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Geometry Level 2

Let A B C D ABCD be a square, and let E E and F F be points on A B AB and B C , BC, respectively. The line through E E parallel to B C BC and the line through F F parallel to A B AB divide A B C D ABCD into two squares and two non-square rectangles. The sum of the areas of the two squares is 9 10 \frac{9}{10} of the area of square A B C D . ABCD.

Find A E E B + E B A E . \frac{AE}{EB}+\frac{EB}{AE}.


The answer is 18.

Ayush G Rai by Ayush G Rai , 20, India

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2 solutions

Andreas Wendler
May 31, 2016

Calculating A E E B + E B A E = A E 2 + E B 2 A E ⋅ E B \frac{AE}{EB}+\frac{EB}{AE}=\frac{AE^{2}+EB^{2}}{AE\cdot EB} Numerator is 0.9 ⋅ A B 2 0.9\cdot AB^{2} and for denominator write A E ⋅ E B = 1 2 ( A B 2 − A E 2 − E B 2 ) = 0.05 ⋅ A B 2 AE\cdot EB=\frac{1}{2}(AB^{2}-AE^{2}-EB^{2})=0.05\cdot AB^{2} Put into wished expression and get 18 \boxed{18}

3 Helpful 0 Interesting 1 Brilliant 1 Confused

good one...(+1)

Ayush G Rai - 5 years ago
Henry Yu
Mar 24, 2019

Let A E = x AE = x and let E B = y EB = y This give us x 2 + y 2 = 9 10 ( x 2 + 2 x y + y 2 ) x^{2} + y^{2} = \frac{9}{10}(x^{2} + 2xy + y^{2}) . Solving, we get x 2 + y 2 = 18 x y x^{2} + y^{2} = 18xy . We wish to find x y + y x \frac{x}{y} + \frac{y}{x} . This simplifies to x 2 + y 2 x y \frac{x^{2} + y^{2}}{xy} Dividing both sides of the earlier equation by x y xy , we find that x 2 + y 2 x y = 18 \frac{x^{2} + y^{2}}{xy} = 18 .

0 Helpful 1 Interesting 1 Brilliant 0 Confused

This is Q3 on the 2013 AIME I.

Henry Yu - 2 years ago

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