Two Reflections in a Row Are the Same as One Half Turn
Learn the coordinate rules for reflecting a point in the axes and the origin, reflecting in a line parallel to an axis, rotating about the origin, and naming the figure formed.
What happens if you reflect a point twice?
You get a rotation — and that is the neatest surprise in this chapter.
Take the point . Reflect it in the -axis and it lands at . Now reflect that in the -axis and it lands at .
Compare the start and the finish: became . Both signs changed, which is exactly what a half turn about the origin does. Two mirrors, applied one after the other, have produced a rotation with no mirror in sight.
Once points carry coordinates, every reflection and rotation becomes a rule about signs, and those rules can be combined and checked by arithmetic rather than by folding paper. This page covers the second part of the ICSE Class 8 Mathematics chapter on symmetry.
Take the point . Reflect it in the -axis and it lands at . Now reflect that in the -axis and it lands at .
Compare the start and the finish: became . Both signs changed, which is exactly what a half turn about the origin does. Two mirrors, applied one after the other, have produced a rotation with no mirror in sight.
Once points carry coordinates, every reflection and rotation becomes a rule about signs, and those rules can be combined and checked by arithmetic rather than by folding paper. This page covers the second part of the ICSE Class 8 Mathematics chapter on symmetry.
Formula
What are the coordinate rules for reflection in the axes and the origin?
Three rules cover every reflection in this chapter:
How to remember which sign flips. Reflecting in the -axis moves the point up or down across a horizontal mirror, so the ****-coordinate changes sign and stays put. Reflecting in the -axis moves it left or right, so ** changes sign. The coordinate that flips is the one measured perpendicular to the mirror.
Worked example 1.** Find the images of .
- In the -axis:
- In the -axis:
- In the origin:
Worked example 2. Find the images of .
- In the -axis:
- In the -axis:
- In the origin:
Notice the -axis image has a positive : changing the sign of gives , not again. Mechanically applying put a minus sign in front is the standard error, and it fails on every point that already has a negative coordinate.
Worked example 3 — the points that do not move. Reflect in the -axis:
The point is unchanged, because it already lies on the mirror. Every point of the -axis is invariant under reflection in the -axis, and every point of the -axis is invariant under reflection in the -axis.
Under reflection in the origin, only one point is invariant — the origin itself. A reflection always fixes exactly the points lying on its mirror, which is why the axes have infinitely many invariant points and the origin has one.
How to remember which sign flips. Reflecting in the -axis moves the point up or down across a horizontal mirror, so the ****-coordinate changes sign and stays put. Reflecting in the -axis moves it left or right, so ** changes sign. The coordinate that flips is the one measured perpendicular to the mirror.
Worked example 1.** Find the images of .
- In the -axis:
- In the -axis:
- In the origin:
Worked example 2. Find the images of .
- In the -axis:
- In the -axis:
- In the origin:
Notice the -axis image has a positive : changing the sign of gives , not again. Mechanically applying put a minus sign in front is the standard error, and it fails on every point that already has a negative coordinate.
Worked example 3 — the points that do not move. Reflect in the -axis:
The point is unchanged, because it already lies on the mirror. Every point of the -axis is invariant under reflection in the -axis, and every point of the -axis is invariant under reflection in the -axis.
Under reflection in the origin, only one point is invariant — the origin itself. A reflection always fixes exactly the points lying on its mirror, which is why the axes have infinitely many invariant points and the origin has one.
How do you reflect a point in a line parallel to an axis?
Measure the distance to the line and step the same distance past it. In coordinates:
The line is a vertical line, so only the -coordinate changes. The line is horizontal, so only changes.
Worked example 1. Reflect in the line .
Check by distance. is units to the left of the line, so the image must be units to the right of it, at . The formula and the picture agree.
Worked example 2. Reflect in the line .
Check. is units above the line, so the image is units below it, at . Correct.
Worked example 3 — a negative line. Reflect in the line .
Check. is units to the right of the line, so the image is units to the left, at . Correct.
Worked example 4 — both negative. Reflect in the line .
The point was units below the line and the image is units above it. Correct.
**Do not confuse the line with the point .** The equation describes a whole vertical line — every point whose first coordinate is . Students who read it as a horizontal line flip the wrong coordinate, and the distance check above catches that instantly.
The axes are just special cases. The -axis is the line , and putting into the formula gives — exactly the rule from the previous section. One formula covers both, so there is really only one idea to learn.
The line is a vertical line, so only the -coordinate changes. The line is horizontal, so only changes.
Worked example 1. Reflect in the line .
Check by distance. is units to the left of the line, so the image must be units to the right of it, at . The formula and the picture agree.
Worked example 2. Reflect in the line .
Check. is units above the line, so the image is units below it, at . Correct.
Worked example 3 — a negative line. Reflect in the line .
Check. is units to the right of the line, so the image is units to the left, at . Correct.
Worked example 4 — both negative. Reflect in the line .
The point was units below the line and the image is units above it. Correct.
**Do not confuse the line with the point .** The equation describes a whole vertical line — every point whose first coordinate is . Students who read it as a horizontal line flip the wrong coordinate, and the distance check above catches that instantly.
The axes are just special cases. The -axis is the line , and putting into the formula gives — exactly the rule from the previous section. One formula covers both, so there is really only one idea to learn.
How do you rotate a point about the origin?
Swap the coordinates and adjust one sign — which one depends on the direction of turn.
Worked example 1. Rotate about the origin.
- anticlockwise:
- clockwise:
- :
Check with distances. A rotation cannot change how far a point is from the centre. For :
And for the image :
The distances match, as they must. This is the single best check on a rotation answer, because a wrong sign or a forgotten swap almost always changes the distance.
Worked example 2. Rotate about the origin.
- anticlockwise:
- clockwise:
- :
Every one of these is from the origin, matching .
Worked example 3 — a right angle at the origin. Take and its image after a anticlockwise turn. Then
Testing Pythagoras on triangle :
So the angle at really is , and triangle is right-angled and isosceles. The coordinate rule has been verified by measurement rather than assumed.
**Rotating and reflecting in the origin are the same transformation.** Both send to , which is why no direction needs to be stated for a half turn — clockwise and anticlockwise give the identical answer. For the direction matters absolutely, and a question that omits it is incomplete.
Worked example 1. Rotate about the origin.
- anticlockwise:
- clockwise:
- :
Check with distances. A rotation cannot change how far a point is from the centre. For :
And for the image :
The distances match, as they must. This is the single best check on a rotation answer, because a wrong sign or a forgotten swap almost always changes the distance.
Worked example 2. Rotate about the origin.
- anticlockwise:
- clockwise:
- :
Every one of these is from the origin, matching .
Worked example 3 — a right angle at the origin. Take and its image after a anticlockwise turn. Then
Testing Pythagoras on triangle :
So the angle at really is , and triangle is right-angled and isosceles. The coordinate rule has been verified by measurement rather than assumed.
**Rotating and reflecting in the origin are the same transformation.** Both send to , which is why no direction needs to be stated for a half turn — clockwise and anticlockwise give the identical answer. For the direction matters absolutely, and a question that omits it is incomplete.
How do you name the figure formed by a point and its images?
Plot every point, then measure the sides and diagonals before naming anything.
Worked example 1 — a rectangle. Plot , its reflection in the -axis , the reflection of in the -axis , and the reflection of in the -axis .
The four points span from to and from to , so
All four angles are right angles, so is a rectangle:
Its diagonal is units, and both diagonals pass through the origin.
It is a rectangle, not a square. The sides are and , which are different. Only a starting point with would give a square — for instance and its three images form a square of side .
Worked example 2 — a rhombus. Plot , its reflection in the -axis , then and its reflection in the -axis .
The diagonals are units along the -axis and units along the -axis, crossing at the origin at right angles and bisecting each other. That makes a rhombus:
Worked example 3 — an isosceles triangle. Plot , its reflection in the -axis , and the origin .
Two equal sides, so the triangle is isosceles. Taking as the base, the height is the distance from to the line , which is units:
The property worth stating. The -axis is the perpendicular bisector of — the midpoint of is , which lies on the -axis, and is vertical while the axis is horizontal.
That is true of every reflection: the mirror line is always the perpendicular bisector of the segment joining a point to its image. It gives you a way to find the mirror when you are given only the point and its image, and it is the single fact that ties all four sections of this page together.
Worked example 1 — a rectangle. Plot , its reflection in the -axis , the reflection of in the -axis , and the reflection of in the -axis .
The four points span from to and from to , so
All four angles are right angles, so is a rectangle:
Its diagonal is units, and both diagonals pass through the origin.
It is a rectangle, not a square. The sides are and , which are different. Only a starting point with would give a square — for instance and its three images form a square of side .
Worked example 2 — a rhombus. Plot , its reflection in the -axis , then and its reflection in the -axis .
The diagonals are units along the -axis and units along the -axis, crossing at the origin at right angles and bisecting each other. That makes a rhombus:
Worked example 3 — an isosceles triangle. Plot , its reflection in the -axis , and the origin .
Two equal sides, so the triangle is isosceles. Taking as the base, the height is the distance from to the line , which is units:
The property worth stating. The -axis is the perpendicular bisector of — the midpoint of is , which lies on the -axis, and is vertical while the axis is horizontal.
That is true of every reflection: the mirror line is always the perpendicular bisector of the segment joining a point to its image. It gives you a way to find the mirror when you are given only the point and its image, and it is the single fact that ties all four sections of this page together.
Exam tip
Exam tip: check the distance from the centre after every rotation
Reflection flips the coordinate perpendicular to the mirror. In the -axis, changes sign; in the -axis, changes sign; in the origin, both do.
Change the sign, do not simply write a minus. The -axis image of is .
For a line parallel to an axis, use for and for — then confirm with the distance: two units before the line means two units after it.
Remember that is a vertical line and a horizontal one.
Rotations: anticlockwise gives , clockwise gives , and gives whichever way you turn.
Check every rotation by distance from the origin — it cannot change, and a wrong sign nearly always breaks it.
A rotation and a reflection in the origin are the same transformation.
Invariant points lie on the mirror: the whole -axis for reflection in the -axis, and only the origin for reflection in the origin.
Plot before naming a figure, and justify the name by side lengths and diagonals — a rectangle with sides and is not a square.
And remember the mirror is always the perpendicular bisector of a point and its image.
Change the sign, do not simply write a minus. The -axis image of is .
For a line parallel to an axis, use for and for — then confirm with the distance: two units before the line means two units after it.
Remember that is a vertical line and a horizontal one.
Rotations: anticlockwise gives , clockwise gives , and gives whichever way you turn.
Check every rotation by distance from the origin — it cannot change, and a wrong sign nearly always breaks it.
A rotation and a reflection in the origin are the same transformation.
Invariant points lie on the mirror: the whole -axis for reflection in the -axis, and only the origin for reflection in the origin.
Plot before naming a figure, and justify the name by side lengths and diagonals — a rectangle with sides and is not a square.
And remember the mirror is always the perpendicular bisector of a point and its image.
Did you know
Why a mirror seems to reverse left and right
Stand in front of a mirror, raise your right hand, and the figure opposite raises what looks like its left. People usually say a mirror swaps left and right — but that is not what it does.
A mirror reverses the direction perpendicular to its surface, and nothing else. For a vertical wall mirror that direction is front-to-back, so the reflection is you turned inside out along the line from your nose to the glass. Your left and right are untouched; it is only because you mentally rotate the figure to face you that the swap seems to happen.
The coordinate rules on this page say the same thing more precisely. Reflecting in the -axis changes only the -coordinate. The -coordinate — the direction along the mirror — is left exactly as it was. One coordinate flips, and it is always the perpendicular one.
This also explains a familiar puzzle. Lie a mirror flat on the floor and the image is upside down instead of left-right reversed, because now the perpendicular direction is vertical. The mirror has not changed its behaviour at all; you have changed which direction is perpendicular to it.
And it is why the rules for and needed no new thinking. Choose the mirror, identify the perpendicular direction, flip that coordinate about the line, and leave the other alone.
A mirror reverses the direction perpendicular to its surface, and nothing else. For a vertical wall mirror that direction is front-to-back, so the reflection is you turned inside out along the line from your nose to the glass. Your left and right are untouched; it is only because you mentally rotate the figure to face you that the swap seems to happen.
The coordinate rules on this page say the same thing more precisely. Reflecting in the -axis changes only the -coordinate. The -coordinate — the direction along the mirror — is left exactly as it was. One coordinate flips, and it is always the perpendicular one.
This also explains a familiar puzzle. Lie a mirror flat on the floor and the image is upside down instead of left-right reversed, because now the perpendicular direction is vertical. The mirror has not changed its behaviour at all; you have changed which direction is perpendicular to it.
And it is why the rules for and needed no new thinking. Choose the mirror, identify the perpendicular direction, flip that coordinate about the line, and leave the other alone.
Key takeaways
Reflection and rotation in coordinates: quick revision
- Reflection rules: in the -axis, ; in the -axis, ; in the origin, . The coordinate that flips is the one perpendicular to the mirror.
- gives , and . gives , and — note the -axis image is positive .
- Invariant points lie on the mirror: the whole -axis, the whole -axis, and for the origin only .
- Lines parallel to an axis: reflection in gives ; in gives .
- in gives ; in gives . in gives ; in gives .
- Always verify by distance: two units before the line means two units beyond it. And is a vertical line.
- The axes are the special cases and of the same formula.
- Rotations about the origin: anticlockwise gives , clockwise gives , and gives .
- becomes , or ; becomes , or .
- Distance from the origin is unchanged — and the image is also away. This is the best check available.
- and its image give and , and since the angle at is .
- A ** rotation is the same as reflection in the origin**, so no direction need be stated; for the direction matters.
- Figures formed: with its three images gives a rectangle by — area square units, perimeter units, diagonal units. Not a square, since .
- , , , give a rhombus with diagonals and , side units and area square units.
- , and the origin give an isosceles triangle with , and area square units.
- The mirror is always the perpendicular bisector of a point and its image — the midpoint of here is , on the -axis.
Take one point with a negative coordinate and write down all six images — three reflections and three rotations — then check each with the distance test before looking anything up.
- gives , and . gives , and — note the -axis image is positive .
- Invariant points lie on the mirror: the whole -axis, the whole -axis, and for the origin only .
- Lines parallel to an axis: reflection in gives ; in gives .
- in gives ; in gives . in gives ; in gives .
- Always verify by distance: two units before the line means two units beyond it. And is a vertical line.
- The axes are the special cases and of the same formula.
- Rotations about the origin: anticlockwise gives , clockwise gives , and gives .
- becomes , or ; becomes , or .
- Distance from the origin is unchanged — and the image is also away. This is the best check available.
- and its image give and , and since the angle at is .
- A ** rotation is the same as reflection in the origin**, so no direction need be stated; for the direction matters.
- Figures formed: with its three images gives a rectangle by — area square units, perimeter units, diagonal units. Not a square, since .
- , , , give a rhombus with diagonals and , side units and area square units.
- , and the origin give an isosceles triangle with , and area square units.
- The mirror is always the perpendicular bisector of a point and its image — the midpoint of here is , on the -axis.
Take one point with a negative coordinate and write down all six images — three reflections and three rotations — then check each with the distance test before looking anything up.