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Intermediate

The three rights

A trapezium containing three right angles is split in two. What is the proportion of the areas yellow : blue?

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Solution

The ratio yellow : blue is (1+√5)/2.

From the right angles at B and C we infer that DC and AB are parallel. Therefore we can equate angles ABD and BDC. Setting the two equal sides to 1, we can easily find the bottom side and diagonal as shown.

Now from the Pythagorean theorem we get 1+1/cos2(α)=1/sin2(α). Using simple trigonometry, we find one positive solution sin(α)=1/φ (the Golden Ratio). Since the yellow and blue triangles have the same height, their area proportion is simply their base proportion, which is φ.

Poem

The three rights
The right to know
The right to participate
The right to refuse
Or the three rights
As figure of geometry
They were in love
Touching eachother
One time in angle
One time in corner
They sometimes cut their way
But don’t like to be secant lines
To play

Bella

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