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Identify a pair of points between which diagonal of the cuboid can be drawn.
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Option (A) ๐ด and ๐ถ, option (B) ๐ธ and ๐ท, option (C) ๐น and ๐ถ, option (D) ๐บ and ๐ท, option (E) ๐ด and ๐บ.
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Here, we have a cuboid, which is often sometimes seen as a rectangular prism.
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When weโre asked to draw a diagonal of this cuboid, what weโre looking for is a line which passes through the interior of the cuboid, often called a space diagonal.
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Notice that itโs different to a face diagonal, which is a diagonal in two dimensions.
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So letโs say that the line is drawn between ๐ท and ๐ต.
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We have then created a diagonal of the plane ๐ด๐ต๐ถ๐ท.
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However, as this plane is in two dimensions, we have only created a face diagonal.
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Therefore, this line would not be a diagonal of the cuboid.
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In the same way, we could create the line between ๐ป and ๐ถ.
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However, once again, weโve found a diagonal of the plane, this time ๐ถ๐ท๐ป๐บ.
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And therefore, the diagonal would be a face diagonal but not a diagonal of the cuboid.
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So what would a diagonal of the cuboid actually look like?
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Letโs say we wanted to start at vertex ๐ป and create a diagonal from there.
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Drawing a line to vertex ๐ถ would give us a face diagonal.
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So instead, traveling in three dimensions through the interior of the cuboid would take us to vertex ๐ต.
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We can therefore say that ๐ป๐ต is the diagonal of the cuboid.
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A diagonal of the cuboid starting at vertex ๐ท and traveling through the interior of the cuboid would take us to vertex ๐น.
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And so ๐ท๐น is also a diagonal of the cuboid.
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In fact, there are a total of four space diagonals in a cuboid.
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Although itโs a little trickier to see on this diagram, the line joining ๐ถ and ๐ธ is a diagonal.
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And finally, ๐ด๐บ is our fourth diagonal.
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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), ๐ด and ๐บ.
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The other four options here would all create face diagonals.
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Itโs important to note that when weโre working with the diagonals of three-dimensional shapes, for example, if weโre using the Pythagorean theorem, then the face diagonals will be a different length to the space diagonals.
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For example, the diagonal ๐ต๐ป will be longer than the face diagonal ๐ต๐ธ.