Use geometric descriptions of rigid motions to predict their effect on a figure and use the rigid-motion definition of congruence to decide whether two figures are congruent

Learning goal: Use geometric descriptions of rigid motions (a reflection across a named line, a rotation about a named center through a named angle, a translation by a named segment) to transform figures and to predict, before drawing, the effect of a given rigid motion on a given figure; and, given two figures, use the definition of congruence in terms of rigid motions to decide whether they are congruent.

Rigid motions and their geometric descriptions. A rigid motion is a transformation of the plane that preserves distance and angle measure. The earlier lessons defined three of them geometrically: the reflection across a line m places each image so that m is the perpendicular bisector of the segment joining a point to its image; the rotation about a center C through an angle θ keeps each point on its circle about C and turns it through θ; the translation by a directed segment AB moves every point along a segment parallel to AB, equal to AB and in the same direction. A geometric description names the line, or the center and angle, or the segment. It does not need a coordinate rule, though on a grid a rule is a convenient way to compute. A sequence of rigid motions, applied one after another, is again a rigid motion, because each step preserves every distance and angle and so does the chain.

Predicting the effect before drawing. Each description carries predictions that can be stated in advance and then checked. A reflection fixes every point on its line and moves every other point to the opposite side, reverses orientation (a figure whose vertices read counterclockwise reads clockwise afterwards), and sends any segment perpendicular to the line onto its own line. A rotation that is not the identity (not a multiple of 360°) fixes exactly one point, the center, turns every segment through the angle of rotation (so a rotation of 90° sends every segment to a perpendicular segment, and a rotation of 180° sends every segment to a parallel segment or to one on the same line), and preserves orientation. A translation fixes no point at all unless the segment has length 0, sends every segment to a parallel segment of the same length (or to one on the same line, when the segment points along the translation), and preserves orientation. Predict first, then draw, then compare; a drawing that contradicts a prediction contains an error in one or the other, and finding out which is the learning.

A worked prediction. Rectangle with vertices (1, 1), (5, 1), (5, 3), (1, 3) is to be reflected across the line y = x. Predictions from the description: the vertex (1, 1) lies on the line, so it stays fixed; the horizontal sides are neither parallel nor perpendicular to the line, so they do not stay horizontal; reflecting across y = x swaps the roles of horizontal and vertical, so horizontal sides become vertical; the image has the same side lengths 4 and 2; the orientation reverses. Now compute with the rule (x, y) → (y, x): (1, 1), (1, 5), (3, 5), (3, 1). The vertex (1, 1) is fixed, the side from (1, 1) to (5, 1) of length 4 became the side from (1, 1) to (1, 5), vertical of length 4, and reading the original vertices goes right then up (counterclockwise) while reading the images goes up then right (clockwise). Every prediction holds.

The definition of congruence. Two figures are congruent when there is a sequence of rigid motions that carries one figure exactly onto the other. This is the definition; "same size and shape" is the informal idea it makes precise. Because every rigid motion preserves distance and angle, two congruent figures have all corresponding lengths and angles equal. The definition also gives a method for deciding: try to find the sequence, and verify it on every vertex. If the sequence exists, the figures are congruent. If a distance in one figure has no match in the other, no rigid motion can exist, because rigid motions cannot change any distance, and the figures are not congruent.

A worked decision. Triangle PQR has P(0, 0), Q(4, 0), R(0, 3); triangle P″Q″R″ has P″(6, 5), Q″(6, 1), R″(3, 5). Step 1, compare distances: PQ = 4, PR = 3, QR = √(16 + 9) = 5; P″Q″ = 4 (vertical), P″R″ = 3 (horizontal), Q″R″ = √(9 + 16) = 5. The sets of side lengths match, with P ↔ P″, Q ↔ Q″, R ↔ R″. Step 2, compare orientation: P → Q → R goes right then up, counterclockwise; P″ → Q″ → R″ goes down then left, clockwise. The orientation is reversed, so the sequence must contain a reflection. Step 3, build the sequence: reflect across the x-axis, (x, y) → (x, −y), giving (0, 0), (4, 0), (0, −3); rotate 90° clockwise about the origin, (x, y) → (y, −x), giving (0, 0), (0, −4), (−3, 0); translate 6 right and 5 up, giving (6, 5), (6, 1), (3, 5). Step 4, check every vertex: (6, 5) = P″, (6, 1) = Q″, (3, 5) = R″. A sequence of rigid motions carries PQR onto P″Q″R″, so by definition the triangles are congruent. The first two steps together are the reflection across the line y = −x, so a single reflection followed by the translation also works; sequences are not unique, and any verified sequence settles the question.

When the answer is no. Take the triangle (0, 0), (6, 0), (0, 4) and the triangle (0, 0), (6, 0), (2, 4). The first has sides 6, 4 and √52 ≈ 7.21; the second has sides 6, √20 ≈ 4.47 and √32 ≈ 5.66. The first triangle has a side of length 4 and the second has no side of length 4. A rigid motion would have to carry that side to a side of length 4, which does not exist, so no sequence of rigid motions can work: the triangles are not congruent. Notice that the argument does not try every possible sequence; it uses the fact that rigid motions preserve distance to rule all of them out at once.

Sides alone can mislead for quadrilaterals. The square (0, 0), (5, 0), (5, 5), (0, 5) and the rhombus (0, 0), (5, 0), (8, 4), (3, 4) both have four sides of length 5. They are not congruent: the diagonals of the square are both 5√2 ≈ 7.07, while the diagonals of the rhombus are √80 ≈ 8.94 and √20 ≈ 4.47, and a rigid motion must preserve the distance between opposite vertices as well as between adjacent ones. For triangles, matching the three sides is enough (a later lesson proves this), but for figures with more vertices, the honest check is the definition: find and verify a sequence, or exhibit a distance, between any two points of the figure, that has no match.

The misconception. Students often decide congruence by area, or by "it looks the same". A 2-by-8 rectangle and a 4-by-4 square both have area 16 and are not congruent, because the rectangle has a side of length 8 and the square has none. Congruence is a statement about a rigid motion existing, and the only evidence that counts is a verified sequence or a distance that rules one out.

Worked, predicting a rotation of 180°: Triangle with vertices (2, 1), (6, 1), (2, 4) is to be rotated 180° about the point (4, 1), which is the midpoint of the side from (2, 1) to (6, 1). Predictions: the center is fixed, the side through the center is carried onto itself with its endpoints swapped, every side becomes a parallel side, and the orientation is unchanged. Rule for the half turn about (4, 1): (x, y) → (8 − x, 2 − y). Images: (6, 1), (2, 1), (6, −2). The endpoints (2, 1) and (6, 1) did swap; the vertical side from (2, 1) to (2, 4) became the vertical side from (6, 1) to (6, −2), parallel and of the same length 3. The predictions hold, and the original and image together form a parallelogram with vertices (2, 1), (6, −2), (6, 1), (2, 4).

Worked, predicting a translation: Quadrilateral (0, 0), (3, 0), (4, 2), (1, 2) is translated by the segment from (1, 1) to (3, −2), a run of 2 and a rise of −3. Predictions: no vertex is fixed, every side is parallel to its image, every side keeps its length. Images: (2, −3), (5, −3), (6, −1), (3, −1). The side from (3, 0) to (4, 2), with run 1 and rise 2, became the side from (5, −3) to (6, −1), also run 1 and rise 2: parallel and equal, as predicted, and no point stayed where it was.

Worked, deciding congruence of quadrilaterals: ABCD has A(0, 0), B(3, 0), C(3, 2), D(0, 1); EFGH has E(5, 4), F(5, 1), G(3, 1), H(4, 4). Sides: AB = 3, BC = 2, CD = √(9 + 1) = √10, DA = 1; EF = 3, FG = 2, GH = √(1 + 9) = √10, HE = 1. Diagonals: AC = √13 and EG = √(4 + 9) = √13; BD = √10 and FH = √10. Orientation: A → B → C goes right then up, counterclockwise; E → F → G goes down then left, clockwise, so a reflection is needed. Reflect across the x-axis: (0, 0), (3, 0), (3, −2), (0, −1). Rotate 90° clockwise about the origin by (x, y) → (y, −x): (0, 0), (0, −3), (−2, −3), (−1, 0). Translate 5 right and 4 up: (5, 4), (5, 1), (3, 1), (4, 4). Every vertex lands on E, F, G, H in order, so the quadrilaterals are congruent.

Worked, ruling congruence out: Is the triangle (0, 0), (4, 0), (0, 3) congruent to the triangle (0, 0), (4, 0), (1, 3)? Sides of the first: 4, 3, 5. Sides of the second: 4, √10 ≈ 3.16, √18 ≈ 4.24. The first has a side of length 5 and the second does not; since a rigid motion preserves every distance, no sequence of rigid motions can carry one onto the other. Not congruent. Both triangles share the base from (0, 0) to (4, 0), which shows that sharing a side is no evidence of congruence.

Recording task: On graph paper draw triangle PQR with P(0, 0), Q(4, 0), R(0, 3) and triangle P″Q″R″ with P″(6, 5), Q″(6, 1), R″(3, 5). Before transforming, write three predictions for the reflection across the x-axis (which points are fixed, what happens to orientation, what happens to the horizontal side). Then draw the image after each of the three steps, reflection across the x-axis, rotation of 90° clockwise about the origin, translation 6 right and 5 up, in three colors, recording a table with a row for each vertex and a column for each step; the last column must read (6, 5), (6, 1), (3, 5). Check each prediction against the first image and write whether it held.

Independent performance: Prepare four pairs of figures on grid paper: two pairs that are congruent (one pair needing a reflection, one not) and two pairs that are not (one pair with a mismatched side length, one pair, such as the square and rhombus with sides 5, with matching sides but a mismatched diagonal). Exchange with a partner and decide each pair using only the definition of congruence in terms of rigid motions. Success criteria: for each congruent pair a complete sequence is named with lines, centers, angles and segments, and verified on every vertex; for each non-congruent pair a specific distance is exhibited that appears in one figure and not the other, with the computation; the orientation check is recorded for every pair; a prediction about fixed points and orientation is written before each rigid motion is applied and compared afterward; and a closing paragraph explains why equal area or equal side lengths are not enough to conclude congruence.