I4.1 In figure $1, D E / / B C$. If $A D=4, D B=6, D E=6$ and $B C=a$, find $a$.
I4.2 $\theta$ is an acute angle such that $\cos \theta=\frac{a}{17}$. If $\tan \theta=\frac{b}{15}$, find $b$.
I4.3 If $c^{3}=b^{2}$, find $c$.
I4.4 The area of an equilateral triangle is $c \sqrt{3} \mathrm{~cm}^{2}$. If its perimeter is $d \mathrm{~cm}$, find $d$.
$$
\begin{array}{l}
\triangle A D E \sim \triangle A B C \text { (equiangular) } \\
\frac{a}{6}=\frac{10}{4} \text { (ratio of sides, } \sim \Delta \text { 's) } \\
a=15 \\
b^{2}=17^{2}-15^{2} \\
b=8 \\
c^{3}=8^{2}=64=4^{3} \\
\Rightarrow c=4
\end{array}
$$

Reference: 1985 FSI.4, 1986 FSG.3, 1987 FG6.2, 1988 FG9.1
$$
\begin{array}{l}
\text { Each side }=\frac{d}{3} \mathrm{~cm} \\
\frac{1}{2} \cdot\left(\frac{d}{3}\right)^{2} \sin 60^{\circ}=c \sqrt{3}=4 \sqrt{3} \\
d=12
\end{array}
$$