Floored

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

Pie Cuts

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

Getting an Angle

How can you make an angle of 60 degrees by folding a sheet of paper twice?

Triangle Split

Stage: 3 Short Challenge Level:

Since $SP = SQ$, the triangle $PSQ$ is isosceles. Therefore, $\angle SPQ = \angle PQS$. We denote the measure of those angles by $y$. Similarly, $\angle SQR = \angle QRS = x^\circ$.

Since the sums of the interior angles of $PQR$ is $180^\circ$, $x + y + (x+y) = 180$, so $2x+2y=180$.

Therefore, $\angle PQR = x^\circ+y^\circ=90^\circ$.

This problem is taken from the UKMT Mathematical Challenges.
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