It's not a short proof... I will not give every detail, so some details you must fill in yourself. Or see for a full proof: http://home.kpn.nl/henkreuling/solutions8foxes/130%20-%208foxes%20-%20solution.pdf

It's not hard to see that every time you cut off a square from a triangle similar to triangle ABC then the remaining triangle on the right side is similar with triangle ABC with factor 1/3; and the remaining triangle on the left side is similar with triangle ABC with factor 2/3; The first square has sides 2/3, and area 4/9. On the right side, because of the factor 1/3, the sides of the squares give a geometric sequence with side xn = 2/3*(1/3)^n n=0,1,2,... The area give a geometric sequence (xn)^2 = 4/9*(1/9)^n. The total area is sum(n from 0 to eternity) 4/9*(1/9)^n = 4/9*(1/(1-1/9))=4/9*9/8 = 1/2. Now the left side: the sides of the squares give a geometric sequence with side yn=2/3*(2/3)^n n=0,1,2,... The area give a geometric sequence (yn)^2 = 4/9*(4/9)^n, with sum 4/9*(1/(1-4/9))=4/9*9/5 = 4/5. But the first square with side 2/3 (and area 4/9) is counted on both sides. The total area of all squares is also: 1/2 + 4/5 - 4/9 = 77/90. Because the area of triangle ABC = 1, is the ratio of the area of ABC tot the sum of all squares equal to 90/77. The correct answer is E. (I'm Dutch, so sorry for my poor English...)

It's not a short proof... I will not give every detail, so some details you must fill in yourself.

ReplyDeleteOr see for a full proof:

http://home.kpn.nl/henkreuling/solutions8foxes/130%20-%208foxes%20-%20solution.pdf

It's not hard to see that every time you cut off a square from a triangle similar to triangle ABC then the remaining triangle on the right side is similar with triangle ABC with factor 1/3; and the remaining triangle on the left side is similar with triangle ABC with factor 2/3;

The first square has sides 2/3, and area 4/9.

On the right side, because of the factor 1/3, the sides of the squares give a geometric sequence with side xn = 2/3*(1/3)^n n=0,1,2,...

The area give a geometric sequence (xn)^2 = 4/9*(1/9)^n.

The total area is sum(n from 0 to eternity) 4/9*(1/9)^n = 4/9*(1/(1-1/9))=4/9*9/8 = 1/2.

Now the left side: the sides of the squares give a geometric sequence with side yn=2/3*(2/3)^n n=0,1,2,...

The area give a geometric sequence (yn)^2 = 4/9*(4/9)^n, with sum 4/9*(1/(1-4/9))=4/9*9/5 = 4/5.

But the first square with side 2/3 (and area 4/9) is counted on both sides.

The total area of all squares is also: 1/2 + 4/5 - 4/9 = 77/90.

Because the area of triangle ABC = 1, is the ratio of the area of ABC tot the sum of all squares equal to 90/77.

The correct answer is E.

(I'm Dutch, so sorry for my poor English...)