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Intermediate

The attic room

A blue square is topped with a right triangle. Using line segments to the top, a green quadrilateral is constructed with at least two right angles. What is the area proportion green : blue?

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Solution

The fraction green : blue is 144/1369.

The whole problem is basically a comparison of scales. From similarity of triangles EGH and EBC we find HG/CB=EG/EB. In this way we can set up the chain HG/CB=EG/EB=FG/AB=EF/EA=IF/DA.

Now we know from the Pythagorean triangle and the blue square that CB=AB=DA=5. This implies HG=FG=IF, so the green quadrilateral is in fact also a square.

Now we easily compute triangle height h as 2 times area 6 divided by base 5. So h=12/5. Now the square side proportion HG/CB=h/(h+5)=12/37. The area proportion is the square of this.

Poem

It’s a new house with it’s secrets
I visited it and moved into it
Explored it from the cellar to the attic room
Entered in a fantastic world
With trunks, toys, old clothes
Hidden treasures since years
Remembrances and fantasy
Books, paintings, drawings
A painter’s history

Bella

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