Identify and apply rigid motions (translations, reflections, rotations), describe their effect on coordinates, and use them to explain why two figures are congruent.

What this topic is asking

The Geometry category begins with the G-CO standards: Congruence built on rigid motions. The Grade 10 MCAS expects you to identify and apply translations, reflections, and rotations, to describe how each affects coordinates, and to use a sequence of rigid motions to explain why two figures are congruent. This transformational view of congruence is central to the 2017 framework, so the reasoning ("maps onto by rigid motions") is as important as the coordinate rules.

The three rigid motions

A rigid motion (isometry) is a transformation that preserves distance, so the image is congruent to the original. There are three basic types:

• Translation (a slide): every point moves the same distance in the same direction. Coordinates change by a fixed amount, such as $(x, y) \to (x + 3, y - 2)$.
• Reflection (a flip): the figure is mirrored across a line. The image is the same distance from the line as the original, on the opposite side.
• Rotation (a turn): the figure turns about a fixed center by a given angle, clockwise or counterclockwise.

All three keep lengths and angles unchanged; only the position or orientation changes. A reflection also reverses orientation (a figure and its mirror image), which the others do not.

Coordinate rules

The MCAS often gives a transformation in the coordinate plane and asks for the image's coordinates. The rules to know:

• Reflection across the x-axis: $(x, y) \to (x, -y)$ (the y-coordinate flips sign).
• Reflection across the y-axis: $(x, y) \to (-x, y)$ (the x-coordinate flips sign).
• Translation by $(a, b)$: $(x, y) \to (x + a, y + b)$.
• Rotation $180^\circ$ about the origin: $(x, y) \to (-x, -y)$.
• Rotation $90^\circ$ counterclockwise about the origin: $(x, y) \to (-y, x)$.

Memorizing which coordinate changes for each reflection prevents the most common slips.

Congruence through rigid motions

In the 2017 framework, two figures are congruent precisely when one can be mapped onto the other by a sequence of rigid motions. This is the modern definition, and the MCAS asks you to use it. Because rigid motions preserve lengths and angles, if such a sequence exists, then all corresponding sides and angles are equal, which is the classical statement of congruence.

So to argue two triangles are congruent, you describe the rigid motions (for example "a reflection across line $\ell$ followed by a translation") that carry one exactly onto the other. Conversely, if no sequence of rigid motions works (only a size change does), the figures are similar but not congruent.

Triangle congruence criteria

While the framework defines congruence through rigid motions, the classical shortcuts for triangles still apply and the MCAS uses them. Two triangles are congruent if they match by SSS (three pairs of sides), SAS (two sides and the included angle), ASA (two angles and the included side), or AAS (two angles and a non-included side). A right-triangle special case is HL (hypotenuse and one leg). Note that SSA and AAA are not valid: AAA gives only similarity (same shape, possibly different size), and SSA can produce two different triangles. Choosing a valid criterion is what a congruence argument rests on.

Symmetry as a transformation

A figure has line symmetry if a reflection maps it onto itself, and rotational symmetry if a rotation of less than a full turn does. A square has four lines of symmetry and 90-degree rotational symmetry; a regular hexagon has six lines of symmetry. The MCAS may ask how many lines of symmetry a shape has, or what rotation carries it onto itself; both are questions about which rigid motions leave the figure unchanged, tying symmetry directly to this topic.

Try this

Q1. Reflect $(−2, 7)$ across the x-axis.

• Cue. $(-2, -7)$.

Q2. Translate $(1, 1)$ by $(−3, 5)$.

• Cue. $(-2, 6)$.