Reflection Geometry: Line of Reflection and Coordinate Rules

A reflection in geometry flips a figure across a line called the line of reflection. Each point and its image sit the same perpendicular distance from that line, which is the perpendicular bisector of the segment joining them. On a coordinate plane: over the x-axis, (x, y) maps to (x, −y); over the y-axis, (x, y) maps to (−x, y); over y = x, swap to (y, x). Distance and size stay fixed, but the mirror flip reverses orientation.

What Is a Reflection in Geometry?

A reflection in geometry is a flip: the figure turns over across the line of reflection, also called the mirror line, like your hand looking reversed in a mirror. The original figure is the pre-image, labeled ABC, and its flipped copy is the image, labeled A′B′C′ with prime marks. Math is Fun uses the same mirror intuition: fold the paper along the mirror line and each point stacks onto its image, and prime notation keeps the two figures apart.

Reflections belong to the family of geometric transformations, the topic mapped in the geometry concepts guide. This guide covers line reflections in two dimensions only. Rotations, translations, dilations, point reflection through a center, and three-dimensional mirrors are separate topics; symmetry as a property of a shape is mentioned here only to say it is not covered.

The Line of Reflection: The Mirror Between a Figure and Its Image

The line of reflection is a straight line; the lines in geometry guide treats lines themselves as an object of study, while this page only needs that a line can act as the mirror. Its key property is that it is the perpendicular bisector of every segment joining a point to its image. Two practical facts follow. Each point and its image sit the same perpendicular distance from the line, and a point that already lies on the line reflects onto itself without moving.

To plot an image, drop a perpendicular from the point to the mirror line, measure that distance, and mark the image the same distance on the opposite side. Khan Academy teaches this perpendicular equal-distance method, including a worked point reflected across a horizontal line. The perpendicular distance from a point to the line itself is the measurement covered in depth by the distance formula guide, which connects this perpendicular-bisector property to coordinate calculations.

Coordinate Rules for Reflections on the Coordinate Plane

On a coordinate plane, reflections become coordinate substitutions. The plane geometry guide explains the axes and ordered pairs; here the pre-image point is (x, y) and its image is (x′, y′). The four standard mirrors follow below, followed by the general method for any horizontal line.

Reflecting over the x-axis: (x, y) → (x, −y)

Reflecting over the x-axis flips the figure vertically across the horizontal axis. The x-coordinate does not move; the y-coordinate changes sign. A pre-image point (x, y) lands at (x, −y). A point already on the axis, with y = 0, stays fixed, because changing the sign of zero leaves zero. On a grid this rule checks itself: count the same number of rows above and below the axis.

Reflecting over the y-axis: (x, y) → (−x, y)

Reflecting over the y-axis flips the figure horizontally across the vertical axis. Now the y-coordinate stays fixed, and the x-coordinate changes sign: (x, y) maps to (−x, y). A point on the y-axis, with x = 0, does not move. The rows stay level while the columns reverse, which looks like turning the figure across the vertical center line of the grid.

Reflecting over y = x and y = −x

The diagonal mirrors require a swap. Across y = x, the two coordinates trade places, so (x, y) becomes (y, x). Across y = −x, the coordinates swap and both signs flip, so (x, y) becomes (−y, −x). The two cases are easy to confuse: over y = x only the order changes, while over y = −x the order changes and each coordinate takes the opposite sign. BYJU'S lists these four rules together in a coordinate-plane table, matching the axis rules above.

Reflecting over an arbitrary horizontal line, e.g. y = 4

A mirror that is not an axis needs the distance method rather than a sign shortcut. Take the horizontal line y = 4. Keep the x-coordinate, then move vertically to the matching point on the other side of the line. A point with y = 1 sits three units below the line, so its image sits three units above, at y = 7. For a general horizontal mirror y = k, the shortcut is y′ = 2k − y; when k = 4, that is y′ = 8 − y. The same perpendicular-measurement method works for slanted mirrors as well, by dropping the perpendicular to the line rather than counting along a grid row.

When you need the other transformations side by side, the geometry formulas reference collects coordinate and measurement formulas in one table.

What Stays the Same: Invariants of a Reflection

A reflection is a rigid, isometric motion. Distances, lengths, angles, shape, and size do not change, so the image is congruent to the pre-image; a triangle keeps its side lengths after the flip. What reverses is orientation: the image is a mirror flip. Over the y-axis that flip trades left and right; over the x-axis or a horizontal line it trades up and down. BYJU'S states the same result in coordinate terms, and Math is Fun notes that the image keeps the original size while reversing orientation. This orientation reversal is what separates a reflection from a translation, which slides a figure without flipping it.

Worked Coordinate Examples

Reflect P(2, 3) over the x-axis. Keep x = 2 and flip the sign of y, giving P′(2, −3). The point moves from three rows above the axis to three rows below it, matching the equal-distance property.

Reflect Q(−1, 4) over the line y = x. Swap the two coordinates, giving Q′(4, −1). The new x is the old y, and the new y is the old x, with no extra sign change, because this mirror does not flip signs.

Reflect R(−2, 1) over the horizontal line y = 4. Keep x = −2. The point sits three units below the line, so its image sits three units above it, at R′(−2, 7). Check the result: both points are three grid units from y = 4, one below and one above.

Common Mistakes to Avoid

• Dropping the sign flip. Over the x-axis, (2, 3) must become (2, −3), not stay at (2, 3). Forgetting the minus leaves the point on the same side of the axis.
• Swapping the wrong pair on the diagonals. Over y = x, only the order swaps. Over y = −x, swap and flip both signs. Writing (x, y) → (y, x) for the y = −x mirror is the most common diagonal slip.
• Measuring diagonally instead of perpendicularly. The equal distance is measured at right angles to the mirror line. Counting along a slanted line gives the wrong image even when the count itself looks correct.
• Confusing reflection with rotation or translation. A rotation and a translation both keep the figure's orientation; only a reflection flips it. Sliding a figure or turning it around a point does not produce a mirror image.

How to Practice Reflections

Build the rules in order. First reflect points over the x-axis and y-axis on a grid, plotting pre-image and image together and counting rows to check the equal-distance property. Next add the diagonal mirrors y = x and y = −x, swapping coordinates carefully. Finish with an arbitrary horizontal line such as y = 4, where you count the perpendicular distance and copy it to the other side. Sketch each result once; after a few examples, the coordinate rule should replace the drawing.

Learn Geometry with Gauth

The existing Gauth Geometry course includes Unit 21, Rigid Transformations and Dilations, with lessons on coordinate-plane transformations and reflecting figures, which matches this topic directly. To drill at your own level, open Gauth Math Courses and enter reflections and coordinate rules.

Frequently Asked Questions About Reflections

How is a reflection different from a rotation or translation? A reflection flips orientation: the figure lands across a mirror line as a mirror image. Over the y-axis that trades left and right; over the x-axis or a horizontal line it trades up and down. A translation slides the figure without turning or flipping it, and a rotation turns it around a fixed point; both keep the original orientation. Only the mirror flip reverses orientation, which is why reflected text reads backward.

Given the figure and its image, how do I find the line of reflection? Connect one matching pair of points, such as A and A′. The midpoint of that segment lies on the mirror line, and the line runs perpendicular to the segment. Repeat with a second pair to confirm the same line, or draw the perpendicular bisector with a straightedge and compass.

Sources and Review

• Math is Fun, Reflection, accessed 2026-10-28: mirror-line intuition, prime notation for pre-image and image, the equal-distance folding method, and the x-axis and y-axis rules.
• Khan Academy, Reflecting shapes, accessed 2026-10-28: line of reflection as a mirror swap, the perpendicular equal-distance method for points and arbitrary horizontal lines, and the perpendicular-bisector setup.
• BYJU'S, Reflections in the Coordinate Plane, accessed 2026-10-28: coordinate-rule table for the x-axis, y-axis, y = x, and y = −x, pre-image and image prime notation, and the congruent rigid-motion result.