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## #1 2005-09-23 05:46:21

Michael
Guest

### geometry problem: how to determine area

Hello

I need help with a problem I found in a book called "Geometrical Puzzles". I need to determine what fractal of the area of the outer square the area of the inner square is. (See the image)

Image

Now, I do know it's in fact 1/5, but I also need to prove that. That's what I got so far: (a is the len of the outer square)
DQ² = a² + 1/2a²;   DQ = √(1,5a²)
gQ = 1/5 * √(1,5a²)
hg = 2/5 * √(1,5a²)
It's no problem showing that Dh=hg=2*gQ=gf=ef=eh...using the intercept theorem.
But now, when I want to calculate the area of hg², I get instead of 1/5a² (=0,25a²) only 0,24a². That's a close miss....What the hell am I doing wrong?

Michael

## #2 2005-09-23 12:42:27

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

### Re: geometry problem: how to determine area

line2: y=2x

line1: y = -x/2  + a

line3: y=2x - a

intersection lines 1 and 2 is (0.4a,0.8a)

intersection lines 1 and 3 is (0.8a,0.6a)

small square side length is square root of ((.4² + .2²)a²), which is the square root of 0.2a².

If you square the side length for the area, you get 0.2a²,
which is five times smaller than the big square's area: a².

Last edited by John E. Franklin (2005-09-23 12:48:21)

igloo myrtilles fourmis

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## #3 2005-09-23 22:57:40

Michael
Guest

### Re: geometry problem: how to determine area

Oh my god, I have been stupid...posting 0,25=1/5

Thx for your answer John, but the JPG isn't viewable (Hotlinking seems to be not allowed). You might try http://www.imageshack.us? No registration required.