Topics / Geometry

Transformations & Symmetry

Rotations, reflections, folding, tessellations, symmetric figures

Transformations & SymmetryAMC 10 Step by Step

Name: Date: September 13, 2026

Geometry · 15 problems have this as their main idea

primary-topic problems (1.2% of all)

What you need to know

Rotations, reflections, and translations preserve lengths, angles, and areas. A $90^\circ$ counterclockwise rotation about the origin sends $(x,y)\to(-y,x)$ .
Reflecting over a line: the line is the perpendicular bisector of the segment joining each point to its image. Folding is a reflection over the crease, so the crease is the perpendicular bisector of $PP'$ and folded lengths are preserved.
A regular $n$ -gon has $n$ lines of symmetry and rotational symmetry of order $n$ .

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 symmetric colorings of a grid.
Rotate a segment or triangle about a vertex by $60^\circ$ or $90^\circ$ ; find the swept area or the overlap with the image.

Standard approaches

For folds, name the image point $P'$ , mark all preserved lengths, and write one Pythagorean equation for the unknown crease position.
When a specific point lands on a specific point, use coordinates and the perpendicular bisector.
For swept areas, decompose into sectors centered at the rotation center 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. The crease is the perpendicular bisector of $\overline{AM}$ , so $PA = PM$ and $QA = QM$ . Let $PA = x$ ; then $PB = 8 - x$ and $BM = 4$ , so $x^2 = (8-x)^2 + 16$ , giving $x = 5$ . Let $DQ = y$ ; then $y^2 + 64 = (8-y)^2 + 16$ , giving $y = 1$ . With $A = (0,0)$ , $P = (5,0)$ and $Q = (1,8)$ , so
$PQ = \sqrt{4^2 + 8^2} = 4\sqrt5.$
The answer is $\boxed{\textbf{(B)}\ 4\sqrt5}$ .

Pitfalls

Treating the crease as the segment $AM$ itself rather than its perpendicular bisector.
Inventing new unknowns for the folded flap instead of reusing the preserved lengths.
Rotating $(x,y)$ to $(y,-x)$ for a counterclockwise turn; that is the clockwise image.

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)

Problems, easiest first

2023 AMC 10B · #7Transformations & Symmetry

Square $ABCD$ is rotated $20^{\circ}$ clockwise about its center to obtain square $EFGH$ , as shown below. What is the degree measure of $\angle{EAB}$ ?

2019 AMC 10A · #8Transformations & Symmetry

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? - some rotation …

2016 AMC 10A · #16Transformations & Symmetry

A triangle with vertices $A(0, 2)$ , $B(-3, 2)$ , and $C(-3, 0)$ is reflected about the $x$ -axis, then the image $\triangle A'B'C'$ is rotated counterclockwise about the origin by $90^{\circ}$ to produce $\triangle A''B''C''$ . Which of the following transformations will return $\triangle A''B''C''$ to $\triangle ABC$ …

2015 AMC 10B · #8Transformations & Symmetry

The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$ -axis, and then rotated a half turn around the origin. What is the final image?

2001 AMC 10 · #5Transformations & Symmetry

How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?

2024 AMC 10A · #13Transformations & Symmetry

Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane: - a translation $2$ units to the right, - a $90^{\circ}$ -rotation counterclockwise about the origin, - a …

2018 AMC 10A · #13Transformations & Symmetry

A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$ . What is the length in inches of the crease?

2016 AMC 10B · #20Transformations & Symmetry

A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$ . What distance does the origin $O(0,0)$ , move under this transformation?

2022 AMC 10A · #18Transformations & Symmetry

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$ -axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3,...,T_n$ returns the point …

2021 AMC 10B · #21Transformations & Symmetry

A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$ , and edge $\overline{AB}$ at point $E$ . Suppose that $C'D = \frac{1}{3}$ . What is the perimeter of triangle …

2020 AMC 10A · #23Transformations & Symmetry

Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$ -axis, and reflection across the …

2014 AMC 10A · #23Transformations & Symmetry

A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$ . The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line …

2013 AMC 10A · #20Transformations & Symmetry

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?

2008 AMC 10A · #19Transformations & Symmetry

Rectangle $PQRS$ lies in a plane with $PQ=RS=2$ and $QR=SP=6$ . The rectangle is rotated $90^\circ$ clockwise about $R$ , then rotated $90^\circ$ clockwise about the point $S$ moved to after the first rotation. What is the length of the path traveled by point $P$ ?

2025 AMC 10B · #25Transformations & Symmetry

Square $ABCD$ has sides of length $4$ . Points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{CD}$ , respectively, with $AP=\frac{8}{5}$ and $DQ=\frac{10}{3}$ . A path begins along the segment from $P$ to $Q$ and continues by reflecting against the sides of $ABCD$ (with congruent incoming and outgoing angles). If …