Circle theorems question I can't solve involving dodecagon...
This is my last resort, as I can't figure it out at all!
This worksheet was given to us (GCSE Maths, top set, yr 11, UK) about a month or two ago in a lesson on circle theorems. The question we all had trouble with was the second-from-bottom one on the second page, marked skull-and-cross-bones (b). Nobody, not even the teacher could solve it. The closest I got was by assuming that the non-circle shape was a regular dodecagon and therefore had interior angles of 150 degrees. Thus, I was able to conclude that the shape invloving the portion of the dodecagon inside the circle, as well as angle X, was a hexagon. I then worked out that a+b+x = 270 degrees, where a and b are the angles at those points between the lines of the dodecagon inside the circle and the line linking to angle x. However, I was not able to work out any more! Any help would be greatly appreciated!
$\endgroup$ 21 Answer
$\begingroup$Draw lines from $A$ and $B$ to the circle centre $O$. This encloses a hexagon to the left of the newly drawn lines; its internal angles must sum to $720^\circ$. Three of those angles are internal angles of the dodecagon, hence $150^\circ$ each; two more are at $A$ and $B$ and are $60^\circ$ each – $150^\circ$ minus the right angle that appears because the new lines are circle radii. This leaves the last angle as $150^\circ$. As this is the angle subtended by a chord from the centre, the angle subtended at any point at the edge – the desired $x$ – is half that, $75^\circ$.
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