Red triangle

[bonanova]

What fraction of the black square does the red right triangle occupy?

[gavinksong]

Applying the Pythagorean theorem on the right triangle formed from the circle's center and the red square's right side,

1 = R/4 + R

where R is the area of the red square, gives us R = 4/5. The red triangle is a fourth of this: 1/5.

 

Applying the Pythagorean theorem for the right triangle formed from a corner of the black square and a segment of the circle's diameter,

1 = B/4 + B/4

where B is the area of the black square, gives us B = 2.

 

So the red triangle occupies a 10th of the black square.

[bonanova]

I had thought of making this a different problem: Given the black square (only) construct a shape inside the square that is 1/10 of its area.

It would entail making the circumcircle, the red square inscribed in the upper semicircle (the hard part) and finally the two red diagonals.

I wonder [1] if anyone would have gotten that solution or [2] if there is a simpler way.

[gavinksong]

I had thought of making this a different problem: Given the black square (only) construct a shape inside the square that is 1/10 of its area.

It would entail making the circumcircle, the red square inscribed in the upper semicircle (the hard part) and finally the two red diagonals.

I wonder [1] if anyone would have gotten that solution or [2] if there is a simpler way.

 

I agree. That would have been a much harder problem.

I probably wouldn't have been able to solve it.

I didn't realize this was an original puzzle. Good job!

[bonanova]

Well I found the image. But then what kind of puzzle to make of it?