Group Event 9
$B C, C A, A B$ are divided respectively by the points $X, Y, Z$ in the ratio $1: 2$. Let
area of $\triangle A Z Y$ : area of $\triangle A B C=2: a$ and
area of $\triangle A Z Y$ : area of $\triangle X Y Z=2: b$.

G9.1 Find the value of $a$.
G9.2 Find the value of $b$.
G9.3 Find the value of $x$.
G9.4 Find the value of $y$.
area of $\triangle A Z Y=\frac{2}{3}$ area of $\triangle A C Z$ (same height)
$=\frac{2}{3} \times \frac{1}{3}$ area of $\triangle A B C$ (same height)
$$
\Rightarrow a=9
$$

Reference: 2000 FI5.3
Similarly, area of $\triangle B Z X=\frac{2}{9}$ area of $\triangle A B C$; area of $\triangle C X Y=\frac{2}{9}$ area of $\triangle A B C$
area of $\triangle X Y Z=$ area of $\triangle A B C-$ area of $\triangle A Z Y-$ area of $\triangle B Z X-$ area of $\triangle C X Y$ $=\frac{1}{3}$ area of $\triangle A B C$
$2: b=$ area of $\triangle A Z Y:$ area of $\triangle X Y Z=\frac{2}{9}: \frac{1}{3}$
$$
\Rightarrow b=3
$$

A die is thrown 2 times. Let $\frac{x}{36}$ be the probability that the sum of numbers obtained is 7 or 8 and $\frac{y}{36}$ be the probability that the difference of numbers obtained is 1 .
Favourable outcomes are $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1),(2,6),(3,5),(4,4),(5,3),(6,2)$
$$
\begin{array}{l}
\mathrm{P}(7 \text { or } 8)=\frac{x}{36} \\
\Rightarrow x=11
\end{array}
$$

Favourable outcomes are $(1,2),(2,3),(3,4),(4,5),(5,6),(6,5),(5,4),(4,3),(3,2),(2,1)$.
$\mathrm{P}($ difference is 1$)=\frac{y}{36}$
$\Rightarrow y=10$