Lesson Video: Reflections on the Coordinate Plane Mathematics

Video Transcript

In this video, we will learn how to find the images of points, lines, and shapes after their reflection in the 𝑥- or 𝑦-axis on the coordinate plane.

Reflection is a type of geometric transformation. Geometric transformations are processes that map one geometric figure, such as a point, line segment, or shape, onto another. In the case of a reflection, a geometric figure, which we call the object, is mapped onto a congruent figure called the image. We will consider three types of reflections here: firstly, reflection in the 𝑥-axis; secondly, reflection in the 𝑦-axis; and finally, reflection in the origin. We’ll begin with an example of identifying a reflection in the 𝑥-axis.

Which pair of triangles represents a reflection in the 𝑥-axis?

To answer this question, we need to imagine placing a mirror along the 𝑥-axis. When we do this, all objects will be reflected onto the opposite side of the mirror. Objects that were originally above the mirror line will be reflected below it. And objects that were originally below the mirror line will be reflected above it. The image of each object will be the same size as the original. And each vertex will be the same distance away from the mirror, just in the opposite direction. By visualizing the effect reflection in the 𝑥-axis will have, we can see that triangles 𝐴 and 𝐵 represent a reflection in the 𝑥-axis.

We can also look more closely at each vertex. Taking triangle 𝐴 to be the object, we can see that the vertex negative two, two is originally two units above the 𝑥-axis. The image of this vertex is at negative two, negative two. The vertex is in the same horizontal position but is now two units below the 𝑥-axis. So it’s the same vertical distance from the mirror, but in the opposite direction. The same is true for each of the other two vertices. The pair of triangles that represents a reflection in the 𝑥-axis is 𝐴 and 𝐵.

In the previous example, we identified that the point negative two, two was mapped to the point negative two, negative two by reflection in the 𝑥-axis. For the other two vertices of the triangle, the point negative three, five was mapped to negative three, negative five. And the point negative five, three was mapped to negative five, negative three. We can observe a pattern in these mappings. In each case, the 𝑥-coordinate is unchanged, but the 𝑦-coordinate has changed sign or, in other words, has been multiplied by negative one.

We can express this as a general result. A reflection in the 𝑥-axis maps a point 𝑃 with coordinates 𝑥, 𝑦 to the point 𝑃 prime with coordinates 𝑥, negative 𝑦. Recalling this result gives a useful method for finding the image of a point under reflection in the 𝑥-axis without needing to perform the transformation graphically.

Next, let’s consider what happens when we reflect an object in the 𝑦-axis.

Find the coordinates of the images of the points 𝐴, 𝐵, 𝐶, and 𝐷 after reflection in the 𝑦-axis.

To reflect these points in the 𝑦-axis, we need to imagine placing a mirror along the 𝑦-axis. The mirror is vertical, so the effect of reflecting in this line will be horizontal. Any point in the coordinate plane will be reflected onto the opposite side of the mirror and will appear the same distance behind the mirror as it originally was in front of the mirror.

Let’s start with point 𝐴, which has coordinates eight, six. This point is eight units to the right of the mirror. So its image will appear eight units to the left of the mirror at the same vertical height. The image of point 𝐴 will therefore be negative eight, six.

Point 𝐵 has coordinates eight, one and is directly below point 𝐴. The image of point 𝐵 will therefore be directly below the image of point 𝐴, so with an 𝑥-coordinate of negative eight, and will have the same 𝑦-coordinate as the original point 𝐵. The image of point 𝐵 is the point negative eight, one.

The coordinates of points 𝐶 and 𝐷 are two, one and two, six, respectively. Each of these points is two units to the right of the mirror. So their image will be two units to the left of the mirror. The images of points 𝐶 and 𝐷 are therefore at negative two, one and negative two, six, respectively. If we were to join the images of the four vertices together, we can see that the shape created is congruent to the original rectangle 𝐴𝐵𝐶𝐷.

It isn’t immediately obvious because this shape is symmetrical, but its orientation has also changed. The coordinates of the images of points 𝐴, 𝐵, 𝐶, and 𝐷 are negative eight, six; negative eight, one; negative two, one; and negative two, six.

Let’s now generalize what we saw in the previous example. Looking at the coordinates of each point and its image following reflection in the 𝑦-axis, we can observe that the 𝑦-coordinate remains the same. But this time the 𝑥-coordinate changes sign. Equivalently, we can say that the 𝑥-coordinate is multiplied by negative one. We can write a general rule to describe this. A reflection in the 𝑦-axis maps a point 𝑃 with coordinates 𝑥, 𝑦 to a point 𝑃 prime with coordinates negative 𝑥, 𝑦. This result enables us to find the coordinates of the image of a point under reflection in the 𝑦-axis without performing the transformation graphically.

Now that we’ve seen examples of reflections in both the 𝑥- and 𝑦-axes, let’s see how we can use this to solve problems.

Three points 𝐴, 𝐵, and 𝐶 with coordinates one, three; one, two; and four, one, respectively, are reflected in the 𝑥-axis to the points 𝐴 prime, 𝐵 prime, and 𝐶 prime. Determine the coordinates of 𝐴 prime, 𝐵 prime, and 𝐶 prime. Is the measure of angle 𝐴𝐵𝐶 less than, greater than, or equal to the measure of angle 𝐴 prime 𝐵 prime 𝐶 prime?

We recall first that a reflection in the 𝑥-axis maps a point with coordinates 𝑥, 𝑦 to the point with coordinates 𝑥, negative 𝑦. The 𝑥-coordinate stays the same, and the 𝑦-coordinate is multiplied by negative one. We can then apply this transformation to each point separately. Point 𝐴, which has coordinates one, three, is mapped to the point one, negative three. Point 𝐵 with coordinates one, two is mapped to one, negative two. And point 𝐶 with coordinates four, one is mapped to four, negative one.

To answer the second part of the question, it may be helpful to sketch the points 𝐴, 𝐵, and 𝐶 together with their images 𝐴 prime, 𝐵 prime, and 𝐶 prime on a coordinate grid. The angles we’re interested in are angle 𝐴𝐵𝐶 and angle 𝐴 prime 𝐵 prime 𝐶 prime, which are marked on the figure. We can see that these are both obtuse angles, which appear to be of equal measure. If we recall that reflections map a geometric figure to a congruent geometric figure, then we can deduce that angles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime must be of equal measure, as they are corresponding angles in congruent triangles.

So we’ve completed the problem. We’ve found the coordinates of 𝐴 prime, 𝐵 prime, and 𝐶 prime and determined that the two angles are of equal measure.

This example demonstrates one of the key properties of reflection in a line, which we’ll now consider in more detail.

There are four key properties of reflection in a line. Firstly, reflection in a line preserves the lengths of line segments. For example, if the line segment 𝐴𝐵 is reflected in the 𝑦-axis, its image 𝐴 prime 𝐵 prime is of equal length. Secondly, it preserves the measures of angles, as we saw in the previous example. Thirdly, reflection in a line preserves the betweenness property. This means that if point 𝐵 lies between points 𝐴 and 𝐶, then its image lies between the images of points 𝐴 and 𝐶. Finally, reflection in a line preserves parallelism. If the line segment 𝐴𝐵 is parallel to the line segment 𝐶𝐷, then the line segment 𝐴 prime 𝐵 prime is parallel to 𝐶 prime 𝐷 prime.

The final type of reflection we’re going to consider here is reflection in a point, which is a little different to reflecting in a line such as the 𝑥- or 𝑦-axis. Suppose we have an object 𝐴 that we want to reflect in a point 𝑀. The image of point 𝐴 is the point 𝐴 prime such that 𝑀 is the midpoint of the line segment connecting 𝐴 and 𝐴 prime. The point 𝑀 is called the center of reflection and is the only point that is unchanged under this particular reflection. On a coordinate grid, the only point we need to be able to reflect objects in is the origin.

Let’s consider what happens to this triangle when we reflect it in the origin. Taking each vertex in turn, we can draw the line connecting this point to the origin. We then find the point on the opposite side of the origin such that the origin is exactly halfway between this point and the original. We could do this either by measuring or by considering that in this case the original point is one square right and two squares up from the origin. So, if we do this in reverse and go one square left and two squares down from the origin, the distance from the origin will be the same.

We can repeat this process for each vertex and then join the three points together to give the image of the orange triangle following reflection in the origin. Considering the coordinates of each point and its image, we can observe that in every case, both the 𝑥- and 𝑦-coordinates have changed sign. We can generalize this result as reflection in the origin maps a point 𝑃 with coordinates 𝑥, 𝑦 to a point 𝑃 prime with coordinates negative 𝑥, negative 𝑦.

Reflection in the origin has an interesting property that differs from reflection in a line, which will become apparent if we label corresponding vertices in the two triangles in our figure with letters. Note that in the original triangle 𝐴𝐵𝐶, the vertices are labeled in clockwise order, and the same is true of the image 𝐴 prime 𝐵 prime 𝐶 prime. This leads to an important property. Reflection about the origin preserves orientation of the vertices. It’s important to understand that we don’t mean the orientation of the shape itself is preserved. But the labeling of the vertices in a clockwise or counterclockwise direction is preserved.

Let’s now look at an example of reflection in the origin.

Which graph represents reflecting the triangle 𝐴𝐵𝐶 about the origin?

We’re asked about the effect of reflecting triangle 𝐴𝐵𝐶 in a point, which is the origin. So triangle 𝐴𝐵𝐶 is the object, and we need to determine which of the four triangles is the correct image for this transformation.

We recall that reflection about the origin maps a general point 𝑃 with coordinates 𝑥, 𝑦 to the point 𝑃 prime with coordinates negative 𝑥, negative 𝑦. Or in other words, 𝑃 and 𝑃 prime have the same coordinates but with opposite signs. The two points are the same distance from the origin, but on opposite sides of it in a straight line.

We can see immediately then that graph (A) cannot be the correct graph, because whilst the 𝑥-coordinates have changed sign, the 𝑦-coordinates have not. Also, if we draw straight lines from each vertex of triangle 𝐴𝐵𝐶 to be the origin and continue these lines, we can see that the image of triangle 𝐴𝐵𝐶 after reflection about the origin should be in the third quadrant. For this reason, we can actually eliminate both (A) and (B) because the image is in the incorrect quadrant. In graphs (C) and (D), however, both images are in the third quadrant.

We can now recall a key property of reflection in the origin, which is that orientation of the vertices is preserved. This means that if the vertices of the object are labeled in, for example, the clockwise direction, the same will be true for the vertices of the image. Looking at triangle 𝐴𝐵𝐶, we can see that the ordering of the vertices is in fact counterclockwise, so the same must be true for its image.

On graph (C), the direction of labeling is also counterclockwise. But on graph (D), the direction is clockwise. This allows us to rule option (D) out. Only graph (C) remains. And we can check that this is indeed correct by connecting corresponding vertices and observing that they are indeed equal distance from the origin but on opposite sides in a straight line. So option (C) is the correct graph to represent a reflection of triangle 𝐴𝐵𝐶 about the origin.

Let’s now summarize the key points from this video. A reflection on the coordinate plane transforms a geometric figure, such as a point, line segment, or shape, to a congruent geometric figure, which we call the image. For a general point 𝑃 with coordinates 𝑥, 𝑦, reflection in the 𝑥-axis maps the point 𝑃 to the point 𝑃 prime with coordinates 𝑥, negative 𝑦. Or in other words the 𝑦-coordinate changes sign. Reflection in the 𝑦-axis changes the sign of the 𝑥-coordinate. So the point 𝑃 is mapped to the point 𝑃 prime with coordinates negative 𝑥, 𝑦.

Reflection in the origin changes the sign of both the 𝑥- and 𝑦-coordinates. The point 𝑃 is mapped to the point 𝑃 prime with coordinates negative 𝑥, negative 𝑦. Reflection in a line preserves the lengths of line segments, the measures of angles, the betweenness property, and parallelism. Reflection about the origin preserves the orientation of vertices, labeled in either a clockwise or counterclockwise direction.