Question Video: Transforming a Shape Using Its Coordinates Mathematics

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.