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# The paper hat

A rectangle and several triangles. The orange triangle has the same area as the green trapezium. What’s the total coloured fraction?

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## Solution

The coloured fraction is 1/√2, which is approximately 0,71.

For convenience we assume the orange area to be 1/2. If its base has size x, its height must be 1/x. From similarity of triangles HIC and FGC and the area proportion 1 : 2, we deduce that base FG is x√2 and the height JC is √2/x.

Now triangles DEC and ABC are also similar and we know from the above that their heights have proportion 1 : √2. Therefore their bases must also have this proportion, so the base AB equals 2x.

Finally, we find that triangle ABC has area 1/2*2x*√2/x=√2. The required fraction therefore is (1/2+1/2)/√2=1/√2.

## Poem

It could be a paper hat
It’s Mirangu who says that
Why not a Xmas tree
With it’s orange brown crown
May be an ill tree
Loosing it’s needles
Why?
The world, the forests are polluted
Trees are suffering
Trees are dying
And what are we doing ?

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