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 June 8th, 2010, 12:44 AM #1 Newbie   Joined: Nov 2009 Posts: 12 Thanks: 0 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.
 June 8th, 2010, 02:26 AM #2 Senior Member   Joined: Apr 2010 Posts: 105 Thanks: 0 Re: geometrical question about congruent rectangles How can I understand eight shared points, if there is an intersection of the areas?
 June 8th, 2010, 06:14 AM #3 Newbie   Joined: Nov 2009 Posts: 12 Thanks: 0 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 ?!?
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)..."

 June 8th, 2010, 06:28 AM #5 Senior Member   Joined: Apr 2010 Posts: 105 Thanks: 0 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.
 June 8th, 2010, 08:44 AM #6 Global Moderator   Joined: Dec 2006 Posts: 20,931 Thanks: 2207 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.
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.

 June 22nd, 2010, 12:04 AM #8 Newbie   Joined: Nov 2009 Posts: 12 Thanks: 0 Re: geometrical question about congruent rectangles no further ideas so far?
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.

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.

### finding the area of a second rectangle that is larger than the first but congruent

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