Point Reflection — Definition, Formula & Examples

Point reflection is a transformation that flips every point of a figure through a single fixed point, called the center. Each image point lands on the opposite side of the center, the same distance away.

A point reflection about center $P$ maps each point $A$ to the point $A'$ such that $P$ is the midpoint of $\overline{AA'}$ . The center maps to itself. A point reflection is an isometry, so it preserves distances and angles. It is equivalent to a rotation of $180^\circ$ about $P$ .

Key Formula

(x,\,y) \;\longmapsto\; (2a - x,\; 2b - y)

Where:

$(x, y)$ = coordinates of the original point
$(a, b)$ = coordinates of the center of reflection
$(2a - x,\, 2b - y)$ = coordinates of the image point

Worked Example

Problem: Reflect the point

A(3, 5)

through the center

P(1, 2)

. Find the image

A'

Step 1: Find the new x-coordinate. Use

x' = 2a - x

with

a = 1

and

x = 3

x' = 2(1) - 3 = -1

Step 2: Find the new y-coordinate. Use

y' = 2b - y

with

b = 2

and

y = 5

y' = 2(2) - 5 = -1

Step 3: Check that the center

P

is the midpoint of

A

and

A'

\left(\frac{3 + (-1)}{2},\; \frac{5 + (-1)}{2}\right) = (1,\, 2)

Answer: The image is

A'(-1, -1)

Why It Matters

Figures that look the same after a point reflection have point symmetry, like the letters S and N or a parallelogram about the intersection of its diagonals. In Algebra 2, odd functions such as

y = x^3

have graphs that are symmetric under point reflection through the origin, which is why

f(-x) = -f(x)

Common Mistakes

Mistake: Using

x' = a - x

or simply negating the coordinates when the center is not the origin.

Correction: The rule

(x, y) \to (-x, -y)

works only when the center is

(0, 0)

. For a center

(a, b)

, you must travel past the center by the same distance, which gives

(2a - x, 2b - y)

. Check your answer: the center should be the midpoint of the point and its image.

Related Terms

Reflection — a closely related concept
Axis of Reflection (Line of Reflection) — Reflection across a line instead of a point
Center of Rotation — Point reflection is a 180° rotation about it
Glide Reflection — Another isometry built from a reflection
Image of a Transformation — Names the reflected point or figure
Dilation of a Geometric Figure — Point reflection equals dilation with scale factor −1