Reflection (Geometry)

Sommaire

In short: Reflection is a flip over a chosen line that sends each point to a partner on the opposite side at an equal distance, so the image is a mirror twin of the figure.

8th GradeGeometry8.G.A.1

What is Reflection?

Stand a figure in front of a mirror and its reflection appears behind it. Every point is paired with an image point across the mirror line, equally far away and joined to it by a segment that meets the line at a right angle.

The two coordinate axes make handy mirrors. A flip over the y-axis sends \((x, y)\) to \((-x, y)\), and a flip over the x-axis sends \((x, y)\) to \((x, -y)\). Sizes and angles survive, but orientation reverses: vertices read clockwise become counterclockwise.

How to spot it

The wording says flip, mirror image, or reflect over the x-axis, the y-axis or some other line.
A point lying on the mirror line stays put while all other points swap sides.
A picture shows a figure and a twin facing opposite ways with a dashed line between them.
A letter or shape looks unchanged in a mirror, which hints at symmetry.

The method

Name the mirror: the y-axis, the x-axis or another line.
Over the y-axis, negate each x-coordinate; over the x-axis, negate each y-coordinate.
Plot the images and join them in the order used for the original corners.
Confirm each point and its partner are equally far from the mirror and joined by a segment perpendicular to it.

Each vertex swaps to the other side of the y-axis at the same distance from it.

Reflection: solved example

Problem

A kite-shaped sail PQR has corners \(P(2, 4)\), \(Q(6, 1)\) and \(R(2, 1)\). Flip it over the y-axis and give the new corners.

Solution

A flip over the y-axis negates every x-coordinate: \((x, y) \to (-x, y)\).

So \(P^{\prime} = (-2, 4)\), \(Q^{\prime} = (-6, 1)\), \(R^{\prime} = (-2, 1)\). The mirror twin has corners \((-2, 4)\), \((-6, 1)\) and \((-2, 1)\), congruent to PQR yet facing the other way.

Try it

Question 1

Reflect \((-4, 3)\) over the x-axis.

Show the answer

Only the y-coordinate changes sign: \((-4, -3)\).

Question 2

The point \((3, 4)\) is reflected over the x-axis, then the image is reflected over the y-axis. Where does it end up?

Show the answer

First \((3, -4)\), then \((-3, -4)\).

Reflection vs. Rotation

A reflection flips the figure over a line and reverses orientation, so the picture becomes a mirror twin. A rotation turns the figure about a point and keeps the orientation. A clue: if tracing the vertices clockwise becomes counterclockwise, it was a reflection; if the order stays the same, it was a rotation or a slide.

Rotation (Geometry)

Keep going

Lessons

Worksheets and quizzes

Related words

Whole dictionary: A–Z · 8th Grade · Geometry

Voter.. page