Draw the image of a figure under a given rigid motion and specify a sequence of transformations carrying one figure onto another

Learning goal: Given a geometric figure and a rotation, reflection or translation, draw the transformed figure accurately using graph paper, tracing paper or geometry software, and, given two congruent figures, specify a sequence of transformations that will carry one onto the other, checking every vertex.

Drawing an image on graph paper. A rigid motion takes segments to segments, so the image of a polygon is determined by the images of its vertices: find each image vertex, then join them in the same order. For a translation, add the run and rise to every vertex. For a reflection across a horizontal or vertical line or across y = x, use the definition: each image vertex is the same perpendicular distance from the line on the other side. For a rotation about the origin, the rules (x, y) → (−y, x) for 90° counterclockwise, (−x, −y) for 180° and (y, −x) for 90° clockwise apply; for a rotation about another center, subtract the center, rotate, and add the center back. Always check one image vertex against the definition, not just the rule, because a sign error in the rule is the most common mistake.

A worked rotation about a point that is not the origin. Rotate triangle DEF with D(3, 2), E(6, 2), F(3, 4) through 90° counterclockwise about the center C(2, 1). Displacements from C: D − C = (1, 1), E − C = (4, 1), F − C = (1, 3). Rotate each by (x, y) → (−y, x): (−1, 1), (−1, 4), (−3, 1). Add C back: D′(1, 2), E′(1, 5), F′(−1, 2). Check E′ against the definition: CE = √(16 + 1) = √17 and CE′ = √(1 + 16) = √17, equal, and the dot product of (4, 1) and (−1, 4) is −4 + 4 = 0, so the angle at C is 90°. The side DE was horizontal of length 3; D′E′ runs from (1, 2) to (1, 5), vertical of length 3, a quarter turn as expected.

Drawing with tracing paper. Trace the figure and the center or line. For a rotation, hold the tracing at the center with a pencil point and turn until a marked reference ray has swept the required angle, then press through the vertices. For a reflection, flip the tracing over along the line so the traced line lands on itself. For a translation, slide the tracing so a marked point lands on its known image, keeping the grid lines aligned. Tracing paper is less precise than coordinates, so treat it as a check on the picture and use arithmetic for the numbers.

Specifying a sequence. Two figures are given and you must state transformations, in order, that carry the first exactly onto the second. A workable method: match the vertices using side lengths (the longest side of one must go to the longest side of the other); compare orientation by reading the matched vertices around each figure, since a reversed orientation means the sequence must include a reflection; use a translation to send one vertex to its match; then use a rotation about that vertex, or a reflection across a line through it, to bring the rest into place. Finally apply the whole sequence to every vertex and confirm each lands on its match. A sequence that has not been checked on every vertex is a guess.

A worked sequence. Triangle ABC has A(1, 1), B(4, 1), C(1, 3). Triangle A″B″C″ has A″(7, 0), B″(7, 3), C″(5, 0). Match sides: AB is horizontal of length 3, and A″B″ is vertical of length 3; AC is vertical of length 2, and A″C″ is horizontal of length 2. So A ↔ A″, B ↔ B″, C ↔ C″. Orientation: going A → B → C turns counterclockwise (right, then up-left), and A″ → B″ → C″ goes up, then down-left, also counterclockwise, so no reflection is needed. The side AB pointing right must end up pointing up, a quarter turn counterclockwise, so start with a rotation of 90° counterclockwise about the origin: A′(−1, 1), B′(−1, 4), C′(−3, 1). Now A′(−1, 1) must reach A″(7, 0): translate by (x, y) → (x + 8, y − 1). Check all three: A′ → (7, 0) = A″; B′(−1, 4) → (7, 3) = B″; C′(−3, 1) → (5, 0) = C″. The sequence "rotate 90° counterclockwise about the origin, then translate 8 right and 1 down" carries ABC onto A″B″C″.

Sequences are not unique, and order matters. The same job could be done by a rotation of 90° counterclockwise about A followed by a different translation, or by a translation first and a rotation about A″ second. Any fully specified, checked sequence is correct. But the order within a chosen sequence is not interchangeable: translating first by (x + 8, y − 1) and then rotating about the origin sends A(1, 1) to (9, 0) and then to (0, 9), nowhere near A″. When you write a sequence, number the steps.

Specifying a reflection when orientation is reversed. If triangle ABC above were to be carried onto the triangle with vertices (−1, 1), (−4, 1), (−1, 3), matching lengths gives A ↔ (−1, 1), B ↔ (−4, 1), C ↔ (−1, 3), and reading around the target goes left, then up-right, which is clockwise: reversed. A single reflection across the y-axis, (x, y) → (−x, y), sends A(1, 1) to (−1, 1), B(4, 1) to (−4, 1), C(1, 3) to (−1, 3). One step suffices, and it must be a reflection; no combination of rotations and translations can reverse orientation.

The misconception. Students frequently move a single vertex onto its match and stop, or describe a motion loosely ("rotate it and slide it over"). A specification must give the center, angle and direction of each rotation, the line of each reflection and the run and rise of each translation, in order, and it must be verified on every vertex. Another error is to assume a rotation about the origin when the figure clearly turns about one of its own vertices; the angle may be right and the image still wrong.

Worked, drawing a reflection across y = x: Quadrilateral with vertices (1, 0), (4, 0), (4, 2), (1, 3), reflected across y = x by (x, y) → (y, x): (0, 1), (0, 4), (2, 4), (3, 1). Check one vertex against the definition: the segment from (1, 3) to (3, 1) has midpoint (2, 2), on the line, and slope −1, perpendicular to the line of slope 1. Join the image vertices in order to draw the image; the side from (4, 0) to (4, 2), vertical of length 2, becomes the side from (0, 4) to (2, 4), horizontal of length 2.

Worked, drawing a rotation of 180° about (3, 2): Segment PQ with P(1, 1) and Q(5, 3). The half turn about (3, 2) is (x, y) → (6 − x, 4 − y): P′(5, 3), Q′(1, 1). The segment is carried onto itself with its endpoints swapped, because (3, 2) is the midpoint of PQ. A half turn about the midpoint of a segment always does this, which is worth remembering when specifying sequences.

Worked, specifying a sequence and testing the orientation call: Triangle with vertices (0, 0), (5, 0), (0, 2) onto the triangle with vertices (2, −1), (2, −6), (4, −1). Lengths: 5, 2 and √29 in both, so match (0, 0) ↔ (2, −1), (5, 0) ↔ (2, −6), (0, 2) ↔ (4, −1). Orientation of the original: from the first vertex, the 5-unit side points right and the 2-unit side points up, a counterclockwise turn from the first side to the second. Orientation of the target: the 5-unit side points down and the 2-unit side points right; turning from "down" to "right" is also counterclockwise. Same orientation, so no reflection. The 5-unit side must turn from pointing right to pointing down, a quarter turn clockwise. Step 1: rotate 90° clockwise about the origin, (x, y) → (y, −x): (0, 0), (0, −5), (2, 0). Step 2: translate by (x + 2, y − 1): (2, −1), (2, −6), (4, −1). All three vertices land. Had the orientation been misread as reversed, a reflection would have been inserted and the check on the third vertex would have failed, for example reflecting across the x-axis first and then rotating gives (2, −1), (2, −6), (0, −1), with (0, −1) instead of (4, −1). The vertex-by-vertex check is what catches a wrong orientation call.

Worked, checking a partner's sequence: A partner claims that "rotate 90° counterclockwise about (2, 1), then translate 3 left" carries triangle (2, 1), (5, 1), (2, 3) onto (−1, 1), (−1, 4), (−3, 1). Apply the rotation: displacements (0, 0), (3, 0), (0, 2) rotate to (0, 0), (0, 3), (−2, 0), giving (2, 1), (2, 4), (0, 1). Translate 3 left: (−1, 1), (−1, 4), (−3, 1). All three match, so the sequence is verified.

Recording task: On graph paper draw triangle ABC with A(1, 1), B(4, 1), C(1, 3) and triangle A″B″C″ with A″(7, 0), B″(7, 3), C″(5, 0). Draw the intermediate image after rotating 90° counterclockwise about the origin, A′(−1, 1), B′(−1, 4), C′(−3, 1), in a second color, and the final image after translating 8 right and 1 down in a third. Record a three-column table: vertex, after the rotation, after the translation. Then trace ABC on tracing paper, turn it a quarter turn counterclockwise about the origin and slide it, confirming it covers A″B″C″. Finally record what happens if the translation is applied first and the rotation second: A goes to (9, 0) and then to (0, 9), which shows the order matters.

Independent performance: Draw a scalene quadrilateral with grid-point vertices. Produce three images of it: one by a rotation of 90° about a vertex of the figure (not the origin), one by a reflection across a horizontal line that does not pass through the figure, and one by a rotation of 180° about a point outside the figure followed by a translation. Give each image to a partner as a list of coordinates only, and in return specify a complete sequence carrying the original onto each image you receive. Success criteria: every image you draw is correct by the definitions, with one vertex per image checked explicitly against the definition (distance and angle for rotations, perpendicular bisector for reflections); every sequence you specify names each transformation fully, in numbered order; every specified sequence is verified on all four vertices; the orientation check is written down and its conclusion matches the presence or absence of a reflection; and one sentence explains why a partner cannot accept a sequence checked on only one vertex.