G3.3 In Figure 2, a rectangle intersects a circle at points $B, C, E$ and $F$. Given that $A B=4 \mathrm{~cm}, B C=5 \mathrm{~cm}$ and $D E=3 \mathrm{~cm}$. If $E F=c \mathrm{~cm}$, find the value of $c$.
$A B=4 \mathrm{~cm}, B C=5 \mathrm{~cm}$ and $D E=3 \mathrm{~cm} . E F=c \mathrm{~cm}$
Draw $B G \perp D F, C H \perp D F$
$D G=A B=4 \mathrm{~cm}, G H=B C=5 \mathrm{~cm}$
$E G=D G-D E=4 \mathrm{~cm}-3 \mathrm{~cm}=1 \mathrm{~cm}$
Let $O$ be the centre.
Let $M$ be the foot of perpendicular of $O$ on $E F$ and produce $O M$ to $N$ on $B C$.
$O N \perp B C$ (corr. $\angle \mathrm{s} A C / / D F$ )
$B N=N C=2.5 \mathrm{~cm}(\perp$ from centre bisect chord $)$
$M F=E M(\perp$ from centre bisect chords $)$
$=E G+G M=1 \mathrm{~cm}+B N=1 \mathrm{~cm}+2.5 \mathrm{~cm}=3.5 \mathrm{~cm}$
$E F=2 E M=7 \mathrm{~cm}$