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
I did, assuming side lengths of 2 for the square Line 1 is y=x/2+1 Line 2 is y=-2x+4 They intersect at x=(4-1)/(0.5+2) , y=0.5(x)+1 (1.2 , 1.6) The white 5gon and the little triangle at the very top is 1×2÷2=1 area The right hand triangle can be cut into two triangles horizontally From the intersection point earlier, the two new tris are 0.8×0.4÷2 = 0.16 and 0.8×1.6÷2=0.64 4 - 1 - 0.64 - 0.16 = 2.2 2.2/4 is 0.55 So 55%? Or 11/20
![Why you didn't learn tetration in school[Tetration]](/img/n.gif)
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)
I used integration just to be _that_ guy.
f(x) = (L+x)/2
g(x) = 2(L-x)
intersecting x value?
(L+x)/2 = 2(L-x)
x = 3L/5
Green Area
= [ 0 ~ 3L/5 ] ∫ f(x) dx + [ 3L/5 ~ L ] ∫ g(x) dx
= L(3L/5)/2 + (3L/5)(3L/5)/4 + 2LL - LL - 2L(3L/5) + (3L/5)(3L/5)
= 11LL/20
Answer = 55%
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
I did, assuming side lengths of 2 for the square
Line 1 is y=x/2+1
Line 2 is y=-2x+4
They intersect at x=(4-1)/(0.5+2) , y=0.5(x)+1
(1.2 , 1.6)
The white 5gon and the little triangle at the very top is 1×2÷2=1 area
The right hand triangle can be cut into two triangles horizontally
From the intersection point earlier, the two new tris are 0.8×0.4÷2 = 0.16 and 0.8×1.6÷2=0.64
4 - 1 - 0.64 - 0.16 = 2.2
2.2/4 is 0.55
So 55%?
Or 11/20