Transformation of Shapes

A figure may be translated, rotated, flipped or resized, or several figures may be the same shape in different positions. The official syllabus lists translation, rotation, scaling, mirroring, assembling and grouping. All of them can be done reliably in two ways: by imagining the motion, which is fast but error-prone, and by tracking a few labelled points with coordinates, which is slower but certain. Use the first and check with the second.

Orientation is the test: translations and rotations keep the cyclic order of labelled corners (clockwise stays clockwise), while a reflection reverses it. That test separates a piece that has been turned from its mirror image.

Rigid motions. Translations, rotations and reflections (and their combinations) keep every length and angle, so the image is congruent to the original. Scaling keeps angles and multiplies lengths by $k$ and areas by $k^2$, so the image is similar.

Orientation test. List three labelled corners $A, B, C$ of the original and of the image. Compute the sign of

For three non-collinear corresponding labelled points, positive means $A\to B\to C$ runs counterclockwise and negative means clockwise. Equal nonzero signs mean orientation is preserved; opposite signs mean orientation is reversed. Only after congruence (length preservation) is established does preserved orientation identify a rotation or translation rather than, for example, positive scaling.

Rotation about another centre. To rotate about a point $P$, translate $P$ to the origin, rotate, and translate back.

Composition facts. Two reflections in parallel mirrors a distance $d$ apart make a translation by $2d$ perpendicular to them. Two reflections in mirrors meeting at angle $\theta$ make a rotation by $2\theta$ about the meeting point. A rotation followed by a reflection reverses orientation once. For a nonidentity rotation (angle not 0 modulo 360 degrees), it is a reflection when the centre lies on the mirror, and a glide reflection otherwise. An identity rotation followed by a reflection is just that reflection.

Why rotation by $90^\circ$ is $(x, y) \to (-y, x)$. A point at distance $r$ from the origin at angle $\phi$ is $(r\cos\phi, r\sin\phi)$. Adding $90^\circ$ gives $(r\cos(\phi + 90^\circ), r\sin(\phi + 90^\circ)) = (-r\sin\phi, r\cos\phi) = (-y, x)$.

Why reflection reverses orientation. The orientation expression above changes sign when $x$ is replaced by $-x$ throughout (one factor in each product changes sign), and is unchanged under $(x, y) \to (-y, x)$ (both products are transformed into each other with the same overall sign). This is why turning a figure can never produce its mirror image in the plane.

Scaling and area. Under $(x, y) \to (kx, ky)$ the orientation expression, which is twice the signed area of triangle $ABC$, is multiplied by $k^2$.

Mirror clocks. In a vertical mirror the clock hands at angle $\alpha$ from 12 appear at $360^\circ - \alpha$. A time of $h$ hours $m$ minutes therefore appears as the time that adds with it to 12:00: the mirror image of 3:40 shows 8:20.

Use the triangle $A(1, 1)$, $B(4, 1)$, $C(1, 3)$, which has area 3 and orientation value $3 \times 2 - 0 \times 0 = 6$ (counterclockwise).

Example 1 (translation). By $(2, -1)$: $A'(3, 0)$, $B'(6, 0)$, $C'(3, 2)$. Same shape, same orientation.

Example 2 (rotation about the origin). $90^\circ$ counterclockwise, $(x, y) \to (-y, x)$: $A'(-1, 1)$, $B'(-1, 4)$, $C'(-3, 1)$. Orientation value still 6.

Example 3 (reflection). In the $y$-axis, $(x, y) \to (-x, y)$: $A'(-1, 1)$, $B'(-4, 1)$, $C'(-1, 3)$. Orientation value $-6$: the labels now run clockwise.

Example 4 (scaling). By 2 about the origin: $A'(2, 2)$, $B'(8, 2)$, $C'(2, 6)$. Sides double; area goes from 3 to 12.

Example 5 (rotation about another point). Rotate $Q(5, 2)$ by $90^\circ$ counterclockwise about $P(2, 1)$. Relative to $P$, $Q$ is at $(3, 1)$; rotating gives $(-1, 3)$; adding $P$ back gives $(1, 4)$.

Example 6 (two reflections). Reflect the point $(0, 0)$ in the line $x = 1$, giving $(2, 0)$, then in the line $x = 4$, giving $(6, 0)$. The mirrors are 3 apart, and the net effect is a translation by $2 \times 3 = 6$ to the right, for every point.

Example 7 (mirror clock). A clock reads 3:40. In a vertical mirror it appears to read $12{:}00 - 3{:}40 = 8{:}20$.

Example 8 (mirror letters). In a vertical mirror, the capital letters A, H, I, M, O, T, U, V, W, X and Y look unchanged (each has a vertical line of symmetry). A word made only of these letters appears in the mirror as the same letters in reverse order: TOMATO appears as OTAMOT.

Example 9 (grouping and odd one out). Four L-shaped pieces are drawn with corners labelled in order 1, 2, 3 (corner of the L, end of the long arm, end of the short arm). Pieces 1, 2 and 4 have 1, 2, 3 counterclockwise; piece 3 has them clockwise. Pieces 1, 2 and 4 are rotations of one another; piece 3 is the mirror image and is the odd one out (Figure 2).

Example 10 (rotational symmetry). The order of rotational symmetry (the number of positions in a full turn where the figure looks the same) is 4 for a square, 3 for an equilateral triangle, 2 for a non-square rectangle and 6 for a regular hexagon. The letters N, S and Z have order 2 and no line of symmetry.

Example 11 (rotation sequence). An arrow points north and turns $45^\circ$ clockwise at each step. After 7 steps it has turned $315^\circ$ clockwise, so it points north-west; after 8 steps it is back to north.

Example 12 (water image). A water image is a reflection in a horizontal line below the object: $(x, y) \to (x, -y)$ when the water line is the $x$-axis. Left and right are kept; up and down swap. An arrow from $(0, 0)$ to $(2, 1)$ (pointing up and to the right) has water image from $(0, 0)$ to $(2, -1)$, pointing down and to the right. For a written word, a water image keeps the letters in their order but turns each one upside down, while a vertical mirror keeps each letter upright but reverses the order (Example 8). The orientation test still works: the triangle of Example 1 has orientation value 6, and its water image $A'(1, -1)$, $B'(4, -1)$, $C'(1, -3)$ has $-6$.

Example 13 (rotating a grid pattern). A $3 \times 3$ grid has shaded cells at (row, column) $(1, 1)$, $(1, 2)$ and $(3, 3)$, with row 1 at the top. After a $90^\circ$ clockwise rotation, the cell $(r, c)$ moves to $(c, 4 - r)$: the shaded cells become $(1, 3)$, $(2, 3)$ and $(3, 1)$. A $90^\circ$ counterclockwise rotation would instead give $(3, 1)$, $(2, 1)$ and $(1, 3)$; comparing the two lists is how the options for "rotated clockwise" and "rotated counterclockwise" are told apart.

Example 14 (mirror versus rotation of a grid). Reflecting the same pattern in the vertical centre line sends $(r, c)$ to $(r, 4 - c)$: $(1, 3)$, $(1, 2)$, $(3, 1)$. This set is not a rotation of the original: the original's two adjacent shaded cells and the lone cell form a shape whose mirror image, like the L-pieces of Example 9, cannot be reached by turning.

Example 15 (assembling). Which three of the following pieces can form a $6 \times 6$ square of area 36? (A) a $6 \times 2$ rectangle (area 12); (B) a $4 \times 4$ square (area 16); (C) a $4 \times 2$ rectangle (area 8); (D) a $5 \times 2$ rectangle (area 10). First check area: $12 + 16 + 8 = 36$, while any trio that includes D has area 38, 30 or 34, none equal to 36, so D is out. The remaining three fit: place A as a $6 \times 2$ strip along one side of the square; the leftover region is a $6 \times 4$ rectangle, which is exactly B and C placed side by side ($4 + 2 = 6$). Area first, then matching edge lengths, is the reliable order.