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Intermediate

The triangle quartet

A square is divided into four triangles. The red area equals the blue area. What is the angle α?

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Solution

The angle α is 30°.

Scale the square to have area 1 and name the outer triangle areas A, B and C. Fitting A to C on the bottom of the square, and using SAS to show congruency, it follows that the inner triangle has area A+C. We have two equations: A+B=C and 2A+B+2C=1. Eliminating B gives A+3C=1.

This gives rise to the equation tan(45-α)+3tan(α)=2. Using trigonometry, this can be worked out to a quadratic equation in tan(α) having one positive root tan(α)=1/√3.

Poem

The triangles have many sides
A heart inside with red, blue and white
with a very special shape
with points and lines straight
looking inside them and their shape
you can add all my angles
you can also see my vertices
the geometry loves me and my
quadrangles and degrees.

Bella

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