Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem13

Level 5&6 Video Solutions 2015 problem13

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Question number 13. Which of these five nets cannot be the net of the pyramid?  And here we have a hint. A net is a flat shape that can be folded up to form a three-dimensional object.  Or, following this backwards, once we have a three-dimensional object, and let's say we can imagine it made out of paper,  unfolding this into a flat sheet of paper would produce a net, so a diagram like this.  So our job is to determine which of these blueprints, if you will, cannot be folded back up to produce a pyramid.  Now let's imagine a pyramid viewed from the top, looking at it from quite far away. What we would see is a square base, and then we would see the four sides all at the same time,  and they would be appearing to cross like this, and the point where they cross, that represents the apex of the pyramid.  And so let's look at each of our diagrams over here, and we can see pretty clearly in diagram A that we can fold up these sides in and produce the top view here that I have drawn.  So I can shade in these sides to make that a little clearer. The sides in red would respond to those sides when viewed from the top,  and then the sides in white would be responding to the sides in white. So that is possible, and let's put a checkmark here by A. In B, we have to think about this a little harder.  If we were to take this entire piece consisting of four triangles and fold it up, what we would have is the triangle all shaded in red from whence the arrow emanates.  That would be corresponding to this piece. On the sides, we would have a wraparound of the orange triangles,  and finally the white triangle we can actually wrap all the way around and position it here on the bottom side, if you will, of the pyramid. So that is also possible.  Let's put a checkmark here by B, and let's move on to C. What happens in C? Let me move this picture so we can better understand what's going on.  In C, we have two initial folds to make. This triangle would be folded inward, so it would be that one, and those then in red would have the following correspondence.  Red, red corresponds to red. We can fold in one of the remaining triangles here,  and that one likewise, and what we would have is the following correspondence. That orange triangle represents the orange face here on the bottom,  but it also so happens that the bottom triangle, once folded in, goes in the same direction and overlays on top of the orange side here, face of the pyramid that I have shaded,  and there is really nothing in that net that will allow us to take care of the white face. So we have not completed the pyramid.  The orange triangles in C, because of a lack of symmetry, they fold on top of one another, and we have here a missing face of the pyramid.  So C does not represent a valid net for a pyramid, and that is our answer here at number 13, C.

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