Video Transcript
In the figure, line segment 𝐴𝐵
has been rotated 90 degrees counterclockwise about the origin. Is the length of the image of line
segment 𝐴 prime 𝐵 prime resulting from this transformation greater than, less
than, or the same as the length of line segment 𝐴𝐵?
In this problem, we are given the
original line segment 𝐴𝐵. It has been rotated by 90 degrees
counterclockwise about the origin to give us the image of line segment 𝐴 prime 𝐵
prime. We are asked to consider how the
length of the original line segment and its image compare. Is one longer than the other or are
they both the same? They do appear to be the same, and
there is a property that we can recall to back this up.
We can say that rotations are a
rigid transformation, which means that distances are preserved through the
transformation. So, if the length of line segment
𝐴𝐵 was 𝑥 length units, then the length of line segment 𝐴 prime 𝐵 prime would
also be 𝑥 length units. And if a polygon was rotated
instead, we would know that the lengths of the sides would all be the same between
those of the original shape and those of its image. A rotation does not make any
lengths larger or smaller. And if we wanted to be thorough, we
could confirm this by noting that the line segments are both the diagonals of a
rectangle with dimensions of two and three length units, so would be the same
length.
Therefore, we can give the answer
that the lengths of line segments 𝐴𝐵 and 𝐴 prime 𝐵 prime are the same.
