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Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
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**Why do this problem?**

This problem encourages students to appreciate that there can be more than one way to tackle a problem. The methods suggested here use symmetry, congruence, and areas. Each method has a picture which students are encouraged to use in order to complete a proof.

**Possible approach**

*This problem featured in an NRICH Secondary webinar in March 2022.*

Project the geogebra app onto a white board. Use the slider to rotate the red square around and ask students what fraction of the blue square they think is covered. In particular, look at when the red square is horizontal and when it makes an angle of $45^{\circ}$ with the horizontal. Do they think that the red square always covers the same fraction of the blue square?

There are 4 different proof methods to consider. You might find it helpful to provide students with a printout of the 4 pictures behind the "hide and reveal" buttons.

These printable worksheets could also be useful: *Method 2, Method 3, Method 4*. Each worksheet consists of a series of diagrams which form a proof, and space for students to add the explanations
necessary to complete the proof.

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

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.