Video Transcript
Find the image of square 𝐴𝐵𝐶𝐷,
where 𝐴 has coordinates one, three; 𝐵 has coordinates three, three; 𝐶 has
coordinates three, one; and 𝐷 has coordinates one, one, after a geometric
transformation that maps 𝑥, 𝑦 onto negative 𝑦, 𝑥. Option (A) 𝐴 prime is negative
one, three; 𝐵 prime negative three, three; 𝐶 prime negative three, one; and 𝐷
prime negative one, one. Option (B) 𝐴 prime is one,
negative three; 𝐵 prime three, negative three; 𝐶 prime three, negative one; and 𝐷
prime one, negative one. Option (C) 𝐴 prime is three,
negative one; 𝐵 prime three, negative three; 𝐶 prime one, negative three; and 𝐷
prime one, negative one. Or option (D) 𝐴 prime is negative
three, one; 𝐵 prime negative three, three; 𝐶 prime negative one, three; and 𝐷
prime negative one, one.
We’re given the rule that describes
this transformation. Every point 𝑥, 𝑦 is mapped to the
point negative 𝑦, 𝑥. To find out which of the options
given represents the image of square 𝐴𝐵𝐶𝐷, we substitute the coordinates of each
of its vertices into the rule of transformation to get the coordinates of the image
with vertices 𝐴 prime, 𝐵 prime, 𝐶 prime, and 𝐷 prime. The point 𝐴 with coordinates one,
three is mapped to the point 𝐴 prime. The coordinates of 𝐴 prime are
found by changing the sign of the 𝑦-coordinate and then swapping it with the
𝑥-coordinate. We let 𝑥 equal one and 𝑦 equal
three. We change the 𝑦-coordinate to
negative three then swap positions with the 𝑥-coordinate. So, the coordinates of 𝐴 prime are
negative three, one.
According to the same rule, the
point 𝐵 with 𝑥-coordinate three and 𝑦-coordinate three is mapped to the point 𝐵
prime with coordinates negative three, three. The point 𝐶 with 𝑥-coordinate
three and 𝑦-coordinate one is mapped to the point 𝐶 prime with coordinates
negative one, three. And finally, the point 𝐷 with
𝑥-coordinate one and 𝑦-coordinate one is mapped to the point 𝐷 prime with
coordinates negative one, one. Therefore, the vertices of the
image have coordinates 𝐴 prime negative three, one; 𝐵 prime negative three, three;
𝐶 prime negative one, three; and 𝐷 prime negative one, one, which is option
(D).
We can demonstrate what
transformation has taken place on square 𝐴𝐵𝐶𝐷 by plotting its coordinates and
the coordinates of its image on an 𝑥𝑦-coordinate plane. First, let’s plot the coordinates
of 𝐴𝐵𝐶𝐷, which are 𝐴 one, three; 𝐵 three, three; 𝐶 three, one; and 𝐷 one,
one. Then, we join up the vertices with
edges to obtain a sketch of the original square. If we also plot the coordinates of
the image, then we get the following. After joining up the vertices with
edges, we obtain a sketch of the image.
At first glance, this may appear to
be a reflection in the 𝑦-axis. However, if we look carefully at
the vertices, we see that they had been rotated 90 degrees counterclockwise about
the origin. We have marked the rotation applied
to vertex 𝐴, and we could do the same for the other three vertices. In fact, we can remember that this
transformation rule will always give a 90-degree rotation about the origin.
