### 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?

# Weekly Problem 27 - 2013

##### Stage: 3 Challenge Level:

The two angles marked $y ^{\circ}$ are equal because they are in an isosceles triangle. For the same reason, the angles $z^{\circ}$are equal. Since an exterior angle of a triangle is the sum of the two interior and opposite angles, it follows that $a=2y$ and $b=2z$. Now $a^{\circ}+ b^{\circ} = 180^{\circ}$ since they are the base angles of a parallelogram. So $2y + 2z = 180$ giving $y+z=90$. But, from the angle sum of a triangle $x+y+z=180$; hence $x =90$.

This problem is taken from the UKMT Mathematical Challenges.

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