### 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?

##### Stage: 3 Challenge Level:

This problem invited students to consider the relationship between opposite angles of cyclic quadrilaterals. Often, with these types of problems, it is helpful to draw diagrams; several students submitted diagrams as part of their solution - well done.

The problem is divided into two parts: the first part contains questions that form "building blocks" to help meet the final challenge in the second part.

Nick, from St Stephen's at Carramar summed up his solution:

The sum of the angles at opposite vertices of a cyclic quadrilateral is $180^\circ$. This is the same for all cyclic quadrilaterals, regardless of the positioning of the centre dot.

Click here to see his full solution with diagrams.

Well done also to the following students, who also submitted similar (and correct!) answers to this problem: Andre, Laura, Sascha, Chris and Sailesh from St. Stephen's School, Marcus and Kye from St Philip's Primary School, and Natasha.

Now that you have completed this problem, you could try the following problems as an extension: Subtended Angles and/or Right Angles.