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A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

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Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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Six Discs

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?

Of All the Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem challenges understandings concerning area on the way to working in a context which leads to non-linear sequences that need explanation. Learners will need to use their visualising skills to help them to create equilateral triangles and find their areas.

Possible approach

You might want to look at the problem Tilted Squares first.
You may need isometric paper, which can be found here.

Without speaking, draw equilateral triangles on the board, starting as shown in the problem. Write the areas of the first two or three triangles and place question marks next to the rest.

Allow time for reflection and discussion drawing out ideas such as the use of non-standard units and the interesting result of square numbers. You might want to spend some time asking learners to try to explain why (see Picturing Square Numbers ).

Present the idea of tilted triangles, discussing how this might be defined before setting the challenge posed in the second part of the problem. A good point to discuss is how we know the triangles are equilateral - those who are convinced that the triangles are equilateral could explain their reasoning to those who aren't.

It is worthwhile giving the class some time to draw out the diagrams and try to come up with their own methods for finding the areas of the tilted triangles, and to share the methods that they find, but if they are struggling to find an efficient way, the pictures in the hints section might be useful as prompts.
Once a few areas have been found, encourage the learners to make conjectures about the areas of much larger triangles with tilt 1, and to justify their ideas.

Key Questions

How do you know the tilted triangles are equilateral?
How can you find the area of a tilted triangle in terms of the unit equilateral triangle?
Can you find a generalisation for the area of a tilted triangle?

Possible Extension

Can you find a general rule for finding the area of any sized triangle with any tilt?
Are there any areas that it's impossible to make with a tilted triangle?

Possible Support

Use the problem Tilted Squares first.
Focus on the justification of equilateral triangles and the calculation of areas rather than seeking generalisations.