How do rigid motions move a figure without changing its size or shape, and how do they define congruence?
Represent and perform reflections, rotations, and translations using rules and the coordinate plane; recognize that rigid motions preserve distance and angle; and define congruence of two figures as the existence of a sequence of rigid motions mapping one onto the other.
A NY Regents Geometry answer on rigid motions: performing reflections, rotations, and translations on the coordinate plane, why they preserve distance and angle, and how a sequence of rigid motions defines congruence.
Reviewed by: AI editorial process; not yet individually human-reviewed
Have a quick question? Jump to the Q&A page
Jump to a section
What this topic is asking
The Regents Geometry exam (the Congruence, G-CO, cluster) wants you to perform the three rigid motions, reflections, rotations, and translations, using coordinate rules, to know that they preserve distance and angle, and to use them to define congruence: two figures are congruent exactly when a sequence of rigid motions carries one onto the other. This transformational definition of congruence is foundational to the whole Geometry course.
The three rigid motions
A rigid motion (or isometry) is a transformation that preserves distance, so the image is congruent to the original. There are three basic types.
- A reflection flips a figure over a line of reflection; each point and its image are equidistant from that line, on opposite sides.
- A rotation turns a figure about a center point by a directed angle; every point stays the same distance from the center.
- A translation slides every point the same distance in the same direction, described by a vector.
Because each preserves distance and angle, the image is always congruent to the pre-image, which is the property that makes these motions special.
Coordinate rules
On the coordinate plane the rigid motions become algebraic rules you must know cold (they are not on the reference sheet).
Rigid motions and congruence
The Regents defines congruence through rigid motions: two figures are congruent if and only if there is a sequence of reflections, rotations, and translations that maps one exactly onto the other. This is why a transformation question so often ends with "explain why the figures are congruent": the expected answer names the rigid motion and states that rigid motions preserve distance and angle, so corresponding parts are equal.
A clarifying point is the difference between rigid motions and dilations. A dilation changes size (unless the scale factor is 1) and so is not a rigid motion; it produces a similar figure, not a congruent one. Reflections, rotations, and translations are the only congruence-preserving transformations, while a dilation belongs to the similarity topic. Keeping this distinction sharp prevents the common error of calling a resized figure "congruent".
Try this
Q1. Find the image of under a reflection over the -axis. [1 credit]
- Cue. Negate the : .
Q2. Find the image of under a 180-degree rotation about the origin. [1 credit]
- Cue. Negate both: .
Exam-style practice questions
Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Regents (style)2 marksPart I (multiple choice). The point is reflected over the -axis. What are the coordinates of its image ? (1) (2) (3) (4) Show worked answer →
The correct answer is (1).
A reflection over the -axis negates the -coordinate and keeps the -coordinate: . So . Choice (2) reflects over the -axis instead. Remember that reflecting over the -axis (the vertical axis) changes the left-right position, which is the -coordinate.
Regents (style)2 marksPart II (constructed response). Triangle is mapped onto triangle by a rotation of 180 degrees about the origin. Explain why triangle is congruent to triangle .Show worked answer →
A 2-credit question: 1 credit for identifying the rigid motion, 1 for the congruence reasoning.
A rotation of 180 degrees about the origin is a rigid motion (an isometry), which preserves all distances and angle measures. Because a rigid motion maps the figure onto an exact copy, every side and angle of triangle equals the corresponding side and angle of triangle . Two figures are congruent precisely when a sequence of rigid motions maps one onto the other, so triangle is congruent to triangle . A response that only says "they look the same" with no mention of distance-preservation earns 1 credit.
Related dot points
- Use the triangle congruence criteria (SSS, SAS, ASA, AAS, HL) to prove two triangles congruent, and use CPCTC (corresponding parts of congruent triangles are congruent) to justify further equal sides or angles.
A NY Regents Geometry answer on triangle congruence: the SSS, SAS, ASA, AAS, and HL criteria, why SSA and AAA fail, and using CPCTC to conclude further equal parts once triangles are proven congruent.
- Perform compass-and-straightedge constructions: copy a segment and an angle; bisect a segment (perpendicular bisector) and an angle; construct a perpendicular and a parallel line; construct an equilateral triangle; and explain why each construction produces the intended figure.
A NY Regents Geometry answer on compass-and-straightedge constructions: copying segments and angles, perpendicular and angle bisectors, perpendicular and parallel lines, the equilateral triangle, and why each one works.
- Prove theorems about lines and angles (vertical angles, the angle relationships from parallel lines cut by a transversal) and about triangles (the angle sum is 180 degrees, the exterior angle theorem, the isosceles triangle base angles, and the midsegment theorem).
A NY Regents Geometry answer on proving angle and triangle theorems: vertical angles, parallel-line angle pairs, the triangle angle sum, the exterior angle theorem, isosceles base angles, and the midsegment theorem.
- Perform dilations on the coordinate plane and describe their effect on lengths and angles; define similarity through a sequence of rigid motions and a dilation; and prove triangles similar using AA (and SAS, SSS similarity), then use proportions to find missing lengths.
A NY Regents Geometry answer on dilations and similarity: performing a dilation about a center, why angles are preserved while lengths scale, the AA similarity criterion, and using proportions to find missing sides.
- Use the distance, midpoint, and slope formulas to analyze figures; determine parallel and perpendicular lines from slope; partition a directed segment into a given ratio; and write equations of lines satisfying given conditions.
A NY Regents Geometry answer on coordinate geometry: the distance, midpoint, and slope formulas, parallel and perpendicular slope conditions, partitioning a segment in a ratio, and writing equations of lines.
Sources & how we know this
- Regents Examination in Geometry — NYSED (2024)
- New York State Next Generation Mathematics Learning Standards — NYSED (2017)