Tri-split

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?

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.

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:

In this problem we are going to be thinking about the areas of equilateral triangles.
The important thing to keep in mind is that, to make life easy for ourselves, we will measure our areas in terms of equilateral triangles too.

Here are some equilateral triangles.

If the area of the first triangle is 1, what are the areas of the others?
Do you see any patterns?
Are you surprised?
Will these patterns continue? Why?

All the triangles you have just been looking at had horizontal bases, but of course equilateral triangles can be tilted.

Here are some equilateral triangles with a tilt of 1.

Can you explain what I mean by a tilt of 1?
How do I know they are equilateral triangles? Can you convince yourself that they are?

Can you find their areas? If you need some help with this why not use the geoboard environment below and/or use the hints .

FULL SCREEN VERSION
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Can you extend your ideas to find the areas of other equilateral triangles with a tilt of 1?
Is there a general rule? Can you explain it?

How about the areas of triangles with a tilt of 2, like the ones below?

What about other tilts?