Three congruent rectangles consisting of two squares each are placed as shown. What is the blue area?
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Solution
The blue area is 50.
The essential ingredient is to recognize that the right triangle in the lower right corner is similar to and two times smaller than the one in the upper right corner. We have added the corresponding side lengths.
From the large square we find that both 2+2z and 2y+z equal 2x. This gives us y=(x+1)/2 and z=x-1. Now we can apply the Pythagorean theorem: y2+z2=x2. This gives x2-6x+5=0. This has roots 1 and 5. If x=1, then z=0, so we can discard this root. Hence x=5. The requested area is 2x2.
Poem
We were pleased to receive the following poem:
Ce sont des dominos
Ils sont cool avec leur théorie
Ils ne vont pas tomber mais se relèvent unis
Ils sont de couleurs ou bien noirs et blancs
Ils chantent la vie pour les gagnants
Et les combinaisons se rient des perdants
Le jeu de dés est un art
A apprendre tôt ou tard
