Challenge Level

Draw a circle with radius $1$ unit. Choose any four points on
the circumference and join them together to form a
quadrilateral.

Label the sides: $a$ is the shortest, $b$ is the next shortest, then $c$, and finally $d$ is the longest side (it is possible to have two sides of equal length).

What is the maximum length that the shortest side $a$ could be?

Side $b$ must be less than a certain value - what value?

What is the maximum length that the longest side $d$ could be? Is it possible for $c$ and $d$ both to be this maximum length?