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Lesson Explainer: Reflections on the Coordinate Plane Mathematics • 11th Grade

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

Recall that a geometric transformation is a process that maps one geometric figure (such as a point, line segment, or shape) onto another.

Definition: Transformation

A transformation transforms every point 𝑃 in a plane onto an image point 𝑃′ in the same plane.

Some well-known types of transformation are translations, rotations, and reflections. We will only discuss reflections here.

A reflection on the coordinate plane takes a geometric figure such as a point, line segment, or shape and transforms it into a congruent geometric figure called the image. In this explainer, we will focus on three different types of reflection on the coordinate plane:

  1. Reflection in the π‘₯-axis
  2. Reflection in the 𝑦-axis
  3. Reflection about the origin

We will look at each of these in turn and work through some accompanying examples.

Let us start with reflection in the π‘₯-axis. In what follows, we will use the notation 𝑃(π‘₯,𝑦) to refer to a general point on the coordinate plane.

Definition: Reflection in the π‘₯-Axis

A reflection in the π‘₯-axis maps 𝑃(π‘₯,𝑦)→𝑃′(π‘₯,βˆ’π‘¦).

In other words, a reflection in the π‘₯-axis maps point 𝑃 onto its image point 𝑃′ by keeping the π‘₯-coordinate the same and changing the sign of the 𝑦-coordinate. The effect is that the position of 𝑃′ will mirror that of 𝑃, on the opposite side of the π‘₯-axis (which is the line with equation 𝑦=0) and at the same perpendicular distance from the π‘₯-axis as 𝑃. Since the π‘₯-axis acts like a mirror, it is called the line of reflection (or mirror line).

For example, the diagram below shows point 𝐴(2,3) reflected in the π‘₯-axis to give its image 𝐴′(2,βˆ’3). Similarly, point 𝐡(βˆ’2,βˆ’1) has the image 𝐡′(βˆ’2,βˆ’(βˆ’1))=𝐡′(βˆ’2,1).

Note however that any point on the π‘₯-axis would remain unchanged under this reflection, because it lies on the mirror line.

If we have more complex geometric figures to reflect, such as line segments or polygons, the simplest way to do this is to reflect each individual endpoint or vertex. Then, we connect the image points in the same sequence as they appear in the original figure. The result is that the geometric figure will be transformed into its congruent mirror image on the opposite side of the mirror line.

Let us now look at an example to test if we can recognize a reflection in the π‘₯-axis.

Example 1: Determining the Pair of Triangles That Represents a Reflection in the π‘₯-Axis

Which pair of triangles represents a reflection in the π‘₯-axis?

Answer

Recall that, for a general point 𝑃(π‘₯,𝑦), a reflection in the π‘₯-axis maps 𝑃(π‘₯,𝑦)→𝑃′(π‘₯,βˆ’π‘¦).

The effect is that the position of 𝑃′ will mirror that of 𝑃, on the opposite side of the π‘₯-axis and at the same perpendicular distance from the π‘₯-axis as 𝑃.

Since this rule applies to the vertices of triangles, then in the diagram above, we would expect the pair of reflected triangles to appear with one above the other, on opposite sides of the π‘₯-axis and at the same perpendicular distance from it.

If triangle 𝐢 was one of the reflected triangles, then the π‘₯-axis would act like a mirror to give the image shown below.

However, as there is no such triangle, then 𝐢 has not been reflected in the π‘₯-axis, so the reflected triangles must be 𝐴 and 𝐡. Although this should be obvious from the diagram, we will use the definition of a reflection in the π‘₯-axis to prove this.

Reading off from the diagram, we see that triangle 𝐴 has vertices with coordinates (βˆ’3,5),(βˆ’2,2), and (βˆ’5,3). Using the fact that 𝑃(π‘₯,𝑦)→𝑃′(π‘₯,βˆ’π‘¦), we find the image points as follows: (βˆ’3,5)β†’(βˆ’3,βˆ’5),(βˆ’2,2)β†’(βˆ’2,βˆ’2),(βˆ’5,3)β†’(βˆ’5,βˆ’3).

It is easy to check that these image points are precisely the vertices shown for triangle 𝐡. Moreover, if we had started with the vertices of triangle 𝐡, we would have found that the image points were the vertices of triangle 𝐴.

We conclude that the pair of triangles 𝐴 and 𝐡 represent a reflection in the π‘₯-axis.

Next, we shall examine reflection in the 𝑦-axis, which follows a very similar pattern to reflection in the π‘₯-axis.

Definition: Reflection in the 𝑦-Axis

A reflection in the 𝑦-axis maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,𝑦).

In other words, a reflection in the 𝑦-axis maps point 𝑃 onto its image point 𝑃′ by changing the sign of the π‘₯-coordinate and keeping the 𝑦-coordinate the same. The effect is that the position of 𝑃′ will mirror that of 𝑃, on the opposite side of the 𝑦-axis (which is the line with equation π‘₯=0) and at the same perpendicular distance from the 𝑦-axis as 𝑃. Since the 𝑦-axis acts like a mirror, it is called the line of reflection (or mirror line).

For example, the diagram below shows point 𝐴(βˆ’1,3) reflected in the 𝑦-axis to give its image 𝐴′(βˆ’(βˆ’1),3)=𝐴′(1,3). Similarly, point 𝐡(4,βˆ’2) has the image 𝐡′(βˆ’4,βˆ’2).

Note however that any point on the 𝑦-axis would remain unchanged under this reflection, because it lies on the mirror line.

Here is an example involving reflection in the 𝑦-axis.

Example 2: Finding the Coordinates of the Vertices of a Quadrilateral after Reflection

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

Answer

Recall that, for a general point 𝑃(π‘₯,𝑦), a reflection in the 𝑦-axis maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,𝑦).

This means it maps point 𝑃 onto the image point 𝑃′ by changing the sign of the π‘₯-coordinate and keeping the 𝑦-coordinate the same.

Here, our first step is to read off the coordinates of the vertices of the quadrilateral. We can then use the above transformation to work out the coordinates of the corresponding image points.

Observe that vertex 𝐴 has an π‘₯-coordinate of 8 and a 𝑦-coordinate of 6, so it can be written as the point 𝐴(8,6). Similarly, we get 𝐡(8,1), 𝐢(2,1), and 𝐷(2,6). Applying the transformation 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,𝑦) to all four vertices gives 𝐴(8,6)→𝐴′(βˆ’8,6),𝐡(8,1)→𝐡′(βˆ’8,1),𝐢(2,1)→𝐢′(βˆ’2,1),𝐷(2,6)→𝐷′(βˆ’2,6).

Remember that a reflection in the 𝑦-axis takes a geometric figure and maps it to its congruent mirror image on the opposite side of the 𝑦-axis, with each vertex of the image at the same perpendicular distance from the 𝑦-axis as the corresponding vertex of the original figure. Plotting the above image points and connecting them in the correct order, we get the image of the quadrilateral shown in the diagram below.

Hence, we have shown that the coordinates of the images of the points 𝐴,𝐡,𝐢, and 𝐷 after reflection in the 𝑦-axis are 𝐴′(βˆ’8,6), 𝐡′(βˆ’8,1), 𝐢′(βˆ’2,1), and 𝐷′(βˆ’2,6).

Now, we return to reflection in the π‘₯-axis. It is always important to read the question carefully so that we apply the correct transformation.

Example 3: Solving a Problem about Points Reflected in the π‘₯-Axis

Three points 𝐴,𝐡, and 𝐢 with coordinates (1,3),(1,2), and (4,1), respectively, are reflected in the π‘₯-axis to the points 𝐴′,𝐡′, and 𝐢′.

  1. Determine the coordinates of 𝐴′,𝐡′, and 𝐢′.
  2. Is the measure of angle 𝐴𝐡𝐢 less than, greater than, or equal to the measure of angle 𝐴′𝐡′𝐢′?

Answer

Recall that, for a general point 𝑃(π‘₯,𝑦), a reflection in the π‘₯-axis maps 𝑃(π‘₯,𝑦)→𝑃′(π‘₯,βˆ’π‘¦), so we obtain the image 𝑃′ from 𝑃 by keeping the π‘₯-coordinate the same and changing the sign of the 𝑦-coordinate.

Part 1

For the first part of the question, we must apply the above transformation to the points 𝐴(1,3), 𝐡(1,2), and 𝐢(4,1); this will give us the coordinates of the image points. Therefore, we have 𝐴(1,3)→𝐴′(1,βˆ’3),𝐡(1,2)→𝐡′(1,βˆ’2),𝐢(4,1)→𝐢′(4,βˆ’1), as required.

Part 2

For the second part, we need to compare the measure of angle 𝐴𝐡𝐢 with the measure of angle 𝐴′𝐡′𝐢′. It helps to sketch the situation, by drawing triangles for 𝐴𝐡𝐢 and 𝐴′𝐡′𝐢′ and highlighting the relevant angles, as shown in the diagram below.

Under reflection in the π‘₯-axis, triangle 𝐴′𝐡′𝐢′ is the mirror image of triangle 𝐴𝐡𝐢. Since βˆ π΄β€²π΅β€²πΆβ€² of triangle 𝐴′𝐡′𝐢′ corresponds to ∠𝐴𝐡𝐢 of triangle 𝐴𝐡𝐢, we conclude that the measure of angle 𝐴𝐡𝐢 is equal to the measure of angle 𝐴′𝐡′𝐢′.

Reflection in the π‘₯-axis and reflection in the 𝑦-axis are two particular examples of the most common type of reflection: reflection in a line. In all such cases, a straight line (called the line of reflection or mirror line) acts like a mirror so that any geometric figure on one side of the line will be transformed into its congruent mirror image on the other side of the line.

Furthermore, reflection in a line has four key properties, which we will illustrate through the diagram below.

Here, the quadrilateral 𝐴𝐡𝐢𝐷 has been reflected in the mirror line to give its image, the congruent quadrilateral 𝐴′𝐡′𝐢′𝐷′. Thus, the image 𝐴′𝐡′𝐢′𝐷′ has exactly the same shape and size as the original quadrilateral 𝐴𝐡𝐢𝐷 but has been flipped over, and each vertex of the image is at the same perpendicular distance from the mirror line as the corresponding vertex of the original quadrilateral.

Note also that point 𝐸, which lies on the mirror line, will be unchanged under this reflection, so its image will be itself. This is true in general of any points lying on a mirror line.

Properties: Reflection in a Line

  • Reflection preserves the lengths of line segments.
    For example, the length of 𝐴𝐡 is the same as the length of 𝐴′𝐡′.
  • Reflection preserves the measures of angles.
    For example, the interior angle measure at vertex 𝐴 is the same as the interior angle measure at vertex 𝐴′.
  • Reflection preserves the betweenness property.
    For example, as vertex 𝐡 lies between 𝐴 and 𝐢, then vertex 𝐡′ lies between 𝐴′ and 𝐢′.
  • Reflection preserves parallelism.
    For example, as 𝐡𝐢 is parallel to 𝐴𝐷, then 𝐡′𝐢′ is parallel to 𝐴′𝐷′.

The final type of reflection we consider here is reflection about the origin. This is different to the previous two types because it involves reflection about a point rather than a line. However, as we shall see, it is straightforward to define and describe. Recall again that we use the notation 𝑃(π‘₯,𝑦) to refer to a general point on the coordinate plane.

Definition: Reflection about the Origin

A reflection about the origin maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,βˆ’π‘¦).

In other words, a reflection about the origin maps point 𝑃 onto its image point 𝑃′ by changing the signs of both of its coordinates.

Another way to think of reflection about the origin is that it maps 𝑃 onto its image 𝑃′ so that the origin is the midpoint of 𝑃𝑃′. The effect is that 𝑃 and 𝑃′ will be at the same distance from the origin but on opposite sides of it in a straight line.

For example, the diagram below shows point 𝐴(3,1) reflected about the origin to give its image 𝐴′(βˆ’3,βˆ’1). Similarly, point 𝐡(βˆ’2,4) has the image 𝐡′(2,βˆ’4). The line segments between 𝐴 and 𝐴′, and between 𝐡 and 𝐡′, are drawn in to show that, in both cases, the origin is their midpoint.

Note that the only point to remain unchanged under this reflection is the origin itself, because this is the point about which all other points are reflected.

It is easiest to see how reflection about the origin works by trying an example.

Example 4: Finding the Coordinates of a Point after Reflection about the Origin

Which of the following represents the image of point 𝐴(1,3) after a reflection about the origin?

  1. 𝐴′(1,βˆ’3)
  2. 𝐴′(βˆ’1,3)
  3. 𝐴′(βˆ’3,βˆ’1)
  4. 𝐴′(βˆ’3,1)
  5. 𝐴′(βˆ’1,βˆ’3)

Answer

Recall that, for a general point 𝑃(π‘₯,𝑦), a reflection about the origin maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,βˆ’π‘¦).

This means it maps point 𝑃 onto its image point 𝑃′ by changing the signs of both of its coordinates.

Therefore, we must apply this transformation to point 𝐴(1,3) to obtain its image 𝐴′, as follows: 𝐴(1,3)→𝐴′(βˆ’1,βˆ’3).

We conclude that 𝐴′(βˆ’1,βˆ’3) represents the image of point 𝐴(1,3) after a reflection about the origin, so option E is the correct answer.

Reflection about the origin has an interesting property that makes it different from reflection in a line. This will become apparent when we look at its effect upon a triangle, as shown in the next diagram.

Notice that the original triangle 𝐴𝐡𝐢 has its vertex letter labels in clockwise order; the same is true of its image 𝐴′𝐡′𝐢′. So, in other words, the orientation of the vertices is preserved. This is different to what happens with reflection in a line, where the order of the vertices is preserved but the direction of labeling (i.e., clockwise or counterclockwise) is reversed.

Property: Reflection about the Origin

  • Reflection about the origin preserves orientation.
    For example, if the vertices of a polygon are labeled in clockwise order, then the corresponding vertex labels in the image will also be ordered clockwise.

Our final example will help us to recognize reflections about the origin.

Example 5: Reflecting a Triangle about the Origin

Which graph represents reflecting the triangle 𝐴𝐡𝐢 about the origin?

Answer

Recall that, for a general point 𝑃(π‘₯,𝑦), a reflection about the origin maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,βˆ’π‘¦).

Therefore, 𝑃′ will have the same coordinates as 𝑃 but with opposite signs, so 𝑃 and 𝑃′ will be at the same distance from the origin, but on opposite sides of it in a straight line. Recall also that reflection about the origin preserves orientation, while a reflection in the π‘₯-axis or 𝑦-axis does not.

Here, we need to examine the graphs one by one to find the reflection about the origin. In graph A, triangle 𝐴𝐡𝐢 and its image 𝐴′𝐡′𝐢′ are on opposite sides of the 𝑦-axis, with corresponding vertices at the same perpendicular distance from the 𝑦-axis. For every vertex of 𝐴𝐡𝐢, its image in 𝐴′𝐡′𝐢′ has the same 𝑦-coordinate but the π‘₯-coordinate has the opposite sign. We recognize this as a reflection of triangle 𝐴𝐡𝐢 in the 𝑦-axis.

In graph B, triangle 𝐴𝐡𝐢 and its image 𝐴′𝐡′𝐢′ are on opposite sides of the π‘₯-axis, with corresponding vertices at the same perpendicular distance from the π‘₯-axis. For every vertex of 𝐴𝐡𝐢, its image in 𝐴′𝐡′𝐢′ has the same π‘₯-coordinate but the 𝑦-coordinate has the opposite sign. We recognize this as a reflection of triangle 𝐴𝐡𝐢 in the π‘₯-axis.

In graph C, triangle 𝐴𝐡𝐢 and its image 𝐴′𝐡′𝐢′ are on opposite sides of the origin. Corresponding vertices are at the same distance from the origin but on opposite sides of it in a straight line. For every vertex of 𝐴𝐡𝐢, its image in 𝐴′𝐡′𝐢′ has the same coordinates but with opposite signs. Triangle 𝐴𝐡𝐢 is labeled counterclockwise and so is 𝐴′𝐡′𝐢′, so the orientation of the vertices is preserved as well as the order. We recognize this as a reflection of triangle 𝐴𝐡𝐢 about the origin, so graph C is the correct answer.

It is interesting to note that graph D actually shows triangle 𝐴𝐡𝐢 reflected in the downward-sloping line that passes through the origin at 45∘ to both the π‘₯-axis and 𝑦-axis. This line has the equation 𝑦=βˆ’π‘₯, and we sketch it below.

However, since reflections of this type go beyond the scope of this explainer, we do not explore it in any more detail here.

Thus, we have shown that graph C represents reflecting the triangle 𝐴𝐡𝐢 about the origin.

Let us finish by recapping some key concepts from this explainer.

Key Points

  • A reflection on the coordinate plane is a type of transformation. It takes a geometric figure such as a point, line segment, or shape and transforms it into a congruent geometric figure called the image.
  • When reflecting geometric figures such as line segments or polygons, the simplest way to do this is to reflect each individual endpoint or vertex and then connect the image points in the same sequence as they appear in the original figure.
  • For a general point 𝑃(π‘₯,𝑦), a reflection in the π‘₯-axis maps 𝑃(π‘₯,𝑦)→𝑃′(π‘₯,βˆ’π‘¦). The effect is that the position of 𝑃′ will mirror that of 𝑃, on the opposite side of the π‘₯-axis and at the same perpendicular distance from the π‘₯-axis as 𝑃.
  • For a general point 𝑃(π‘₯,𝑦), a reflection in the 𝑦-axis maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,𝑦). The effect is that the position of 𝑃′ will mirror that of 𝑃, on the opposite side of the 𝑦-axis and at the same perpendicular distance from the 𝑦-axis as 𝑃.
  • Reflection in a line preserves the lengths of line segments, the measures of angles, the betweenness property, and parallelism.
  • For a general point 𝑃(π‘₯,𝑦), a reflection about the origin maps 𝑃(π‘₯,𝑦)→𝑃′(βˆ’π‘₯,βˆ’π‘¦). The effect is that 𝑃 and 𝑃′ will be at the same distance from the origin, but on opposite sides of it in a straight line.
  • Reflection about the origin preserves orientation.

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