Four congruent right triangles form a rectangle. What fraction is red?

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## Solution

1/4 of the rectangle is red.

First the triangle with sides a, b and e is isosceles, since a and b both are the short side of one of the four right triangles. Naming the angle α, one can easily propagate it using this fact and the congruency as show. Thus the lower right corner is covered by three isosceles triangles.

Now we can follow the chain a=b=c=d=e. So we have one equilateral triangle and two others of the same size, since α=30°. Setting a=1, their area is √3/4. Since the rectangle has sides √3 and 2, its area is 2√3. It follows that the red area must be √3/2.

## Poem

Belladonna kindly composed this geometric poem:

Le vitrail de l’église

Aux carreaux fait la bise

De rouge il est paré

Pour le jour s’allumer

Un peu comme une étoile

Qui doucement largue ses voiles,

Un phénix idéal

Un prodige intégral.

Belladonna

## Visual solution

We include a purely visual solution to The ace of diamonds from David Andriana.