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Long Short

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
Quadrilateral in a circle
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?