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.

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Mixing More Paints

Why do this problem?

This problem follows on from Mixing Paints and encourages students to think about ratio in new ways, explore using an interactive environment, and come up with some generalisations and proofs. The most general case will require perseverance to discover and prove.

Possible approach

This problem could be used in a follow-up lesson after working on Mixing Paints, or as an extension activity for some students.

Students could start by using the interactivity to experiment, and then gradually move to pen-and-paper methods, only using the interactivity to check.

Key questions

If I am mixing $1:4$ paint with $1:5$ paint, why might it be useful to start with 30 litres of paint?
If I want to mix $1:x$ paint with $1:y$ paint, what would I start with instead of 30 litres?

Possible support

Make sure students are secure in any general strategies they came up with for Mixing Paints before embarking on this task.

Possible extension

Coming up with a general strategy for making any ratio from any paint is a challenging extension task.