Welcome to today's mathematics lesson. We are going to explore Reflection in coordinate geometry. This is a fascinating topic in coordinate geometry where we discover how points transform when flipped across lines and points. By the end of this lesson, you will understand reflection in the x-axis, y-axis, the origin, and in vertical and horizontal lines. You will also learn about invariant points — special points that refuse to change.
Let us begin with the foundation. In coordinate geometry, we locate points using two perpendicular number lines called coordinate axes. The horizontal line is the x-axis, and the vertical line is the y-axis. They intersect at the origin, denoted as (0, 0).
Any point P in the plane is written as (x, y). The first number, x, is called the abscissa — it tells us how far left or right the point lies from the y-axis. The second number, y, is called the ordinate — it tells us how far up or down the point lies from the x-axis. Remember, the abscissa always comes first, followed by the ordinate.
Now, what exactly is reflection? Imagine standing before a mirror. Your image appears behind the mirror, at the same distance as you stand in front of it. In mathematics, reflection works the same way.
Given a point P and a line AB, the reflection of P in AB is a point P' on the opposite side of AB, such that AB is the perpendicular bisector of the segment PP'. In other words, AB cuts PP' at right angles, exactly at its midpoint. We call AB the mirror line.
We denote reflection using the symbol M with a subscript indicating the mirror. So Mₓ means reflection in the x-axis, Mᵧ means reflection in the y-axis, and M₀ means reflection in the origin.
Let us start with reflection in the x-axis, which is the line y = 0.
When we reflect a point (x, y) in the x-axis, something interesting happens. The x-coordinate stays exactly the same, but the y-coordinate changes its sign. The image becomes (x, -y).
Symbolically, Mₓ(x, y) = (x, -y). So the rule is: reflection in the x-axis changes the sign of the ordinate.
Let us see this in action. Take the point (2, 3). Its reflection in the x-axis is (2, -3). Similarly, Mₓ(2, -3) gives us back (2, 3). And Mₓ(-5, 7) equals (-5, -7).
Next, reflection in the y-axis, which is the line x = 0.
Here, the pattern reverses. When we reflect (x, y) in the y-axis, the y-coordinate remains unchanged, but the x-coordinate changes its sign. The image becomes (-x, y).
Symbolically, Mᵧ(x, y) = (-x, y). The rule is: reflection in the y-axis changes the sign of the abscissa.
For example, Mᵧ(2, 3) equals (-2, 3). And Mᵧ(-4, -5) gives us (4, -5).
Now comes reflection in the origin — the most dramatic transformation of all.
When we reflect a point in the origin, both coordinates change their signs. The point (x, y) transforms to (-x, -y).
Symbolically, M₀(x, y) = (-x, -y). Both the abscissa and ordinate flip.
Let us verify: M₀(2, 3) equals (-2, -3). And M₀(2, -3) equals (-2, 3). Notice that reflecting in the origin is equivalent to reflecting first in the x-axis and then in the y-axis, or vice versa.
Let us work through a complete example to see how these transformations combine.
Consider triangle ABC with vertices at (1, 2), (4, 4), and (3, 7). First, we reflect this triangle in the x-axis, which is the line y = 0, to get triangle A'B'C'. Then we reflect triangle A'B'C' in the origin to get triangle A''B''C''.
For the first reflection, using Mₓ(x, y) = (x, -y): A at (1, 2) goes to A' at (1, -2). B at (4, 4) goes to B' at (4, -4). C at (3, 7) goes to C' at (3, -7).
For the second reflection, using M₀(x, y) = (-x, -y) on triangle A'B'C': A' at (1, -2) goes to A'' at (-1, 2). B' at (4, -4) goes to B'' at (-4, 4). C' at (3, -7) goes to C'' at (-3, 7).
Notice something remarkable: A'' is simply the reflection of A in the y-axis. This makes sense because reflecting in the x-axis then the origin is equivalent to a single reflection in the y-axis.
Now we turn to a special concept: invariant points.
An invariant point is any point that remains unchanged under a given transformation. In other words, the point is its own image.
Consider the point (5, 0). When reflected in the x-axis, the y-coordinate becomes minus zero, which is still zero.
So the point remains (5, 0). So (5, 0) is invariant under reflection in the x-axis.
Similarly, any point on the x-axis — that is, any point of the form (a, 0) — is invariant under reflection in the x-axis. Likewise, any point on the y-axis — any point (0, b) — is invariant under reflection in the y-axis.
The origin (0, 0) is truly special: it is invariant under reflection in the x-axis, the y-axis, and the origin itself.
Here is the key principle: a point is invariant under reflection in a line if and only if it lies on that line. The line is its own mirror, so every point on it stays fixed.
Finally, let us explore reflection in vertical and horizontal lines other than the axes.
Consider reflection in the vertical line x = a. This line is parallel to the y-axis, passing through all points where the x-coordinate equals a.
To find the image of a point P, we draw a perpendicular from P to the line x = a, and extend it equally far on the other side. If P has coordinates (x, y), and the line is x = a, then the image P' has coordinates (2a - x, y).
Let us verify with an example. Reflect the point (-1, 3) in the line x = 2. Here, a equals 2 and x equals minus 1. So the image has x-coordinate 2 × 2 − (−1), which equals 5. The y-coordinate stays 3. Thus P' is at (5, 3).
Similarly, for reflection in the horizontal line y = a, the rule is: the x-coordinate stays fixed, while the y-coordinate becomes 2a - y.
Take the point (2, 1) reflected in the line y = -3. Here, a equals minus 3 and y equals 1. The new y-coordinate is 2 × (−3) − 1, which equals minus 7. The image is at (2, -7).
Let us solidify our understanding with another worked example.
A point P is reflected in the x-axis, and its image is (8, -6). We need to find P, and then find the image of P under reflection in the y-axis.
Since reflection in the x-axis changes the sign of the y-coordinate, the original point P must have y-coordinate 6. The x-coordinate stays 8. Therefore, P is at (8, 6).
Now reflecting P in the y-axis: we change the sign of the x-coordinate. The image is at (-8, 6).
Here is one more elegant problem combining invariant points.
We are told that points (-5, 0) and (4, 0) are invariant under reflection in line L₁. Since invariant points must lie on the mirror line, L₁ must be the x-axis, with equation y = 0.
Similarly, points (0, -6) and (0, 5) are invariant under reflection in line L₂. These both lie on the y-axis, so L₂ is the line x = 0.
Now, reflecting (2, 6) in L₁, which is the x-axis, gives (2, -6). Reflecting the same point in L₂, the y-axis, gives (-2, 6).
Observe what happens if we reflect point Q' at (-8, 3) in the origin. We get (8, -3), which is Q''. This single transformation maps Q' directly to Q''.
Let us recap the key takeaways from this lesson.
First, reflection in the x-axis, given by Mₓ(x, y) = (x, -y), changes the sign of the ordinate while keeping the abscissa fixed.
Second, reflection in the y-axis, given by Mᵧ(x, y) = (-x, y), changes the sign of the abscissa while keeping the ordinate fixed.
Third, reflection in the origin, given by M₀(x, y) = (−x, −y), changes both signs.
Fourth, a point is invariant under reflection in a line if and only if it lies on that line.
Fifth, reflection in the vertical line x = a maps (x, y) to (2a - x, y).
Sixth, reflection in the horizontal line y = a maps (x, y) to (x, 2a - y).
Reflection is a beautiful symmetry operation that appears throughout mathematics and nature. Master these rules, practice with coordinates, and you will find that what once seemed mysterious becomes perfectly clear. Keep exploring, keep questioning, and I look forward to our next mathematical journey together.