Rigid Transformations and Symmetry: Chapter Review

Vocabulary and notation

Transformation, preimage, and image

A transformation assigns each plane point exactly one image. In $P\mapsto P'$, $P$ is the starting preimage point and $P'$ its corresponding image.

Translation and displacement

A translation gives every point one common slide. Displacement $(a,b)$ records horizontal change, then vertical change; it is not a destination. Zero displacement leaves every point in place.

Reflection and mirror

A reflection folds the plane across one mirror line. Off-mirror points pair across that line; points on it stay fixed.

Rotation and directed turn

A rotation uses one center, angle, and direction. Each off-center point follows its own circle. Signed turns are positive counterclockwise and negative clockwise.

Fixed point and unchanged figure

A fixed point maps to itself. A whole figure can occupy exactly its original positions afterward while its individual points exchange places.

Rigid motion

A motion preserving every distance and every angle measure. Translations, reflections, rotations, and their sequences are rigid.

Orientation

The clockwise or counterclockwise order of corresponding noncollinear triangle vertices. Slides and turns preserve it; a reflection reverses it.

Composition and commuting

A composition applies motions in a stated order. They commute if reversing their order gives the same mapping everywhere.

Inverse

A mapping returning every image point to its preimage.

Glide reflection

A reflection combined with a nonzero translation parallel to its mirror.

Congruence and correspondence

Figures are congruent when a rigid-motion sequence carries one whole figure exactly onto the other. Correspondence records which points match, in order.

Symmetry and line symmetry

A symmetry is a rigid motion mapping a whole figure exactly onto itself. A line of symmetry is a mirror whose reflection makes that match.

Point symmetry

A half-turn about a stated point matches the whole figure.

Identity and rotational order

Identity fixes every point; zero and full turns give the same mapping. Finite rotational order counts distinct matching rotations about a stated center in $0^\circ\le\theta<360^\circ$, including identity once.

Formulas and theorems

• Rigid preservation

  $$\begin{gathered} A'B'=AB\\ m\angle A'B'C'=m\angle ABC \end{gathered}$$

  Use when Use the actual correspondence under a rigid motion or sequence. Distance and angle preservation are accepted directly from the slide, fold, and turn pictures. Straight sides map to straight sides; a translation also preserves parallel relationships.

• Translation coordinates and recovery

  $$\begin{gathered} (x,y)\mapsto(x+a,y+b)\\ a=x'-x\\ b=y'-y \end{gathered}$$

  Use when The same fixed $a,b$ apply everywhere. One pair determines a translation; every additional specified pair must agree. The inverse displacement is $(-a,-b)$.

  e.g. Translate $(-6,4)$ by $(2,-3)$: the image is $(-4,1)$.

• Reflection connector

  $$\begin{gathered} PP'\perp m\\ PH=HP'\\ PP'=2PH \end{gathered}$$

  Use when For $P$ off mirror $m$, $H$ is the connector's intersection with $m$; $P,P'$ lie on opposite sides. Thus $m$ perpendicularly bisects this connector. For $P$ on $m$, $P'=P$; no nonzero connector or right-angle mark applies.

• Horizontal and vertical mirrors

  $$\begin{gathered} x=h:\ (2h-x,y)\\ y=k:\ (x,2k-y)\\ y\text{-axis}:\ (-x,y)\\ x\text{-axis}:\ (x,-y) \end{gathered}$$

  Use when Each row gives the image of $(x,y)$ across its named mirror. A vertical mirror keeps height; a horizontal mirror keeps horizontal position.

• Diagonal mirrors

  $$\begin{gathered} y=x:\ (y,x)\\ y=-x:\ (-y,-x) \end{gathered}$$

  Use when Images of $(x,y)$ across these exact diagonals. The first exchanges coordinates; the second also reverses both signs. Points lying on the chosen diagonal stay fixed.

• Rotation radius and angle

  Every off-center pair satisfies $OP'=OP$ and the same directed turn from $OP$ to $OP'$.

  Use when Use one center $O$ and $P\ne O$. Set $O'=O$ separately. Different radii need not give equal travel distances. Adding full turns about the same center leaves the final image unchanged.

• Origin rotation rules

  Counterclockwise $90^\circ$ (clockwise $270^\circ$) gives $(-y,x)$. A half-turn gives $(-x,-y)$. Clockwise $90^\circ$ (counterclockwise $270^\circ$) gives $(y,-x)$.

  Use when Images of $(x,y)$ with the origin as center and axes fixed. Either half-turn direction gives the same image.

• Fixed-point cases

  A nonzero translation fixes no point; zero translation fixes every point. Reflection fixes exactly its mirror. A nonidentity rotation fixes only its center; identity fixes every point. A nonzero glide fixes no point.

  Use when These classify known motions of the whole plane. A nonidentity rotation's turn is not an integer multiple of $360^\circ$. A fixed point then identifies its center; identity does not isolate a center.

• Undoing a sequence

  Reverse the order and invert each motion.

  Use when Invert a translation by opposite displacement, a reflection by the same mirror, and a rotation by the opposite directed turn about the same center. Half-turns and identity undo themselves.

• Two parallel reflections

  The composition translates perpendicular to the mirrors through twice their gap, from the first mirror toward the second.

  Use when Use two distinct parallel mirrors and their perpendicular separation $d>0$; travel is $2d$. Reversing order reverses direction. Coincident mirrors instead give identity.

• Two intersecting reflections

  The composition rotates about the intersection through twice the directed angle from the first mirror to the second.

  Use when Choose a ray along each mirror to specify the directed angle. Opposite ray choices change the doubled turn by a full turn, preserving the image. Reversing order reverses the turn; perpendicular mirrors commute because both orders give a half-turn.

• Glide commuting and zero case

  Reflection and translation parallel to its mirror give the same mapping in either order.

  Use when A glide requires nonzero displacement. Zero displacement gives just reflection. The commuting claim does not extend to a translation across the mirror.

• Finite rotational order

  $$\begin{gathered} n\alpha=360^\circ\\ \alpha=\frac{360^\circ}{n}\\ n=\frac{360^\circ}{\alpha} \end{gathered}$$

  Use when Finite order $n$ about the same center, including identity once; $\alpha$ is the least positive matching angle. Matches in one turn are $0,\alpha,\ldots,(n-1)\alpha$; $n-1$ are nonidentity. Order one gives $\alpha=360^\circ$ and no nontrivial rotational symmetry.

• Finite order and point symmetry

  Point symmetry occurs exactly when the rotational order is even.

  Use when The order is finite and both tests use the same center. A half-turn requires $n/2$ least-angle steps. Odd order, including order one, excludes that half-turn.

• Familiar unmarked figures

  An unequal-sided rectangle has two center-line mirrors and order $2$, least angle $180^\circ$. A square has four mirrors (two center lines and two diagonals) and order $4$, least angle $90^\circ$. The triangle joining three equally spaced points on a circle has three mirrors and order $3$, least angle $120^\circ$.

  Use when Use the pictured or precisely specified unmarked figures and their stated centers. The triangle's guide circle is excluded. A rectangle has four right angles; a square also has four equal sides. Do not assign this triangle's symmetries to every triangle.

• Circle exception

  An unmarked circle has infinitely many symmetry lines through its center and infinitely many matching rotations about that center, including a half-turn. It has no least positive matching angle.

  Use when A circle has positive radius. Offset mirrors fail. Any proposed positive least angle can be halved; the finite-order formula does not apply. Included marks require a new whole-figure test.

Problem types, step by step

Exam traps

• Trap Confusing travel with size

  Fix Compare $AB$ with $A'B'$, not with $AA'$. Equal travel lengths alone do not establish a translation; the directions must also agree.

• Trap Adding horizontal and vertical travel counts

  Fix Those changes describe different directions. Their sum need not equal the straight displacement length.

• Trap Treating an unchanged figure as fixed point by point

  Fix A line or outline can occupy its old positions while points exchange places. A fixed point must map to itself.

• Trap Using equal mirror distances without alignment

  Fix Off-mirror reflection pairs must also share one perpendicular connector. On-mirror points stay fixed and need no right-angle connector mark.

• Trap Mixing axis and diagonal signs

  Fix The horizontal axis changes height; the vertical axis changes horizontal position. The rising diagonal swaps coordinates; the falling diagonal swaps and negates both.

• Trap Turning the axes or adding a turn to a corner angle

  Fix Keep the coordinate reference fixed and use the actual center. The directed turn changes placement; each corner angle keeps its measure, even when a reflection reverses orientation.

• Trap Accepting a candidate or a few samples as a global rule

  Fix Check the common directed angle at a candidate center. Tested matches alone do not identify an arbitrary transformation everywhere.

• Trap Assuming reversing order always changes the image

  Fix Compute the ordered steps. Perpendicular-mirror reflections commute, as do reflection and translation along its mirror, including zero displacement.

• Trap Calling a zero or across-mirror slide a glide

  Fix Require a nonzero translation parallel to the mirror. Zero gives reflection alone; an across-mirror slide lacks the glide commuting guarantee.

• Trap Letting a component decide the whole figure's symmetry

  Fix Include decorations and exact letter boundaries. Equal areas or one matched corner are insufficient; point symmetry requires the complete half-turn match.

• Trap Counting identity twice or applying finite order to a circle

  Fix Count $0^\circ$ and $360^\circ$ once. Order one has no nontrivial turn. An unmarked circle has infinitely many center rotations and no least positive angle.

• Trap Treating pencil measurements as exact proof

  Fix Ruler, protractor, and compass drawings have limited precision. Check the intended construction against the exact motion definition; measured examples supply approximate evidence.