This unit is about moving and resizing figures with precise rules: sliding, flipping, turning, and scaling them on the coordinate plane. By the end, you should be able to apply any transformation to a figure's coordinates, describe its symmetry, and identify a single equivalent transformation when a sequence can be simplified.
Read the overview first for the big picture, then work through each topic's notes and try the checkpoint question before checking the answer.
Transformations give geometry a precise way to say two figures are congruent or similar. Instead of eyeballing shapes, you describe exactly how one figure maps onto another using coordinate rules.
Honors Geometry Unit 9 covers transformations of figures in the plane and their connection to congruence, similarity, and symmetry. You learn dilations and similarity transformations, the three rigid motions (translations, reflections, and rotations) with their coordinate rules, line, rotational, and plane symmetry in 2D and 3D figures, and compositions, where two or more transformations are applied in sequence. Expect problems that ask you to find image coordinates, identify a single transformation equivalent to a sequence, and explain why two figures are congruent or similar.
Translations, reflections, and rotations move a figure without stretching it, so the image is congruent to the original. This is the formal definition of congruence in this course. If a sequence of rigid motions maps one figure onto another, the two figures are congruent.
A dilation with nonzero scale factor multiplies every distance from the center by . Angles stay the same and side lengths also scale by , so the image is similar to the original. Combining a dilation with rigid motions gives a similarity transformation.
A figure has symmetry when a reflection or rotation maps it exactly onto itself. A composition chains transformations together, and some chains collapse into a single move, like two reflections across parallel lines acting as one translation.
Treat each transformation as a rule like that you apply to every vertex. Then ask what it preserves. If distances and angles are preserved, the image is congruent. If a dilation with a nonzero scale factor and rigid motions map one figure onto another, the figures are similar.
A dilation resizes a figure from a fixed point called the center of dilation using a scale factor k.
These three are rigid motions, also called isometries, because they preserve distance and angle measure.
A figure has symmetry when some reflection or rotation maps it exactly onto itself.
A composition applies one transformation, then applies another to the result.
Open this guide for a closer review of the topic.
Open this guide for a closer review of the topic.
A dilation resizes a figure from a fixed point called the center of dilation using a scale factor . For a nonzero scale factor, the figure enlarges if , shrinks if , and keeps its size if . If , the image also lands on the opposite side of the center. A scale factor of 0 sends every point to the center, so it does not produce a similar figure.
With the center at the origin, the rule is . With a center at , use . For , angles stay the same, side lengths multiply by , and areas multiply by . A side that doesn't pass through the center maps to a parallel side.
Worked example. Dilate by about the center . First find the offset from the center, . Multiply by 2 to get . Add the center back to get .
A similarity transformation is a dilation with a nonzero scale factor followed by any number of rigid motions. Two figures are similar if one can be mapped onto the other this way, which means corresponding angles are congruent and corresponding sides share one ratio. A 3-4-5 triangle and a 6-8-10 triangle are similar with scale factor 2.
Check yourself: A square with side length 6 is dilated by . What are the new side length and area? (Answer: The side becomes , and the area goes from 36 to 4, which is .)
These three are rigid motions, also called isometries, because they preserve distance and angle measure. The image is always congruent to the pre-image.
A translation slides every point by the same vector , so . A reflection flips a figure over a line, and each image point is the same distance from the line as its pre-image, on the other side. A rotation turns a figure about a center by a given angle, and rotations are counterclockwise unless the problem says otherwise.
Orientation is the order of the vertices around the figure. Translations and rotations keep it, so if go counterclockwise, do too. Reflections reverse it. This gives you a quick test when a problem asks which transformation produced an image.
Worked example. Reflect over . The rule swaps the coordinates and negates both, so . Rotating the same point 90° counterclockwise about the origin uses , which gives .
If you forget a rule, test the point . A 90° counterclockwise turn should send it to , and does exactly that.
Check yourself: Where does land after a 270° counterclockwise rotation about the origin? (Answer: The rule is , so the image is .)
| Transformation | Coordinate rule | Keeps orientation? |
|---|---|---|
| Translation by $(a, b)$ | $(x + a, y + b)$ | Yes |
| Reflect over x-axis | $(x, -y)$ | No |
| Reflect over y-axis | $(-x, y)$ | No |
| Reflect over $y = x$ | $(y, x)$ | No |
| Reflect over $y = -x$ | $(-y, -x)$ | No |
| Rotate 90° CCW about the origin | $(-y, x)$ | Yes |
| Rotate 180° about the origin | $(-x, -y)$ | Yes |
| Rotate 270° CCW (90° CW) about the origin | $(y, -x)$ | Yes |
A figure has symmetry when some reflection or rotation maps it exactly onto itself. Symmetry is just a transformation that leaves the figure looking unchanged.
Line symmetry means a reflection across some line maps the figure onto itself, so folding along that line matches the halves. Rotational symmetry means a rotation by some angle between 0° and 360° maps the figure onto itself. The order of rotational symmetry counts how many positions in one full turn match the original, including the 360° turn.
A regular -gon has lines of symmetry and rotational symmetry of order , with a smallest angle of . A regular hexagon has 6 lines and turns onto itself every 60°. The two kinds of symmetry don't always come together. A parallelogram that isn't a rectangle or rhombus has no lines of symmetry but still has rotational symmetry of order 2.
In 3D, a plane of symmetry splits a solid into two mirror-image halves, and an axis of rotational symmetry is a line about which a rotation greater than 0° and less than 360° maps the solid onto itself. A cube has 9 planes of symmetry and 13 rotation axes (3 through face centers, 4 through opposite vertices, 6 through midpoints of opposite edges). A sphere has infinitely many of both.
Check yourself: How many lines of symmetry does a non-square rhombus have, and what is its order of rotational symmetry? (Answer: 2 lines of symmetry, along the diagonals, and order 2, since a 180° turn maps it onto itself.)
| Figure | Lines of symmetry | Rotational order | Smallest rotation angle |
|---|---|---|---|
| Equilateral triangle | 3 | 3 | 120° |
| Square | 4 | 4 | 90° |
| Non-square rectangle | 2 | 2 | 180° |
| Parallelogram (no right angles, unequal sides) | 0 | 2 | 180° |
| Regular hexagon | 6 | 6 | 60° |
| Circle | Infinitely many | Infinite | No smallest positive angle |
A composition applies one transformation, then applies another to the result. In notation, means do first, then , so you read right to left just like function composition in algebra.
Order usually changes the answer, so track which step comes first. Some compositions simplify to a single transformation. Two translations combine by adding vectors. Two rotations about the same center combine by adding angles. Two dilations from the same center combine by multiplying scale factors. Two reflections across parallel lines give a translation of twice the distance between the lines, perpendicular to them. Two reflections across intersecting lines give a rotation about the intersection point by twice the directed angle from the first line to the second. A translation followed by a reflection across a line parallel to the translation is called a glide reflection.
Worked example. Take . Reflect over the x-axis to get , then translate by to get . Now reverse the order. Translate first to get , then reflect to get . The two answers differ, which shows order matters.
For parallel lines, reflect over to get , then over to get . The lines are 3 apart, and the point moved 6 units right.
Check yourself. Reflect across the x-axis, then across a line through the origin at 40° counterclockwise from the positive x-axis. What single transformation has the same result? (Answer: A rotation of counterclockwise about the origin.)
| Term | Definition |
|---|---|
| Transformation | A rule that maps every point of a figure, called the pre-image, to a point of a new figure, called the image. |
| Isometry | Another name for a rigid motion, meaning a transformation that preserves distance. |
| Translation | A rigid motion that slides every point of a figure by the same vector $(a, b)$. |
| Reflection | A rigid motion that flips a figure across a line, reversing its orientation. |
| Rotation | A rigid motion that turns a figure about a fixed center by a given angle, counterclockwise unless stated otherwise. |
| Dilation | A transformation with scale factor $k$ that multiplies every distance from a center point by $|k|$. It produces a similar figure when $k \neq 0$. |
| Congruent figures | Figures where one can be mapped onto the other by a sequence of rigid motions. |
| Similar figures | Figures where one can be mapped onto the other by a dilation with a nonzero scale factor plus rigid motions, so corresponding angles match and sides are proportional. |
| Rotational symmetry | The property of a figure that maps onto itself after a rotation by some angle between 0° and 360°. |
Both rules swap the coordinates, so they look alike. A 90° counterclockwise rotation is , while reflecting over is with no sign change. Test with , which should rotate to and reflect to as well, then try , which rotates to but reflects to .
In , it's tempting to start with because it appears first. The convention is right to left, so happens first. When a problem uses words like "followed by" instead of symbols, follow the words in reading order.
The rule only works for a dilation centered at the origin. For a center at , subtract the center, multiply by , then add the center back. A quick check is that the center itself should map to itself.
Lengths scale by , but area depends on two dimensions, so it scales by . A dilation with makes a triangle's sides 3 times longer and its area 9 times larger. Angles don't change at all.