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The Frieze Tree

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Some local pupils lost a geometric opportunity recently as they surveyed the cars in the car park. Did you know that car tyres, and the wheels that they on, are a rich source of geometry?

...on the Wall

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Calum from Woolmer Hill sent in his solution to the first part of the problem:

Rotate the flag 180 degrees around the point where the 2 lines meet.

Here is our explanation of the transformations required in the general case of mirror lines at any angle :

Mirror lines at angle theta

The resulting transformation is a rotation by $2\theta$ about the point of intersection of the lines. If the direction of the angle $\theta$ is from the first line of reflection to the second, then the direction of rotation is the same as the direction of $\theta$.

You should probably check that the same thing works if you reflect in the further line first (remembering that this means the direction is reversed), and if the flag is between the two lines.

We just need to check the result for a single point, because then every point will rotate by the same amount and so the flag will remain intact (and will have rotated by the same amount).


The diagram says it all, really: the line segment joining the point to the origin has rotated by an angle of $\alpha+\alpha+(\theta-\alpha)+(\theta -\alpha)=2\theta$ in the direction of $\theta$.