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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it? Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Age 14 to 16 Challenge Level:
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$O$ is the centre of the balance, and $X$ is the position of the initially empty hook.

If you leave the hook at $X$ empty, what angle does $OX$ make to the vertical once the balance settles?

If you were to attach a weight of $1$ unit at $X$ , what do you think the angle might be?

Were you right?

If you were to attach a weight of $2$ units at $X$ what do you think the angle might be?

Right that time?

You only need the $1$ and $2$ unit weights for the problem set but the others are there to let you explore, conjecture and test - have fun.