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...)

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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...)

Do not use all of these Private Money Lender here.They are located in Nigeria, Ghana Turkey, France and Israel.My name is Mrs.Ramirez Cecilia, I am from Philippines. Have you been looking for a loan?Do you need an urgent personal or business loan?contact Fast Legitimate Loan Approval he help me with a loan of $78.000 some days ago after been scammed of $19,000 from a woman claiming to be a loan lender from Nigeria but i thank God today that i got my loan worth $78.000.Feel free to contact the company for a genuine financial call/whats-App Contact Number +918929509036 Email:(fastloanoffer34@gmail.com)

ReplyDelete