### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

### Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

### Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

# Chord

##### Age 16 to 18 Challenge Level:

Three circles all of radius 2 cm touch as shown in the diagram.

A straight line $AG$ is tangential to the third circle, meeting the middle circle at $H$ and $I$.

How long is $HI$?

What if there are four or more circles?