I4.1 If the lines $x+2 y+3=0$ and $4 x-a y+5=0$ are perpendicular to each other, find $a$.
I4.2 In Figure $1, A B C D$ is a trapezium with $A B$ parallel to $D C$ and $\angle A B C=\angle D C B=90^{\circ}$. If $A B=a, B C=C D=8$ and $A D=b$, find $b$.

I4.3 In Figure $2, B D=\frac{b}{2}, D E=4, E C=3$. If the area of $\triangle A E C$ is 24 and the area of $\triangle A B C$ is $c$, find $c$.
Reference: 1983 FG9.3, 1984 FSG.3, 1986 FSG.2, 1987 FG10.2, 1988 FG8.2
$$
\begin{array}{l}
-\frac{1}{2} \times \frac{4}{a}=-1 \\
\Rightarrow a=2
\end{array}
$$

Draw a line segment $A E / / B C$, cutting $D C$ at $E$.
$$
\angle B A E=90^{\circ}=\angle A E C \text { (int. } \angle \mathrm{s}, A E / / B C \text { ) }
$$
$A B C E$ is a rectangle
$$
\begin{array}{l}
A E=8, C E=a=2 \text { (opp. sides, } / / \text {-gram) } \\
D E=8-a=6
\end{array}
$$
$b^{2}=8^{2}+6^{2}=100$ (Pythagoras' theorem on $\triangle A D E$ )
$$
b=10
$$
$\triangle A B D, \triangle A D E$ and $\triangle A C E$ have the same height.
The area of $\triangle A B C=c=24 \times \frac{5+4+3}{3}=96$
I4.4 If $3 x^{3}-2 x^{2}+d x-c$ is divisible by $x-1$, find $d$.
$$
\begin{array}{l}
3-2+d-96=0 \\
d=95
\end{array}
$$