Rigid Motions and the Definition of Congruence

A transformation is a function on points

In algebra a function takes a number and returns a number. A transformation takes a point and returns a point.

$$T(P) = P'$$

The input $P$ is called the preimage and the output $P'$ is its image. Written with coordinates, a translation 3 right and 2 up is the function

$$T(x, y) = (x + 3, \; y + 2)$$

Thinking of it as a function is not a formality. It says every point in the plane has exactly one image, which is what lets a transformation be applied to a whole figure at once — every vertex, every edge, every point in between.

Which transformations keep the figure intact

Some transformations move a figure without distorting it. Others change its size or shape.

Transformation	Rule	Distances	Angles
translation	$(x, y) \to (x + 3, y + 2)$	preserved	preserved
reflection over the $y$-axis	$(x, y) \to (-x, y)$	preserved	preserved
rotation $90°$ about the origin	$(x, y) \to (-y, x)$	preserved	preserved
dilation by $2$	$(x, y) \to (2x, 2y)$	doubled	preserved
horizontal stretch	$(x, y) \to (2x, y)$	changed	changed

A transformation that preserves both distance and angle is a rigid motion.

The horizontal stretch is worth a second look. It doubles horizontal distances and leaves vertical ones alone, so a square becomes a rectangle and a $45°$ diagonal becomes a shallower line. Changing one direction and not the other distorts angles as well as lengths, which is why it fails on both counts.

Precise definitions

Informal descriptions — slide, flip, turn — are fine for recognizing a motion. Proofs need definitions stated in terms of segments, angles and lines.

Motion	Defined by
translation along a directed segment $\overrightarrow{AB}$	every point moves the length of $AB$ in the direction from $A$ to $B$, so each segment $PP'$ is parallel to $AB$ and the same length
reflection over line $\ell$	$\ell$ is the perpendicular bisector of each segment $PP'$; points on $\ell$ do not move
rotation about center $O$ by angle $\theta$	$OP' = OP$ and $\angle POP' = \theta$; the center does not move

Each definition pins down exactly one image for every point. Take the reflection: $P'$ must lie on the line through $P$ perpendicular to $\ell$, and at the same distance from $\ell$ on the other side. There is only one such point, so the reflection is a well-defined function.

The rotation’s definition uses a circle: $P'$ lies on the circle centered at $O$ through $P$, at the angle $\theta$ around from it.

Why congruence is defined by motion

The grade 8 idea of congruence — same size, same shape — raises a question it cannot answer by itself: same size measured how, and same shape according to what?

Rigid motions answer both at once.

$$\text{Two figures are congruent} \iff \text{a sequence of rigid motions carries one onto the other}$$

Because every rigid motion preserves every distance and every angle, any measurement taken on the original — a side, a diagonal, an angle, a perimeter — comes out the same on the image. The definition guarantees agreement on every measurement there is, not only the ones someone thought to check.

Deciding congruence is then a concrete task: find the sequence of motions, or show that none exists.

Finding a sequence of motions

Triangle $ABC$ has vertices $A(1, 1)$, $B(4, 1)$, $C(1, 3)$. Triangle $DEF$ has $D(-1, -1)$, $E(-4, -1)$, $F(-1, -3)$. Are they congruent?

Every coordinate of $DEF$ is the negative of the matching one in $ABC$:

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

That is a rotation of $180°$ about the origin. A single rigid motion carries $ABC$ onto $DEF$, with $A \to D$, $B \to E$, $C \to F$, so the triangles are congruent.

When one motion is not enough, combine them. A common strategy:

1. Translate so one vertex lands on its partner.
2. Rotate about that vertex so a side lines up with its partner.
3. Reflect over that side if the third vertex is on the wrong side.

Those three steps are enough for any pair of congruent triangles — which is the idea behind the triangle congruence criteria.

Motions that carry a figure onto itself

Some rigid motions map a figure exactly onto itself. These describe its symmetry.

Figure	Rotations that work	Reflections that work
rectangle (not a square)	$180°$	$2$ — through midpoints of opposite sides
rhombus (not a square)	$180°$	$2$ — along its diagonals
parallelogram (neither of those)	$180°$	none
isosceles trapezoid	none but $360°$	$1$ — through midpoints of the parallel sides
square	$90°$, $180°$, $270°$	$4$
regular $n$-gon	multiples of $\tfrac{360°}{n}$	$n$

For a regular hexagon, the rotations are multiples of $60°$ and there are $6$ lines of reflection.

The rotations always include $360°$, which returns every point to where it started. Counting that one, a regular $n$-gon has exactly $2n$ symmetries — $n$ rotations and $n$ reflections.

Worked examples

Common mistakes

Practice problems