Transformations in Geometry
Geometric transformations describe how figures move, flip, turn, and resize on the coordinate plane. Understanding transformations is essential for proving congruence and similarity, analyzing symmetry, and solving real-world design problems.
This guide covers translations, reflections, rotations, dilations, and compositions of transformations with coordinate rules, SVG diagrams, worked examples, and a practice quiz.
1Introduction
A transformation is a function that maps every point of a figure to a new location, producing an image. The original figure is called the pre-image. Transformations are the mathematical language for describing motion and change in geometry.
Architects use reflections to design symmetric buildings. Animators compose translations and rotations to move characters. GPS satellites use coordinate transformations to convert between reference frames. Textile designers use repeated translations and rotations to create patterns.
Rigid Motions (Isometries)
Translations, reflections, and rotations preserve size and shape. The image is congruent to the pre-image.
Non-Rigid Motions
Dilations change size while preserving shape. The image is similar to the pre-image.
2Key Definitions
Transformation
A function that maps every point of a figure to a new position.
Pre-image / Image
The original figure (pre-image) and the result after the transformation (image, often noted with primes: A').
Isometry (Rigid Motion)
A transformation that preserves distances and angle measures. The image is congruent to the pre-image.
Composition
Performing two or more transformations in sequence. The result depends on the order.
Translation
A slide in a given direction and distance.
Reflection
A flip over a line of reflection.
Rotation
A turn about a center point by an angle.
Dilation
A resize by a scale factor from a center point.
3Translations
A translation slides every point of a figure the same distance in the same direction. It is described by a translation vector <a, b>.
(x, y) → (x + a, y + b)
Every point moves a units horizontally and b units vertically.
Triangle ABC translated by vector <4, 2> to produce image A'B'C'.
Translations preserve size, shape, and orientation. The pre-image and image are congruent, and corresponding sides remain parallel.
4Reflections
A reflection flips a figure over a line called the line of reflection. Each point and its image are equidistant from the line.
Coordinate Rules for Common Reflections
Over x-axis: (x, y) → (x, −y)
Over y-axis: (x, y) → (−x, y)
Over y = x: (x, y) → (y, x)
Over y = −x: (x, y) → (−y, −x)
Triangle ABC reflected over the y-axis. Each point and its image are equidistant from the line.
Reflections reverse orientation. If the pre-image vertices go clockwise, the image vertices go counterclockwise. This is the key difference between reflections and the other rigid motions.
5Rotations
A rotation turns a figure about a fixed point called the center of rotation by a given angle. Positive angles indicate counterclockwise (CCW) rotation.
Coordinate Rules (about the origin)
90° CCW: (x, y) → (−y, x)
180°: (x, y) → (−x, −y)
270° CCW (= 90° CW): (x, y) → (y, −x)
Point P(3, 1) rotated 90° CCW about the origin to P'(−1, 3).
A 90° CCW rotation is the same as a 270° CW rotation. A 180° rotation is the same clockwise or counterclockwise. Always note the direction specified in the problem.
6Dilations
A dilation resizes a figure by a scale factor (k) from a center of dilation. It preserves shape but changes size (unless k = 1).
(x, y) → (kx, ky)
Center at origin. Each coordinate is multiplied by the scale factor k.
k > 1
Enlargement
k = 1
No change (identity)
0 < k < 1
Reduction
Dilation is NOT an isometry. It does not preserve distance. The image is similar to the pre-image but not congruent (unless k = 1 or k = −1).
7Compositions of Transformations
A composition applies two or more transformations in sequence. The second transformation is applied to the image of the first.
Key Composition Facts
- Two reflections over parallel lines = a translation (distance = 2× the gap between lines).
- Two reflections over intersecting lines = a rotation (angle = 2× the angle between lines).
- A glide reflection = a reflection followed by a translation parallel to the line of reflection.
- Order matters! In general, composing T then R gives a different result from R then T.
When working with compositions, always apply transformations from right to left (just like function composition). If the problem says "reflect then translate," apply the reflection first, then translate the result.
8Worked Examples
Example 1: Translation
Translate the point A(4, −2) by the vector <−3, 5>.
Step 1: Apply the rule: (x, y) → (x + a, y + b)
Step 2: A' = (4 + (−3), −2 + 5)
Step 3: A' = (1, 3)
Answer: A' = (1, 3)
Example 2: Reflection over y = x
Reflect the point B(5, −1) over the line y = x.
Step 1: Apply the rule: (x, y) → (y, x)
Step 2: B' = (−1, 5)
Answer: B' = (−1, 5)
Example 3: 90° CCW Rotation + Translation
Point C(2, 6) is first rotated 90° CCW about the origin, then translated by <3, −1>. Find C''.
Step 1: Rotate 90° CCW: (x, y) → (−y, x)
C' = (−6, 2)
Step 2: Translate by <3, −1>: (x, y) → (x + 3, y − 1)
C'' = (−6 + 3, 2 − 1) = (−3, 1)
Answer: C'' = (−3, 1)
Example 4: Dilation
Dilate triangle with vertices D(1, 3), E(4, 3), F(4, 1) by scale factor k = 2, center at origin.
Step 1: Apply (x, y) → (2x, 2y) to each vertex:
D' = (2, 6)
E' = (8, 6)
F' = (8, 2)
Answer: D'(2, 6), E'(8, 6), F'(8, 2). The side lengths are doubled but the shape is preserved.
9Memory Aids
"Slide, Flip, Turn, Resize"
Translation = Slide, Reflection = Flip, Rotation = Turn, Dilation = Resize.
"90 Negate-Y, Swap; 180 Negate Both; 270 Negate-X, Swap"
90° CCW: (−y, x). 180°: (−x, −y). 270° CCW: (y, −x).
"x-axis: negate y; y-axis: negate x; y=x: swap"
The axis name tells you which coordinate stays the same (reflecting over the x-axis keeps x).
"TRR are rigid; D is not"
Translation, Reflection, Rotation preserve congruence. Dilation only preserves similarity.
10Common Mistakes
Mixing up CW and CCW rotations
A 90° counterclockwise rotation uses (−y, x), while a 90° clockwise rotation uses (y, −x). Always check the direction specified in the problem.
Applying composition in the wrong order
If the problem says "reflect then translate," apply the reflection first. Reversing the order usually gives a different result.
Forgetting that reflections reverse orientation
Unlike translations and rotations, reflections flip the figure. If pre-image vertices are ordered clockwise, image vertices are counterclockwise.
Treating dilation as an isometry
Dilation changes distances. The image is similar but not congruent (unless k = ±1).
Negating the wrong coordinate in reflections
Reflecting over the x-axis negates y, not x. Reflecting over the y-axis negates x, not y. The axis name tells you which coordinate is preserved.
Forgetting the center of dilation
The formula (x, y) → (kx, ky) only works when the center is the origin. For other centers, translate to the origin first, dilate, then translate back.
Quick Revision Summary
- ✓A transformation maps every point of a figure to a new location, producing an image.
- ✓Isometries (rigid motions) preserve size and shape: translations, reflections, rotations.
- ✓Translation: (x, y) → (x + a, y + b). Preserves orientation.
- ✓Reflection: flip over a line. Reverses orientation. x-axis: (x, −y). y-axis: (−x, y). y=x: (y, x).
- ✓Rotation: turn about a center. 90° CCW: (−y, x). 180°: (−x, −y). 270° CCW: (y, −x).
- ✓Dilation: (x, y) → (kx, ky). Preserves shape but not size. k > 1 enlarges, 0 < k < 1 reduces.
- ✓Compositions: apply transformations in sequence. Order matters. Two reflections over intersecting lines = rotation.
- ✓Congruent figures can be mapped by rigid motions. Similar figures can be mapped by a dilation followed by rigid motions.
Frequently Asked Questions
- Which transformations preserve congruence?
- Translations, reflections, and rotations are rigid motions that preserve size and shape. Dilations change size, so they produce similar but not congruent figures.
- What is the difference between a rigid motion and a dilation?
- Rigid motions (translations, reflections, rotations) preserve both shape and size — the image is congruent to the pre-image. A dilation changes the size by a scale factor while preserving shape — the image is similar to the pre-image.
- How do you determine the center and angle of rotation?
- Connect corresponding pre-image and image points. The perpendicular bisectors of these segments intersect at the center of rotation. The angle between any pre-image point, the center, and its image point gives the rotation angle.
- Can a composition of two reflections equal a rotation?
- Yes. Two reflections over intersecting lines produce a rotation about the intersection point with an angle equal to twice the angle between the lines.
- What happens when the scale factor of a dilation is negative?
- A negative scale factor produces a dilation combined with a 180° rotation about the center. The image is on the opposite side of the center from the pre-image.
Practice Quiz
Test your understanding of geometric transformations — translations, reflections, rotations, and dilations.
1.Which transformation is NOT an isometry?
2.A point P(5, -3) is translated by the vector < -2, 4 >. What are the coordinates of P'?
3.What is the image of the point (-4, 7) after a reflection over the x-axis?
4.If a triangle is rotated 180° about the origin, what is the coordinate rule applied to its vertices (x, y)?
5.A square with vertices at (0,0), (2,0), (2,2), (0,2) is dilated by a scale factor of 3 centered at the origin. What are the new coordinates of the vertex (2,2)?
6.Which transformation reverses the orientation of a figure?
7.A point (6, -2) is reflected over the line y = x. What are its new coordinates?
8.A figure is dilated by a scale factor of 0.5. This means the figure is:
9.A point A(3, 5) is rotated 90° counterclockwise about the origin, then translated by vector <1, -2>. What are the final coordinates of A''?
10.If two figures are congruent, which of the following statements must be true?
Final Study Advice
- 1.Practice plotting pre-images and images on graph paper to visualize each transformation.
- 2.Memorize the coordinate rules for the four standard reflections and three standard rotations.
- 3.When composing transformations, always apply them in the correct order (right to left).
- 4.Check your work: after a rigid motion, distances should be preserved. After a dilation, distances should be multiplied by k.
- 5.Remember that only reflections reverse orientation. Translations and rotations do not.