9.1 Translations, reflections, and rotations

Transformations in the Coordinate Plane

Geometric transformations let you move, flip, and spin figures around the coordinate plane while following precise coordinate rules. In Honors Geometry, you need to know three rigid transformations: translations, reflections, and rotations. Each one changes a figure's position or orientation, but none of them change its size or shape.

Types of Geometric Transformations

Translations slide every point of a figure the same distance in the same direction. You describe a translation using a vector $(a, b)$, where $a$ is the horizontal shift and $b$ is the vertical shift. A positive $a$ moves right; a negative $a$ moves left. Same logic applies vertically with $b$.

Reflections flip a figure over a line called the line of reflection, producing a mirror image. The most common lines of reflection are the $x$-axis, the $y$-axis, $y = x$, and $y = -x$. Every point in the image is the same distance from the line of reflection as the corresponding point in the pre-image, just on the opposite side.

Rotations turn a figure around a fixed point called the center of rotation. You specify a rotation by its angle (in degrees) and direction (clockwise or counterclockwise). The most common rotation angles are 90°, 180°, and 270°. Unless stated otherwise, rotations are counterclockwise.

Applying Transformations to Figures

Translations

To translate a figure by vector $(a, b)$:

1. Add $a$ to the $x$-coordinate of each point.
2. Add $b$ to the $y$-coordinate of each point.

Reflections

The rule depends on which line you're reflecting over:

• Over the $x$-axis: Negate the $y$-coordinate. $(x, y) \rightarrow (x, -y)$
• Over the $y$-axis: Negate the $x$-coordinate. $(x, y) \rightarrow (-x, y)$
• Over $y = x$: Swap the coordinates. $(x, y) \rightarrow (y, x)$
• Over $y = -x$: Swap the coordinates and negate both. $(x, y) \rightarrow (-y, -x)$

Rotations About the Origin

For common counterclockwise rotations, you don't need the general formula every time. Memorize these three:

• 90° counterclockwise: $(x, y) \rightarrow (-y, x)$
• 180°: $(x, y) \rightarrow (-x, -y)$
• 270° counterclockwise (same as 90° clockwise): $(x, y) \rightarrow (y, -x)$

For a clockwise rotation, you can either use the clockwise rules directly or convert to the equivalent counterclockwise angle (e.g., 90° clockwise = 270° counterclockwise).

The general rotation formula for any angle $\theta$ counterclockwise about the origin is:

$(x, y) \rightarrow (x \cos \theta - y \sin \theta,\; x \sin \theta + y \cos \theta)$

For clockwise rotations, substitute $-\theta$ for $\theta$. In practice, the 90°/180°/270° shortcut rules above come from plugging those specific angles into this formula.

Coordinates After Transformations

When you need to find the image of an entire figure, apply the appropriate rule to every vertex individually. A quick summary of the coordinate rules:

Effects of Transformations on Figures

All three transformations are rigid motions (also called isometries), meaning they preserve the size and shape of the figure. Distances between points and angle measures stay the same. Here's what changes and what doesn't:

• Translations change position only. The figure's orientation stays the same. If a triangle's vertices go A-B-C counterclockwise before the translation, they still go A'-B'-C' counterclockwise after.
• Reflections change position and reverse orientation. A counterclockwise ordering of vertices becomes clockwise in the image. This is the key distinction: reflections produce a mirror image, not just a shifted copy.
• Rotations change position and orientation (the figure is turned), but they do not reverse orientation the way reflections do. The vertex ordering stays the same (counterclockwise stays counterclockwise).

Because all three are rigid motions, the pre-image and image are always congruent. This fact is the foundation for using transformations to prove congruence later in the course.