Below is the answer for POTW #14:
Thursday, December 10, 2015
POTW #15 - Confusing Triangle
There is a math riddle involving the missing piece of the triangle below. The area seems to remain the same yet there is a piece missing when you reorganize the right triangle. BOTH triangles make a 13 x 5 shape. All you've done is reconfigure the parts of the triangle, the sizes of each piece have not changed. How can there be a missing space if it makes the same shape? Explain with details.
Below is the answer for POTW #14:
Below is the answer for POTW #14:
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When I see the original and final phases, I notice that the angles of the 2 triangles are different, and will allow for a more extended surface, potentially taking up the 1x1 cube's missing area
ReplyDeleteAlso, the blue triangle intersects into the top of the shape, which would not count as part of the shape due to the box limiting what we can view. This and the different angle will create enough of a increased the area due to the different angles on the shape.
DeleteIt might be like a tanagram, and just doesn't work sometimes. The green piece didn't move, they overlapped eachother, and the triangles are very different sizes.
ReplyDeleteThe whole triangle is made up of:
ReplyDeleteBlue Triangle = 5 x 2 = 10/2 = 5units2
Red Triangle = 8 x 3 = 24/2 = 12units2
Green Shape = 5 x 2 - 2 = 8units2
Yellow Shape = 5 x 2 - 3 = 7units2
= 32units2
Big Triangle = 13 x 5 = 65/2 = 32.5 units
How the missing square may work, is through that the area of the triangle first which is 32.5 when using the traditional formula, but when finding the area by calculating it through the each individual shape, the outcomes are different at 32. There is a 0.5 difference. That my tie into the reason why there is a open space.
I think that it might not be able to be arranged in another way because it was "cut up" in a specific way so that only a certain piece can go in a certain place. So, kind of like a tangram.
ReplyDeleteI agree with Veronica; it being cut like a tanagram. I cannot see that this could be solved in any other way, other that this: Even though it says in the question it is, maybe isn't the same shape and is slightly bigger.
ReplyDeleteCan I talk about this in class when we take this up? I talked it through with my Dad and I want people to see how we did this. There is a space because the area of the rectangle occupied by the green and yellow is different. In one case the area is 5 by 3 but when the triangles switch, the base and height are different. It is 8 by 2 instead of 5 by 3, so the area is 16 instead of 15. Also, the slopes of the triangles are different.
ReplyDeleteSure Caelan. That would be great. We can share it Wednesday or Thursday on math.
DeleteOne reason for this could be that it is a tangram, and that different figures must be placed where they are to make it fit. I noticed that the yellow figure fits with the triangle above and green figure below, but doesn't with the red triangle, as it's base is for more longer, as it is 8, compared to the green triangle's base of 9.
ReplyDeleteWhen tiling a plane you have to have specific places in which you put the shapes. Each shape has to have a relationship with the another shape in which they have to interlock. If they don't interlock properly than there is going to be a hole, yet it can still make the same shape because most of the shapes are arranged differently giving that you can try to make it work.
ReplyDeleteThe reason is that in a tan-gram, different shapes have different purposes in creating a shape. One possibility is that since the shape is shrinking in size, 2 shapes may fit perfectly in one area with a small width, but when moved down the width the big for the shapes. Another reason is that the big triangle on the top could have been placed in an area where it most effective
ReplyDeleteThere is still be a missing pieces in the triangle because this triangle is designed in a way where all pieces MUST be arranged in an certain order or else you can not make the shape without a missing space, you may be able to make it look like the original shape but it would still be missing a piece.
ReplyDeleteWhen looking closely at the shape before and after, I could tell that the angle of the triangle 1 was different from the angle of triangle 2. Triangle 2 looked like it was bulging, and the longest side of the triangle didn't even look straight, unlike the first triangle. I guess that the angle change would be the cause of the extra missing square.
ReplyDeleteIf I was a bit more analytical about the picture than before i answered this question, the second shape is accurate in lines, but the first shape was a little bit under the line, so i would assume that the rearrangement showed the total area that the line was under. for example, (5,2) clearly showed in picture 1, how it is untouched by the shape, but picture 2 has clearly reached the point.
ReplyDeleteDo mind my poor explanation, i am not much of a visual person.
Because this image is on a screen, it may be harder to tell. But with observation, you can notice that the missing gap has to come out of somewhere. You may think that the blue and red triangles are similar triangle, by they are not. In fact the square unit of missing space is missing from the slant of the triangle. The ratio of the slants for the 2 triangles are different. The whole shape isn't even a triangle, just a messed up quadrilateral. The missing space is distributed evenly on the slant to make it look as less suspicious as possible.
ReplyDeleteI found that the first triangular shapes were slightly below the lines on the chart, and the second red triangle was bulging and didn't look quite straight. I figured that the second picture had that extra space, from the space between the the first two triangles and the line above them.
ReplyDelete