Can This Be True?

[Feathery]

Can This Be True?

 

[drew70]

I've seen this before. It's a good brain buster. I finally figured out the answer, but being the first to respond I'll let others take a shot at it.

 

[drew70]

Well, since nobody else seems interested, I'll give it a shot.

Question: We have what appears to be two identical right triangles composed of the same four geometric parts. Yet the arrangement of the parts in the bottom triangle leave one block of space unaccounted for. Since the parts are all identical, this would seem to be a virtual impossiblity.

Answer: Neither the above nor the below composite polygons are actually triangles, even though they appear to be so. In each of these bogus triangles, the "hypoteneuse" is not a straight line. There's a nearly invisible angle where the red and the green triangles touch. In the top figure, it bends in slightly. In the bottom figure, it bends out slightly, so hence the extra space.

If you still insist they are true right triangles, let's add up the collective areas of the four parts and see if it matches the area of the composite "triangle."

green triangle = 5 blocks
red triangle = 12 blocks
green hexagon = 8 blocks
orange hexagon = 7 blocks
-------------------------
Total parts area = 32 blocks.
In the bottom triangle if you include the "hole" it comes to 33 blocks

Composite "triangle" is...let's see...5 x 13 = 65...divide by 2....uh oh....32.5 blocks! Close, but no cigar!

Bottom line: Though the two composite figures appear to be identically shaped right triangles, they are in fact, differently shaped quadrilaterals.

😀 😀 😀

 

[Feathery]

Well, since nobody else seems interested, I'll give it a shot.

Question: We have what appears to be two identical right triangles composed of the same four geometric parts. Yet the arrangement of the parts in the bottom triangle leave one block of space unaccounted for. Since the parts are all identical, this would seem to be a virtual impossiblity.

Answer: Neither the above nor the below composite polygons are actually triangles, even though they appear to be so. In each of these bogus triangles, the "hypoteneuse" is not a straight line. There's a nearly invisible angle where the red and the green triangles touch. In the top figure, it bends in slightly. In the bottom figure, it bends out slightly, so hence the extra space.

If you still insist they are true right triangles, let's add up the collective areas of the four parts and see if it matches the area of the composite "triangle."

green triangle = 5 blocks
red triangle = 12 blocks
green hexagon = 8 blocks
orange hexagon = 7 blocks
-------------------------
Total parts area = 32 blocks.
In the bottom triangle if you include the "hole" it comes to 33 blocks

Composite "triangle" is...let's see...5 x 13 = 65...divide by 2....uh oh....32.5 blocks! Close, but no cigar!

Bottom line: Though the two composite figures appear to be identically shaped right triangles, they are in fact, differently shaped quadrilaterals.

😀 😀 😀

Wow, Drew... I'm impressed! Are you a mathematician? Thanks for figuring this out!

 

[isabeau]

I've seen this before. It's a good brain buster. I finally figured out the answer, but being the first to respond I'll let others take a shot at it.

are you nuts???? i barely made it thru algebra two in highschool, no way was i going to try and tackle this problem, just the pictures brought back terrifying nightmares lololol

isabeau

 

[nowayjose]

Answer: Neither the above nor the below composite polygons are actually triangles, even though they appear to be so. In each of these bogus triangles, the "hypoteneuse" is not a straight line. There's a nearly invisible angle where the red and the green triangles touch. In the top figure, it bends in slightly. In the bottom figure, it bends out slightly, so hence the extra space.

True.. without actually calculating the lower left angles of the two small component triangles (which is left as an exercise to the reader, requires basic trigonometry), it can be seen simply by the ratios of the adjacent : opposite sides:
In case of the red small triangle, it is 8:3 (2.666), whereas in case of the green small triangle, it is 5:2 (2.5). This means that the red small triangle has a smaller angle than the green one.
When the red triangle is at the bottom left, and the green triangle at the top right (like in the first figure), we have first a smaller inclination (red) followed by a larger inclination (green), which effects the "hypotenuse" (the slanted side) of the composite triangle to "bend in". Whereas if we have first the green, then the red triangle (as in the second figure), we have a larger inclination followed by a smaller one, effectively making the "hypotenuse" of the composite triangle to "bend out", cleverly arranged by the creator of the puzzle to leave enough space for an empty field.

 

[Feathery]

Wow... I take my hat off to you all!! I wasn't the greatest in math either!!

 

[lk70]

What is truth?

 

[locker669]

What is truth?

Truth is those funny commercials about smoking cigarettes and doing drugs.


I would say the red triangle is not the same size in both diagrams.

 

[drew70]

Wow, Drew... I'm impressed! Are you a mathematician? Thanks for figuring this out!

You're welcome. And thanks for the compliment. I'm not a mathematician, but I did get straight A's in 10th-grade Geometry. Interestingly, there was only one other guy who aced the course. He and I were the biggest stoners in the class. Could there be a link between marijuana use and geometric comprehension? Might explain the triangle on the Dark Side of the Moon album. 😀

 

[aun_existe_amor]

I'm extremely impressed!!! 🙂