# Question Video: Transforming a Shape Using Its Coordinates Mathematics

Find the image of square π΄π΅πΆπ·, where π΄(1, 3), π΅(3, 3), πΆ(3, 1), and π·(1, 1), after a geometric transformation (π₯, π¦) β (βπ¦, π₯).

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### Video Transcript

Find the image of square π΄π΅πΆπ·, where π΄ has coordinates one, three; π΅ has coordinates three, three; πΆ 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; πΆ prime negative three, one; and π· prime negative one, one. Option (B) π΄ prime is one, negative three; π΅ prime three, negative three; πΆ prime three, negative one; and π· prime one, negative one. Option (C) π΄ prime is three, negative one; π΅ prime three, negative three; πΆ prime one, negative three; and π· prime one, negative one. Or option (D) π΄ prime is negative three, one; π΅ prime negative three, three; πΆ prime negative one, three; and π· prime negative one, one.

Weβre 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΅πΆπ·, 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, πΆ 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 πΆ with π₯-coordinate three and π¦-coordinate one is mapped to the point πΆ 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; πΆ 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΅πΆπ· by plotting its coordinates and the coordinates of its image on an π₯π¦-coordinate plane. First, letβs plot the coordinates of π΄π΅πΆπ·, which are π΄ one, three; π΅ three, three; πΆ 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.