A unit square with an extended side immersed in another square. What is the area of the red triangle?

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## Solution

The red area is √3/12.

The key is to slide down the tilted square along the extended side such that its vertex coincides with the large square vertex. Since it fitted in horizontally, it will also fit in vertically (90° rotational symmetry).

Now it is clear that MHIJ is a rectangle and moreover the perpendicular line segment extends to its diagonal MI. The other diagonal HJ was already drawn. Clearly, the red triangle is isosceles with its congruent angles equal to the tilt angle 30°. Its area is easily calculated as tan(30°)/4.

## Poem

Head above water

The square controls the tide

In deep water

May be coming from rain clouds

It’s a fresh breath coming out

Just as a secret of my mouth

No muddy water

Everything fresh and clear

My thoughts swim without any fear

For all the souls

Far and near