The diagram contains two circles that touch at $C$. $A$, $B$ and $C$ lie on the smaller circle, whose centre is $X$. $C$, $D$ and $E$ lie on the larger circle, whose centre is $Y$. $AXCYE$ and $BCD$ are straight lines, and $Y\hat{D}E = x^\circ$.
(a)[3]
Prove that triangle $BCX$ is similar to triangle $DCY$, and justify every statement you write.
(b(i))[1]
Find, in terms of $x$, $\angle DC\hat{Y}$.
(b(ii))[1]
Find, in terms of $x$, $\angle BX\hat{A}$.
(c)[3]
With $BC = 3.5\,\text{cm}$, $CX = 3.2\,\text{cm}$ and $CD = 5.6\,\text{cm}$ given, determine the length of $AE$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Angles that are vertically opposite” …