What fraction of the square's area is green?

Easy to see each of the big triangles represent a quarter. Thus fraction that is green = 1/2 + overlap. The overlapping triangle is similar to big triangles, and w.l.o.g we give the square side length of 2 so we can get the hyp of the big triangles to be sqrt5 and the hyp of the small triangle to be 1. If the side length is scaled down sqrt5 then the area is scaled down sqrt5^2 = 5 so the overlap is 1/5 of the big triangle, or one 20th of the whole square or 5% this leaves 55% that is shaded green. (11/20)

1) Let the side of the square be 2a. Its area will be 4a². 2) The legs of the two large triangles are 2a and a, making their areas = a² (i.e. 2a•a/2), so, both together would be 2a². 3) They overlap in a small triangle which is similar to the large triangles (left to the reader.) 4) The hypotenuse of the small triangle is a. The hypotenuse of the large triangles is a√5, making the ratio of the sides 1/√5... 5) Therefore the legs of the small triangle are a/√5, and 2a/√5. 6) therefore the area of the overlap is (2a²/5)/2, or a²/5. 7) So the colored area is given as: (area of square) - (sum of areas of large triangle - from #2 above) +(area of overlap - from #6 above). Or: 4a² - 2a² + a²/5 => 2a² + a²/5 => 11a²/5 8) Finally, to get the colored area's percentage of area of the square, we divide the area from #7 by the area of the square, or: (11a²/5)/4a² => 11/20 9) Converting 11/20 to a percentage yields: 55% So, that's the answer: 55% Now to watch the video to see if I got it right. -s.west