Question Video: Finding the Coordinates of a Square after a Reflection in the 𝑥-Axis Mathematics

Video Transcript

A square 𝐴𝐵𝐶𝐷 is drawn on the coordinate plane. Given the coordinates 𝐴 one, one and 𝐶 three, three, which of the following represents the coordinates of the shape when it is reflected in the 𝑥-axis? Option (A) 𝐴 prime negative one, one; 𝐵 prime negative three, one; 𝐶 prime negative three, three; and 𝐷 prime negative one, three. Option (B) 𝐴 prime negative three, one; 𝐵 prime negative one, one; 𝐶 prime negative one, three; and 𝐷 prime negative three, three. Option (C) 𝐴 prime one, negative three; 𝐵 prime three, negative three; 𝐶 prime three, negative one; and 𝐷 prime one, negative one. Option (D) 𝐴 prime three, one; 𝐵 prime three, three; 𝐶 prime one, three; and 𝐷 prime one, one. Or option (E) 𝐴 prime one, negative one; 𝐵 prime three, negative one; 𝐶 prime three, negative three; and 𝐷 prime one, negative three.

We begin this question by noting that we are told there is a square 𝐴𝐵𝐶𝐷. And we are given the coordinates of two points, 𝐴 and 𝐶. So, let’s draw a coordinate grid and mark on the two points 𝐴 and 𝐶. Because this is a square and the vertices must be labeled in order, we know that the other two points must be plotted at the coordinates one, three and at three, one. There are no other coordinates such that we could create a square from these two points, remembering that there must be four sides congruent and four angles congruent.

We can join the points to create the square 𝐴𝐵𝐶𝐷 like this. However, we don’t know which of these coordinates will be 𝐵 and which will be 𝐷. The vertices of the square may be labeled in a clockwise direction, like this, or in a counterclockwise direction, like this. It is often more common to have the points labeled in a counterclockwise direction, but we can’t be sure. However, we are given five different answer options which we can use to help us. So, let’s consider what the four points on the square would look like when the reflection is performed.

We are told that this is a reflection in the 𝑥-axis. So, when we reflect point 𝐴 in the 𝑥-axis, we obtain the image 𝐴 prime at the coordinates one, negative one. The image of 𝐶, which we can call 𝐶 prime, is at the coordinates three, negative three. And the other two points can be reflected to their images at the coordinates three, negative one and at one, negative three. This would create the image, square 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime.

The only answer option that has the same coordinates for 𝐴 prime and 𝐶 prime is answer option (E). The other two coordinates that we determined of three, negative one and one, negative three match the given points in answer (E) for the coordinates of 𝐵 prime and 𝐷 prime, respectively.

Therefore, we can say that the coordinates of the reflected square 𝐴𝐵𝐶𝐷 are 𝐴 prime at coordinates one, negative one; 𝐵 prime at three, negative one; 𝐶 prime at three, negative three; and 𝐷 prime at the coordinates one, negative three.