Video Transcript
If we rotated line segment 𝐴𝐵
about the origin by an angle of 90 degrees, what would its final position be? Option (A) 𝐴 prime with
coordinates negative one, negative two; 𝐵 prime with coordinates negative three,
negative four. Option (B) 𝐴 prime with
coordinates negative two, one; 𝐵 prime with coordinates negative four, three. Option (C) 𝐴 prime with
coordinates two, negative one; 𝐵 prime with coordinates four, negative three. Option (D) 𝐴 prime with
coordinates negative one, two; 𝐵 prime with coordinates negative three, four. Or option (E) 𝐴 prime with
coordinates one, negative two; 𝐵 prime with coordinates three, negative four.
In this question, we are asked
about a rotation of a line segment. And given that we have the angle of
90 degrees with no direction given, we can recall that a positive degree value
indicates a counterclockwise rotation. Therefore, the rotation of the line
segment will be in this direction. We note the important information
that the center of rotation is given as the origin, which is the coordinates zero,
zero.
So let’s consider the rotation of
point 𝐴, which has coordinates one, two, first. It can be helpful to draw a line
from the center of rotation to the point we are rotating. A 90-degree counterclockwise turn
about the origin would take 𝐴 to this point. The image of 𝐴 will therefore be
at the point 𝐴 prime with coordinates negative two, one. Now let’s perform the same steps to
find the rotation of point 𝐵. We can draw a line segment from 𝐵
to the origin and then rotate this point counterclockwise through a 90-degree angle
measure, which would give us the image 𝐵 prime at the coordinates negative four,
three.
A good check on our answer is to
draw the new line segment 𝐴 prime 𝐵 prime and make sure that it looks correct. For example, we can see that the
lengths of the original line segment and the new line segment are both the same. And that’s a good thing, because in
a rotation we know that the properties of lengths are all preserved. The answer is therefore that given
in option (B).
However, if we want to check our
answers for the coordinates of 𝐴 prime and 𝐵 prime, there is another method we
could use. This method has the advantage that
we don’t need to draw out the rotations we are completing. It is the property that a rotation
of 90 degrees counterclockwise about the origin is equivalent to the coordinate
transformation such that the coordinates 𝑥, 𝑦 map to the coordinates negative 𝑦,
𝑥. However, it’s worth really
highlighting that we can only use this property when the rotation is about the
origin.
So let’s take the coordinates one,
two for point 𝐴. The first step is to switch the 𝑥-
and 𝑦-coordinates. And then the new 𝑥-coordinate is
the negative of the original 𝑦-coordinate, which gives us the complete coordinates
of negative two, one for the point 𝐴 prime, which confirms our first answer for
this point. In the same way, we can take the
original coordinates of 𝐵. We switch the 𝑥- and
𝑦-coordinates. And then, the new 𝑥-coordinate is
the negative of the original 𝑦-coordinate. Therefore, we can give the answer
that after the rotation of line segment 𝐴𝐵, in its final position, the coordinates
of 𝐴 prime are negative two, one and the coordinates of 𝐵 prime are negative four,
three.
