### Video Transcript

Find the image of square π΄π΅πΆπ·,
where π΄ has coordinates one, three; π΅ has coordinates three, three; πΆ has
coordinates three, one; and π· has coordinates one, one, after a geometric
transformation that maps π₯, π¦ onto negative π¦, π₯. Option (A) π΄ prime is negative
one, three; π΅ prime negative three, three; πΆ prime negative three, one; and π·
prime negative one, one. Option (B) π΄ prime is one,
negative three; π΅ prime three, negative three; πΆ prime three, negative one; and π·
prime one, negative one. Option (C) π΄ prime is three,
negative one; π΅ prime three, negative three; πΆ prime one, negative three; and π·
prime one, negative one. Or option (D) π΄ prime is negative
three, one; π΅ prime negative three, three; πΆ prime negative one, three; and π·
prime negative one, one.

Weβre given the rule that describes
this transformation. Every point π₯, π¦ is mapped to the
point negative π¦, π₯. To find out which of the options
given represents the image of square π΄π΅πΆπ·, we substitute the coordinates of each
of its vertices into the rule of transformation to get the coordinates of the image
with vertices π΄ prime, π΅ prime, πΆ prime, and π· prime. The point π΄ with coordinates one,
three is mapped to the point π΄ prime. The coordinates of π΄ prime are
found by changing the sign of the π¦-coordinate and then swapping it with the
π₯-coordinate. We let π₯ equal one and π¦ equal
three. We change the π¦-coordinate to
negative three then swap positions with the π₯-coordinate. So, the coordinates of π΄ prime are
negative three, one.

According to the same rule, the
point π΅ with π₯-coordinate three and π¦-coordinate three is mapped to the point π΅
prime with coordinates negative three, three. The point πΆ with π₯-coordinate
three and π¦-coordinate one is mapped to the point πΆ prime with coordinates
negative one, three. And finally, the point π· with
π₯-coordinate one and π¦-coordinate one is mapped to the point π· prime with
coordinates negative one, one. Therefore, the vertices of the
image have coordinates π΄ prime negative three, one; π΅ prime negative three, three;
πΆ prime negative one, three; and π· prime negative one, one, which is option
(D).

We can demonstrate what
transformation has taken place on square π΄π΅πΆπ· by plotting its coordinates and
the coordinates of its image on an π₯π¦-coordinate plane. First, letβs plot the coordinates
of π΄π΅πΆπ·, which are π΄ one, three; π΅ three, three; πΆ three, one; and π· 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 π¦-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 π΄, 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.