Hello, Dragon!
Hmmm, I got a different answer . . .
$\displaystyle AB$ is a diameter of a circle of radius 1 unit.
$\displaystyle CD$ is a chord perpendicular to $\displaystyle AB$ that cuts $\displaystyle AB$ at $\displaystyle E$.
If the arc $\displaystyle CAD$ is 2/3 of the circumference of the circle,
what is the length of the segment $\displaystyle AE$?
Of course you made a sketch, right? Code:
* * * C
* *
* / | *
* 1/ | *
/ |
* 1 /60° | *
A * - - - - * - - + - * B
* O\ |E *
\ |
* 1\ | *
* \ | *
* *
* * * D
Since major arc $\displaystyle CAD \:= \:\frac{2}{3}\cdot360^o \:=\:240^o$
. . then minor arc $\displaystyle CBD \,=\,120^o$ and central angle $\displaystyle COB \,=\,60^o.$
Right triangle $\displaystyle CEO$ has acute angle $\displaystyle 60^o$ and $\displaystyle hyp = 1.$
Hence: $\displaystyle OE \,= \,adj \,= \,\frac{1}{2}$
Therefore: .$\displaystyle AE\:=\:\frac{3}{2}$