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June 8th, 2010, 12:44 AM   #1
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geometrical question about congruent rectangles

Hello,

I have a geometrical question:

there are two congruent rectangles, now you lay one on top of the other, so that they have eight shared points. It has to be shown that the area of the intersection is bigger than the area of half of the rectangle.

Thanks marten.
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June 8th, 2010, 02:26 AM   #2
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Re: geometrical question about congruent rectangles

How can I understand eight shared points, if there is an intersection of the areas?
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June 8th, 2010, 06:14 AM   #3
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Re: geometrical question about congruent rectangles

Yes, there is an intersection of areas, I tried to upload an image, but it didn't work.
I don't know if shared is the right word, perhaps "in common" is better ?!?
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June 8th, 2010, 06:21 AM   #4
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Re: geometrical question about congruent rectangles

Quote:
Originally Posted by SafeX
How can I understand eight shared points, if there is an intersection of the areas?
I think what he means is equivalent to the following:
"Consider a rectangle in a plane. Rotate the rectangle slightly (let's say 20-30 degrees) from its center so that the edges of the image intersect the edges of the preimage in 8 points. Show that (something about areas)..."
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June 8th, 2010, 06:28 AM   #5
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Re: geometrical question about congruent rectangles

OK, the eight points are shared on the edges. Otherwise you have infinitely points too in the intersection of the areas.
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June 8th, 2010, 08:44 AM   #6
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This is problem 12 from the 4th ASU (All Soviet Union Mathematical Olympiad) 1970.
The original wording is as shown below.

Two congruent rectangles of area A intersect in eight points. Show that the area of the intersection is more than A/2.
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June 8th, 2010, 08:57 AM   #7
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Re:

Quote:
Originally Posted by skipjack
This is problem 12 from the 4th ASU (All Soviet Union Mathematical Olympiad) 1970.
The original wording is as shown below.

Two congruent rectangles of area A intersect in eight points. Show that the area of the intersection is more than A/2.
Wow, this guy must have been thinking about this problem for FORTY YEARS!!!!!!!!!!! :P
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June 22nd, 2010, 12:04 AM   #8
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Re: geometrical question about congruent rectangles

no further ideas so far?
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June 22nd, 2010, 01:57 AM   #9
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Re: geometrical question about congruent rectangles

Quote:
Originally Posted by MG85
no further ideas so far?
Choose two congruent squares.
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June 22nd, 2010, 04:26 AM   #10
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Re: geometrical question about congruent rectangles

Quote:
Originally Posted by SafeX
Quote:
Originally Posted by MG85
no further ideas so far?
Choose two congruent squares.
WHAT?!? No.

You have to choose two arbitrary rectangles that satisfy the conditions and then show
something based on assumptions. Starting with squares leads to a weaker statement.
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