# Triangle with extra area?

1. Jan 26, 2005

### RandallB

Triangle with extra area???

Four puzzle pieces; two right triangles and two six sided polygons with all right angles.
The two six sided polygons can form a perfect 12 by 12 square as shown figure1.
They are put together in two different ways to create triangular shapes of exactly the same height and width (fig 1 & 2).
BUT, some extra uncovered area, a 4 by 4 square, shows up in fig 2!

*Where does this extra space come from???
*You can show you know the solution with one descriptive word.

Without a drawing function it's hard to show in 'text drawing' but with the dimensions given above and figures below it should be easy to make four paper cutouts to help solve if needed.

Last edited: Jan 26, 2005
2. Jan 26, 2005

### dextercioby

SLOPE

THEY ARE NOT TRIANGLES

Daniel.

3. Jan 27, 2005

### RandallB

WRONG
That does not describe the source the extra area.

4. Jan 27, 2005

### Pseudopod

Sure it explains it, just hard to do it in one word. The hypotenuse of the red triangle must have a different slope than the hypotenuse of the black triangle. Specifically, the red triangle's hypotenuse must be a steeper slope than the black triangle's. This would make the first giant "triangle" (really a quadrilateral) have a concave hypotenuse and the lower "triangle" have a convex hypotenuse, accounting for the extra area.

5. Jan 28, 2005

### gerben

here is a nice image:

6. Jan 29, 2005

### Gokul43201

Staff Emeritus
7. Jan 29, 2005

### dextercioby

Nice,Gokul,a page especially for u.Greg must have been in a very good mood that day... :tongue2:

Daniel.

8. Jan 29, 2005

### mattmns

9. Jan 29, 2005

### RandallB

Parallelogram

Give credit to Pseudopod for coming close by mentioning the word quadrilateral.

The two large Triangular Shapes, easily mistaken for triangles, are of course quadrilaterals. Overlaying the small quadrilateral on top of the large quadrilateral will expose another quadrilateral shape as a visible portion from the larger .

Only this quadrilateral has two sides equal in length and both parallel to the hypotenuse of the small triangle and the other two are parallel with hypotenuse of the large triangle. Thus the best one word description of the source of the extra area:
PARALLELOGRAM

Which doesn’t directly give away the explanation,
but at least a less advanced poster was able to give the detailed description.

10. Jan 29, 2005

### Bartholomew

I disagree. Someone who did not understand the problem would not be helped by the word "parallelogram"--paralellogram where? A better answer is "hypotenuse." That at least identifies which region.

11. Jan 29, 2005

### Gokul43201

Staff Emeritus
PARALLELOGRAM would be a poor choice of word. If you take any pair of congruent triangles (like what someone might interpret the two figue=res to be, if he hasn't cracked the problem), flip one triangle over, and stick it on the other, you get a parallelogram.

I prefer SLOPES (the plural being important), "HYPOTENUSE" (within quotes to indicate it is not really), QUADRILATERAL (this is straightforward and requires no tricks), or even SIMILARITY (the fact that the three triangles - the two real ones, and the bogus one - are not similar is a sufficient observation).

12. Jan 31, 2005

### RandallB

None of these directly describes “Where the extra space comes from” those are hints to or gives away the solution that a triangular shape is not necessarily a triangle. And all require more explanation, thus none was given as one word alone.

However to just “show you know the solution” implies doing so without giving away the solution to others.
Done to often by those that already know the answers and want to show off.

Yet once they do solve, they couldn't doubt someone clearly knew that had given the one word.
PARALLELOGRAM
Yet it does the least to give away the subtle misdirection of the puzzle.

PS: Gerben, thanks for the link

Last edited: Jan 31, 2005
13. Jan 31, 2005

### K.J.Healey

"Parallelogram" does not describe teh solution or problem at all... You DO know what a parallelogram IS right?

I think the best answer is "There is no extra area."

14. Jan 31, 2005

### Problem+Solve=Reason

Ratio....

The hypotenuse length of the overall triangle (all shapes put together) is equal to the sum of the hypotenuse length on the II triangle and the base length on the 111 shape.

Because the 00 triangle is 4 spaces shorter in hypotenuse length and height (as seen) than the II triangle, the 4x4 block must be inserted for the pieces to fit correctly, yielding the same measurements as the first whole triangle.

____________________________________________
In seeking wisdom thou art wise; in imagining that thou hast attained it - thou art a fool.
Lord Chesterfield

15. Feb 1, 2005

### RandallB

Never said it described the solution! It describes how figure 2 is larger than figure 1.
Figure 2 has to outline a larger area than figure 1 to allow for the 4x4 hole.
Did you and Lord Chesterfield try cutting out the four shapes?
Your both wrong, just trace them on paper if you have trouble seeing it.

16. Feb 1, 2005

### K.J.Healey

What do you mean Im wrong? I understand completely where the hole comes from. Neither of these two shapes are triangles, I don't see why people would have any trouble seeing this at all, unless the image that shows it is distorted or misleading.

17. Feb 2, 2005

### RandallB

So your saying you finally did find the Parallelogram?
You DO know what a parallelogram IS right?

Follow the link for a better photo, but if you need to SEE the Parallelogram
cut out the four shapes.

18. Feb 2, 2005

### Integral

Staff Emeritus
A parallelogram, by definition has parallel sides that shape is NOT a parallelogram. Nor is it a Triangle. It is a Quadrilateral, It has four sides none of which are parallel to any other.

19. Feb 2, 2005

### Gokul43201

Staff Emeritus
RandallB never claimed that the provided shape wa a parallelogram. I think what he said was that if the two shapes are superimposed, the bigger one will have an excess area that is the shape of a parallelogram. This excess parallelogram has the same area as the removed square.

20. Feb 2, 2005

### K.J.Healey

Ahh, I didn't follow what he was trying to say. I thought he meant somehow it all had to do with laws of parallelograms, not that it would generate that shape... I see now.