# Triangle - how can this be?



## Snelly (Aug 20, 2005)

Try working this one out!


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## Waleem (Nov 1, 2006)

Spooky..... 8O


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## DABurleigh (May 9, 2005)

Because the hypotenuse is bowed in the lower one to make room for the new square.

Dave


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## Snelly (Aug 20, 2005)

DABurleigh said:


> Because the hypotenuse is bowed in the lower one to make room for the new square.
> 
> Dave


I don't know much about animals, but I can't see a hippo bent over?? Please explain Dave...


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## artona (Jan 19, 2006)

Hi Shane

Ask Mr Pythagorouse, the mouse :lol: :lol: :lol: 


stew


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## DABurleigh (May 9, 2005)

The slopes of the red and green triangles are different, so the areas of the total triangles are different, and the difference is equal to one smaller square.


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## Snelly (Aug 20, 2005)

DABurleigh said:


> The slopes of the red and green triangles are different, so the areas of the total triangles are different, and the difference is equal to one smaller square.


They look identical...???


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## DABurleigh (May 9, 2005)

Well they're not


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## Jiggles (Apr 17, 2007)

Oh yes they are.
Count the squares covered by each shape.


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## Snelly (Aug 20, 2005)

I've printed this out, cut out the shapes, compared and measured... they are exactly the same. It must just be one of those things??


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## DABurleigh (May 9, 2005)

Thank heavens I had the wisdom to avoid teaching; I'd be driven up the wall. I repeat - the area of both overall "triangles" (because they aren't) is DIFFERENT - by one small square.


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## Snelly (Aug 20, 2005)

DABurleigh said:


> Thank heavens I had the wisdom to avoid teaching; I'd be driven up the wall. I repeat - the area of both overall "triangles" (because they aren't) is DIFFERENT - by one small square.


HOW THOUGH?? this has stumped me! (and Sal)


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## DABurleigh (May 9, 2005)

I've exhausted my repertoire of variants of "the bottom one is bigger". If you don't get it, give up and we'll all be happy.

Dave


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## Snelly (Aug 20, 2005)

DABurleigh said:


> I've exhausted my repertoire of variants of "the bottom one is bigger". If you don't get it, give up and we'll all be happy.
> 
> Dave


Im not quitting just yet. I have had my ruler out... the area taken up by both triangles is the same, the pieces are the same, but yet a mystery space appears on the bottom arrangement!


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## carol (May 9, 2005)

Well OK, Dave alerted me.... been watching the box...

Look at row 5, on lower one, look where it finishes on the slope.... ok see that, now look at the one above, the line is lower...hence that little bit all the way up, makes for one square.....

visual bit like those old things they used to do at school, put a letter | and on one add a v top and bottom, and on the other add a ^ and then see - one looks longer than the other.

Now I shall go back to watch the box, and thanks Dave for the exercise... 

Snelly, do you get it this time?

Carol


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## Snelly (Aug 20, 2005)

carol said:


> Well OK, Dave alerted me.... been watching the box...
> 
> Look at row 5, on lower one, look where it finishes on the slope.... ok see that, now look at the one above, the line is lower...hence that little bit all the way up, makes for one square.....
> 
> ...


Thank you Carol... I see what Dave was getting at now! It only subtle isn't it...

Dave, ur right.


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## carol (May 9, 2005)

Shane - he always is, glad I'm not married to him.....couldn't stand someone ALWAYS right....trouble is the one I do have, had a mother who told him he was PERFECT...sometimes even worse!!!

Ah well....

Carol


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## Snelly (Aug 20, 2005)

Buty hold on a minute... I cut out the shapes from the top one and was able to reproduce the lower triangle using the top shapes... the top shapes also fit perfectly over the bottom ones... If the triangles were different...?? oh my head!


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## artona (Jan 19, 2006)

Hi

Its something to do with the least/most efficient stacking married to the turquoise triangle rising at a steeper gradient than the red one

stew


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## carol (May 9, 2005)

You're missing the point - the triangles are NOT the same

Carol


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## 88781 (May 9, 2005)

S'easy Shane,. look carefully at the hypotenuse in Dave's illustration,... ignore the triangle but look at the difference in amount of each square covered by colour, it adds up to one square, (the 'missing' one) :lol: :lol:


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## artona (Jan 19, 2006)

Hi

Lets try a different theory. Suppose the triangles are exactly the same. In both set ups one triangle is to the left and one to the right, one higher, one lower, so in effect the height and the width remain constant, regardless of which way around they are. 

In diagram 2 (the lower one) the torquoise triangle uses less floor space, releasing that space to make way for another square. In office design this could mean an extra desk

Not saying my theory is right, its just an alternative explanation


stew


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## Steptoe (Nov 8, 2005)

I've struggled with this for several minutes, at 6am  

Eventually I put a ruler on the slope (even though I haven't got a flat screen monitor :lol: ), as Dave says the extra square is accommodated by bowing the slope in and out as appropriate.

Good one though, by one of those coincidences grandaughter asked for help with homework last night, the class are doing compound shapes, so will print it out & get her to take it to school to check on the teacher's ability :wink:


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## peedee (May 10, 2005)

What your saying is the bottom pic is not really a triangle, the slope is really an arc. Therefore the pieces are not really identical, they just look it. If this is not the case then I don't get it either.

peedee


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## DABurleigh (May 9, 2005)

Neither pic is a triangle (I said this earlier). That's because the slope of the blue triangle is different to the red triangle (said that, too.). Therefore neither "hypotenuse" of the overall pics is a straight line.

If the top pic is overlaid on the bottom pic, a sliver of the bottom "hypotenuse" will show. The area of this sliver is equal to one square.

"Therefore the pieces are not really identical, they just look it."
If by pieces you mean the individual coloured shapes, they ARE identical.

Dave


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## Snelly (Aug 20, 2005)

DABurleigh said:


> Neither pic is a triangle (I said this earlier). That's because the slope of the blue triangle is different to the red triangle (said that, too.). Therefore neither "hypotenuse" of the overall pics is a straight line.
> 
> If the top pic is overlaid on the bottom pic, a sliver of the bottom "hypotenuse" will show. The area of this sliver is equal to one square.
> 
> ...


I agree Dave, thank you and sorry


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## DABurleigh (May 9, 2005)

Nothing to apologise for. It's a puzzle and a bit of teasing banter.


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## peedee (May 10, 2005)

Thanks for the clarification Dave, I can rest easy now  
peedee


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