How do coordinates change under a reflection in an axis?

Reflected in the x axis, a point (x, y) becomes (x, −y): the ordinate changes sign. Reflected in the y axis it becomes (−x, y): the abscissa changes sign. The rule is simple — what changes is the coordinate measured perpendicular to the axis of reflection.

How is a reflection in a point different from a reflection in a line?

A reflection in a line mirrors the figure and reverses its orientation. A reflection in a point rotates it by 180° about that point, so the orientation is preserved. Under a reflection in a point S, every point and its image lie on opposite sides of S at equal distance — that is, S is the midpoint of AA′.

How do you translate a point by a vector?

Add the coordinates of the vector to those of the point: a translation by [a, b] takes (x, y) to (x + a, y + b). The reverse question — which point was taken to a given one — is answered by subtracting the same numbers, not by flipping a sign on one side.

Which figures have a centre of symmetry?

A circle, a square, a rectangle, a rhombus, a parallelogram and a regular hexagon all do. An equilateral triangle and a regular pentagon do not — they have axes of symmetry only. The general rule for regular polygons is that a centre of symmetry belongs to those with an even number of sides.

How do you construct the perpendicular bisector of a segment?

Open the compass to more than half the segment and strike an arc from one endpoint, then another from the other endpoint with the same opening. The arcs meet at two points, and the line through them is the perpendicular bisector: it passes through the midpoint and is perpendicular to the segment. The same construction finds the midpoint without measuring the length at all.

Intermediate level

Geometric transformations

Reflection in a line, reflection in a point, translation and rotation move a figure without changing its size — and each of them comes down to a single operation on coordinates. Plus two compass-and-straightedge constructions: the perpendicular bisector of a segment and the bisector of an angle.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

The coordinate planeTwo perpendicular axes turn the plane into a map on which every point has an address made of two numbers. The order of those numbers is part of the address, their signs name the quadrant, and a segment parallel to an axis is measured by subtraction. Plus the midpoint of a segment, which is the average of the endpoints.
AnglesAngles are measured in degrees: acute below 90°, right exactly 90°, obtuse above it. Meet the kinds of angle, complementary and supplementary pairs, the alternate and corresponding angles at two parallel lines, and the two sum rules — 180° in a triangle and (n − 2) · 180° in a polygon.

Where this is used

Real situations where you count exactly the way this lesson teaches:

2D graphics and moving a sprite
A game character drawn at (120, 80) is to move by the vector [45, −20]. The engine adds the coordinates and draws it at (165, 60). The same operation run backwards — from (165, 60) to the starting point — is a subtraction, and that is what an undo is built on.
Mirrored machining on a CNC
The program cutting the left-hand part has points (30, 12) and (55, 40). The right-hand, mirrored part comes from a reflection in the vertical axis: (−30, 12) and (−55, 40). No second program is needed, only a sign change in one coordinate of every point.
Printing and a mirrored plate
A flexographic plate has to be a mirror image of the print, because it meets the substrate image-side down. A caption with corners (0, 0) and (140, 25) mm goes onto the plate as (0, 0) and (−140, 25) mm — one reflection instead of redrawing by hand.
Wallpaper and tile patterns
A 20 cm tile whose motif is rotated by 180° matches itself in every position, because a 180° rotation is a reflection in the centre of the tile — the point (6, 4) goes to (14, 16) about the centre (10, 10). The tiler therefore never has to check which way up a tile goes.
Woodwork without a protractor
To mark the middle of a board 187 cm long there is no need to divide by two: two arcs struck from both ends with the same compass opening cross at two points, and the line through them lands exactly at the middle and is perpendicular to the edge.

All formulas

Reflection in the x axis
$S_{OX}: (x,\ y) \mapsto (x,\ -y)$
the ordinate changes sign
Reflection in the y axis
$S_{OY}: (x,\ y) \mapsto (-x,\ y)$
the abscissa changes sign
Reflection in the origin
$S_{O}: (x,\ y) \mapsto (-x,\ -y)$
the same thing as a 180° rotation
Reflection in a point S(p, q)
$S_{S}: (x,\ y) \mapsto (2p - x,\ 2q - y)$
S is the midpoint of AA′
Translation by the vector [a, b]
$T_{\vec{v}}: (x,\ y) \mapsto (x + a,\ y + b)$
read backwards it is a subtraction

A geometric transformation assigns to every point of the plane its image. The four transformations in this lesson are isometries — they preserve distances, so a figure and its image are congruent: the position changes, the size and the shape do not.

All four can be written as a single operation on coordinates, and that is the shortest route to the image of any point.

Reflection in a line

Reflecting in a line works like a mirror: the image lies on the other side, at the same distance, on the perpendicular to that line. For the coordinate axes this gives two formulas:

S_{OX}: (x,\ y) \mapsto (x,\ -y), \qquad S_{OY}: (x,\ y) \mapsto (-x,\ y)

The triangle ABC and its image under reflection in the x axis. Every vertex kept its abscissa and flipped the sign of its ordinate.

What changes is the coordinate measured perpendicular to the axis of reflection — a mirror in the horizontal axis swaps "up and down", one in the vertical axis swaps "left and right".

A figure has an axis of symmetry when it coincides with its own image: a square has four, a rectangle two, an equilateral triangle three, and a circle infinitely many.

Reflection in a point

A reflection in a point $S$ takes a point $A$ to the point $A'$ for which $S$ is the midpoint of $AA'$ . It is simply a rotation by $180^\circ$ about $S$ .

For the origin the formula is as short as it gets:

S_{O}: (x,\ y) \mapsto (-x,\ -y)

For an arbitrary point $S = (p,\ q)$ we use the midpoint formula read backwards:

S_{S}: (x,\ y) \mapsto (2p - x,\ 2q - y)

Find the image of A(3, −5) under the reflection in S(−1, 2).

A figure has a centre of symmetry when it coincides with its image under such a reflection — a circle, a square, a rectangle, a rhombus and a parallelogram do, an equilateral triangle does not.

Translation by a vector

A translation moves every point by the same segment in the same direction. That segment with a direction is a vector, and its coordinates are simply added:

T_{\vec{v}}: (x,\ y) \mapsto (x + a,\ y + b), \qquad \vec{v} = [a,\ b]

A′(4, 1) is the image of A under a translation by [6, −3]. Find A.

That "backwards" question is the trickiest thing in the topic: the wording of the exercise looks the same and the operation is the opposite one.

Rotation

A rotation by an angle $\alpha$ about a point $O$ moves every point along a circle centred at $O$ . Two cases have formulas simple enough to be worth knowing:

Rotation about $O$	Image of $(x,\ y)$
$90^\circ$ counter-clockwise	$(-y,\ x)$
$180^\circ$	$(-x,\ -y)$
$270^\circ$ counter-clockwise	$(y,\ -x)$

A rotation by $180^\circ$ is the same transformation as a reflection in the centre of rotation — not two operations, but two names for one.

Compass-and-straightedge constructions

Two classical constructions need no arithmetic at all and give an exact result rather than a measured one.

The perpendicular bisector of a segment is the line perpendicular to it through its midpoint. Strike arcs of equal radius (greater than half the segment) from both endpoints and join the points where they cross.

The crossing points of the arcs are equidistant from A and from B, so the line through them is the set of all such points.

The bisector has a property that makes it useful beyond drawing: it consists of exactly those points that are equidistant from the two endpoints.

An angle bisector cuts an angle into two equal angles. Strike an arc from the vertex, then two arcs of equal radius from the points where it meets the arms.

The bisector of a 70° angle splits it into two angles of 35°. It consists of exactly those points that are equidistant from both arms.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

image under reflection in the axis OX: A(−1, 3)

Common mistakes

Flipping the wrong coordinate — reflecting in the x axis changes the ordinate, because that is what measures the distance from that axis.
Adding the vector instead of subtracting it in a backwards question — the image comes from adding, so the original comes from subtracting.
Confusing a reflection in a point with one in a line — the first preserves the orientation of the figure, the second reverses it.
Giving an equilateral triangle a centre of symmetry — it has three axes of symmetry but no centre.
Using $(-x,\ -y)$ for a reflection in a point other than the origin — there the formula is $(2p - x,\ 2q - y)$ .
Striking arcs of different radii in the bisector construction — the radius has to be the same from both endpoints, or the crossings are not equidistant from them.

Formula card

Topic: Geometric transformations

Reflection in the x axis
$S_{OX}: (x,\ y) \mapsto (x,\ -y)$
the ordinate changes sign
Reflection in the y axis
$S_{OY}: (x,\ y) \mapsto (-x,\ y)$
the abscissa changes sign
Reflection in the origin
$S_{O}: (x,\ y) \mapsto (-x,\ -y)$
the same thing as a 180° rotation
Reflection in a point S(p, q)
$S_{S}: (x,\ y) \mapsto (2p - x,\ 2q - y)$
S is the midpoint of AA′
Translation by the vector [a, b]
$T_{\vec{v}}: (x,\ y) \mapsto (x + a,\ y + b)$
read backwards it is a subtraction

A triangle and its image under reflection in the x axis. The ordinates change sign, the abscissas stay put.

Constructing the perpendicular bisector: two arcs of the same radius struck from both endpoints determine a line through the midpoint, perpendicular to the segment.

Frequently asked questions

A luminous right triangle with squares built on its sides drawn as a constellation, beside a circle with its radius, against a cosmic nebula in violet and magenta

Geometry

Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min