 \sqrt{d}}{5}$, find $d$.
$\begin{array}{l}C P=\sqrt{3^{2}+4^{2}} \mathrm{~cm}=5 \mathrm{~cm}=P X \text { (Pythagoras' theorem) } \\ a=1 \\ C X=\sqrt{6^{2}+4^{2}} \mathrm{~cm}=\sqrt{52} \mathrm{~cm} \text { (Pythagoras' theorem) } \\ b=52 \\ \cos \angle P C X=\frac{\sqrt{52} \div 2}{5}=\frac{\sqrt{13}}{5} \\ \Rightarrow c=13 \\ \sin ^{2} \angle P C X=1-\cos ^{2} \angle P C X=\frac{12}{25} \\ \sin \angle P C X=\frac{2 \sqrt{3}}{5} \\ \Rightarrow d=3\end{array}$