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Axis of Reflection — Definition, Examples & Formula

Axis of Reflection

The "mirror line" of a reflection. That is, the line across which a reflection takes place.

 

Two pentagons labeled "pre-image" and "image" facing each other, separated by a vertical "axis of reflection" line.

 

See also

Pre-image, image, transformation

Key Formula

If P=(a,b) is reflected over the y-axis, then P′=(−a,b).\text{If } P = (a, b) \text{ is reflected over the } y\text{-axis, then } P' = (-a, b).
Where:
  • PP = The original point (pre-image)
  • P′P' = The reflected point (image)
  • aa = The x-coordinate of the original point
  • bb = The y-coordinate of the original point

Worked Example

Problem: Reflect the point A(3, 2) over the line y = 1. Find the coordinates of the reflected point A'.
Step 1: Identify the axis of reflection. Here the mirror line is the horizontal line y = 1.
Step 2: Find the vertical distance from A to the axis. The y-coordinate of A is 2 and the axis is at y = 1, so the distance is 1 unit above the axis.
d=2−1=1d = 2 - 1 = 1
Step 3: Move the same distance to the other side of the axis. Since A is 1 unit above y = 1, the reflected point is 1 unit below y = 1.
y′=1−1=0y' = 1 - 1 = 0
Step 4: The x-coordinate stays the same because the axis of reflection is horizontal. Write the coordinates of A'.
A′=(3,0)A' = (3, 0)
Answer: The reflected point is A'(3, 0). Notice that both A and A' are exactly 1 unit from the axis y = 1, on opposite sides.

Another Example

Problem: Triangle PQR has vertices P(1, 4), Q(3, 4), and R(2, 6). Reflect the triangle over the x-axis and find the vertices of the image triangle P'Q'R'.
Step 1: The axis of reflection is the x-axis, which is the line y = 0. When reflecting over the x-axis, each point's x-coordinate stays the same and its y-coordinate changes sign.
P(x,y)→P′(x,−y)P(x, y) \rightarrow P'(x, -y)
Step 2: Reflect each vertex by negating its y-coordinate.
P(1,4)→P′(1,−4)P(1, 4) \rightarrow P'(1, -4)
Step 3: Apply the same rule to Q and R.
Q(3,4)→Q′(3,−4),R(2,6)→R′(2,−6)Q(3, 4) \rightarrow Q'(3, -4), \quad R(2, 6) \rightarrow R'(2, -6)
Answer: The reflected triangle has vertices P'(1, −4), Q'(3, −4), and R'(2, −6). The x-axis served as the axis of reflection, and every vertex is the same distance from the x-axis as its original, just on the opposite side.

Frequently Asked Questions

How do you find the axis of reflection between two figures?
Connect any point on the original figure to its corresponding point on the image. Find the midpoint of that segment. Repeat for at least one more pair of corresponding points. The axis of reflection is the line that passes through all those midpoints. This line is always the perpendicular bisector of every segment joining a point to its image.
Can the axis of reflection be diagonal or curved?
The axis of reflection can be any straight line — horizontal, vertical, or diagonal (such as y = x). However, it must be a straight line; a curved line would not produce a standard reflection. Common diagonal axes include y = x and y = −x.

Axis of Reflection vs. Axis of Symmetry

An axis of reflection is the line you choose to flip a figure across to create a separate image. An axis of symmetry is a line within a single figure that divides it into two mirror-image halves. If a figure is reflected over its own axis of symmetry, the image lands exactly on the original figure. So an axis of symmetry is a special case of an axis of reflection where the figure maps onto itself.

Why It Matters

Reflections are one of the four rigid transformations (along with translations, rotations, and glide reflections) used throughout geometry to prove congruence and analyze symmetry. Identifying the axis of reflection is essential when describing how a figure was transformed or when constructing reflected images on the coordinate plane. In real life, the concept appears in mirror optics, architectural design, and any situation involving bilateral symmetry.

Common Mistakes

Mistake: Reflecting by changing the wrong coordinate. For example, when reflecting over the x-axis, students sometimes negate the x-coordinate instead of the y-coordinate.
Correction: Reflection over the x-axis changes the sign of y: (x, y) → (x, −y). Reflection over the y-axis changes the sign of x: (x, y) → (−x, y). The coordinate that is perpendicular to the axis is the one that changes.
Mistake: Measuring the distance from the point to the axis along a direction that is not perpendicular to the axis.
Correction: The reflected point must be found by moving perpendicularly to the axis of reflection. The axis is the perpendicular bisector of the segment connecting each original point to its image. Always measure the shortest (perpendicular) distance.

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