### Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

# Perception Versus Reality

### Why do this problem?

This problem challenges students to think about the way data can be presented to support an argument. There is the opportunity for cross-curricular collaboration as students look at communication and real world issues.

### Possible approach

Show the video (and perhaps also the video from the USA linked at the bottom of the problem.
You could pause the video at each stage to allow students to reflect for themselves about their own perception of wealth distribution.

What makes the videos so powerful is the vast difference between people's perception and reality. Ask the class to think of their own areas of interest to find a question they would like to explore, and invite them to prepare presentations similar to the ones in the video. Students will need time to collect data and research their chosen topic.

### Possible extension

Particularly powerful presentations could be shared with the rest of the school and the wider community, and perhaps shared through social media.

### Possible support

Picturing the World invites students to think about data representation of large scale statistics in a slightly less politicised way.