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Reflection over a Line

How a Mirror Line Pairs Points

In a line reflection, the $mirror line$ is the symmetry boundary between a point and its image. A point on the mirror stays fixed. A point outside it maps to the opposite side at the same perpendicular distance.

If lies on the mirror, . Otherwise, draw the perpendicular from to the mirror and call its foot . Continue the segment an equal distance beyond the mirror to locate the image :

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Reflection over a Line

How a Mirror Line Pairs Points

If lies on the mirror, . Otherwise, draw the perpendicular from to the mirror and call its foot . Continue the segment an equal distance beyond the mirror to locate the image :

P

P'=P

P

M

P'

PM=MP',\qquad PP'\perp m

Reflecting a Point across an Oblique Line

Reflect $P(2,4)$ across $m:y=\frac12x+1$ . The $midpoint$ $M$ lies on the mirror, while the segment joining the point to its image is perpendicular to the mirror.

Point Reflection over a Line

Point

P(2,4)

is reflected over line

m

to produce the image

P'(3.6, 0.8)

Finding the Perpendicular Foot

The mirror slope is $\frac12$ . The $perpendicular slope$ is therefore $-2$ , since their product is $-1$ . The perpendicular through $P(2,4)$ has equation:

y-4=-2(x-2)\quad\Longrightarrow\quad y=-2x+8

At the intersection, both equations give the same vertical coordinate:

\frac12x+1=-2x+8\quad\Longrightarrow\quad\frac52x=7\quad\Longrightarrow\quad x=\frac{14}{5}

Finding and Checking the Image

Since $M$ is the $segment midpoint$ , double its coordinates and subtract the coordinates of $P$ to find the image:

P'=2M-P=\left(2\cdot\frac{14}{5}-2,2\cdot\frac{12}{5}-4\right)=\left(\frac{18}{5},\frac45\right)

The distances to the mirror are the segment lengths to $M$ . This calculation checks the equal lengths visible in the diagram:

PM=\sqrt{\left(\frac{14}{5}-2\right)^2+\left(\frac{12}{5}-4\right)^2}=\sqrt{\frac{16+64}{25}}=\frac{4\sqrt5}{5}

Reflection Matrix over Center Point

Reflection over Point

Reflection over a LineConstruct a reflection across any line with perpendicular bisectors and equal distances.

On this page

How a Mirror Line Pairs Points
Reflecting a Point across an Oblique Line
- Finding the Perpendicular Foot
- Finding and Checking the Image

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Reflection over a Line

Published: May 10, 2025. Updated: Sep 8, 2026.