Transformations & Symmetry

What you need to know

• Moves that keep size and shape. Rotations (turns), reflections (flips), and translations (slides) do not change lengths, angles, or areas. A $90^\circ$ counterclockwise rotation about the origin sends $(x,y)$ to $(-y,x)$. Example: $(3,1)$ goes to $(-1,3)$, and both are $\sqrt{10}$ from the origin.
• Reflecting across a line. If $P'$ is the reflection of $P$, the line is the perpendicular bisector of $PP'$: it meets $PP'$ at a right angle, through its midpoint. Example: reflecting $(1,4)$ across the line $y = 1$ gives $(1,-2)$, and the midpoint $(1,1)$ is on the line.
• Folding is reflecting across the crease. If a fold puts point $P$ on point $P'$, the crease is the perpendicular bisector of $PP'$, and every length on the folded flap stays the same.
• Symmetry of regular polygons. A regular polygon with $n$ sides has $n$ lines of symmetry, and rotational symmetry of order $n$: it looks the same after each of $n$ turns of $\frac{360^\circ}{n}$ about its center. Example: a square has $4$ lines of symmetry and matches itself after turns of $90^\circ$, $180^\circ$, $270^\circ$, $360^\circ$.

How AMC 10 tests it

• Problems 12–18: fold a rectangle or square so a corner lands on an edge, a midpoint, or the opposite corner; find the crease length or a folded segment.
• "Which figures have a line of symmetry, or rotational symmetry?" or counting the colorings of a grid that are symmetric.
• Turn a segment or triangle $60^\circ$ or $90^\circ$ about one of its corners; find the area it sweeps out, or its overlap with its image.

Standard approaches

1. For folds, call the landing point $P'$, mark every length the fold keeps, and write one Pythagorean equation for the unknown crease position.
2. When the problem says exactly which point lands on which, use coordinates and the perpendicular bisector.
3. For swept areas, split the region into sectors centered at the center of rotation, plus triangles.

Worked example

Square $ABCD$ has side length $8$. The paper is folded so that vertex $A$ lands on the midpoint $M$ of side $\overline{BC}$. The crease meets $\overline{AB}$ at $P$ and $\overline{CD}$ at $Q$. What is the length of $\overline{PQ}$?

(A) $8$ (B) $4\sqrt5$ (C) $9$ (D) $6\sqrt3$ (E) $10$

Solution. Place the square with $A = (0,0)$, $B = (8,0)$, $C = (8,8)$, $D = (0,8)$, so $M = (8,4)$.

Step 1. The fold sends $A$ to $M$, so the crease is the perpendicular bisector of $\overline{AM}$. Every point on it is as far from $A$ as from $M$, so $PA = PM$ and $QA = QM$.

Step 2. Let $PA = x$. Then $PB = 8 - x$ and $BM = 4$. Right triangle $PBM$ gives
$$x^2 = (8-x)^2 + 4^2 = x^2 - 16x + 80,$$
so $16x = 80$ and $x = 5$. That puts $P$ at $(5,0)$.

Step 3. Let $DQ = y$, so $QC = 8 - y$. Right triangles $QDA$ and $QCM$ give
$$y^2 + 8^2 = (8-y)^2 + 4^2, \quad\text{that is,}\quad y^2 + 64 = y^2 - 16y + 80,$$
so $16y = 16$ and $y = 1$. That puts $Q$ at $(1,8)$.

Step 4. From $P = (5,0)$ to $Q = (1,8)$ is $4$ across and $8$ up:
$$PQ = \sqrt{4^2 + 8^2} = \sqrt{80} = 4\sqrt5.$$
The answer is $\boxed{\textbf{(B)}\ 4\sqrt5}$.

Pitfalls

• Treating the crease as segment $AM$ itself. The crease is the perpendicular bisector of $AM$.
• Making up new unknowns for the folded flap instead of reusing the lengths the fold keeps. In the example, $PM$ is just $PA$.
• Writing $(y,-x)$ for a $90^\circ$ counterclockwise turn. That is the clockwise turn. Example: counterclockwise, $(1,0)$ goes up to $(0,1)$, not down to $(0,-1)$.

Traps that recur

• Tracing the bounces one at a time, losing the thread after four or five reflections and concluding from the absence of an early corner that the path runs forever, which is (E). (2025 AMC 10B #25)
• Checking only even n (which first works at n = 720) and missing that an odd number of transformations, ending on a reflection, already returns the point at n = 359. (2022 AMC 10A #18)
• Assuming AE = EC' or that the crease passes through E; E lies on the folded image of edge CB, not on the crease. (2021 AMC 10B #21)
• Forgetting that a rotation sandwiched between two reflections is reversed, or allowing sequences with one or three reflections. (2020 AMC 10A #23)