Video: Identifying the Pair of Points between Which a Diagonal of the Cuboid Can Be Drawn

Identify a pair of points between which diagonal of the cuboid can be drawn. [A] ๐ด, ๐ถ. [B] ๐ธ, ๐ท. [C] ๐น, ๐ถ. [D] ๐บ, ๐ท. [E] ๐ด, ๐บ.

02:33

Video Transcript

Identify a pair of points between which diagonal of the cuboid can be drawn. Option (A) ๐ด and ๐ถ, option (B) ๐ธ and ๐ท, option (C) ๐น and ๐ถ, option (D) ๐บ and ๐ท, option (E) ๐ด and ๐บ.

Here, we have a cuboid, which is often sometimes seen as a rectangular prism. When weโ€™re asked to draw a diagonal of this cuboid, what weโ€™re looking for is a line which passes through the interior of the cuboid, often called a space diagonal. Notice that itโ€™s different to a face diagonal, which is a diagonal in two dimensions. So letโ€™s say that the line is drawn between ๐ท and ๐ต. We have then created a diagonal of the plane ๐ด๐ต๐ถ๐ท. However, as this plane is in two dimensions, we have only created a face diagonal. Therefore, this line would not be a diagonal of the cuboid. In the same way, we could create the line between ๐ป and ๐ถ. However, once again, weโ€™ve found a diagonal of the plane, this time ๐ถ๐ท๐ป๐บ. And therefore, the diagonal would be a face diagonal but not a diagonal of the cuboid.

So what would a diagonal of the cuboid actually look like? Letโ€™s say we wanted to start at vertex ๐ป and create a diagonal from there. Drawing a line to vertex ๐ถ would give us a face diagonal. So instead, traveling in three dimensions through the interior of the cuboid would take us to vertex ๐ต. We can therefore say that ๐ป๐ต is the diagonal of the cuboid. A diagonal of the cuboid starting at vertex ๐ท and traveling through the interior of the cuboid would take us to vertex ๐น.

And so ๐ท๐น is also a diagonal of the cuboid. In fact, there are a total of four space diagonals in a cuboid. Although itโ€™s a little trickier to see on this diagram, the line joining ๐ถ and ๐ธ is a diagonal. And finally, ๐ด๐บ is our fourth diagonal. Out of the answer options, (A) through (E), that we were given, the only one which is a pair of points between which a diagonal can be drawn is option (E), ๐ด and ๐บ. The other four options here would all create face diagonals.

Itโ€™s important to note that when weโ€™re working with the diagonals of three-dimensional shapes, for example, if weโ€™re using the Pythagorean theorem, then the face diagonals will be a different length to the space diagonals. For example, the diagonal ๐ต๐ป will be longer than the face diagonal ๐ต๐ธ.

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