Geometric transformations describe how a figure (a shape made of points) changes position, orientation, or size.
In coordinate geometry, a transformation can be treated like a function: it takes an input point (or whole shape) and produces exactly one output point (or image).
Definition
Geometric transformation
A rule that maps every point of a figure to a new point, producing an image of the figure.
Transformations Can Be Treated As Functions
A function matches each input with one and only one output.
A geometric transformation does the same thing, but the "inputs" and "outputs" are usually points (or sets of points) in the plane.
If we write $T$ for a transformation, then
Input $x$ can be a point like $(x,y)$, or a whole shape (a set of points).
The mapping is $T$, the rule (for example "translate by $(4,2)$").
Output $y$ is the image point (or image shape).
So we can think of transformation notation like $$\text{image} = T(\text{pre-image})$$
Note
When you transform an entire shape, you are really transforming every point on it.
If every point maps to exactly one point, then the whole shape maps to exactly one image.
Isometric Transformations Preserve Distance And Angle
Isometric transformations change a figure's position (and sometimes orientation), but do not change size or shape. This means:
Lengths stay the same.
Angles stay the same.
The image is congruent to the original.
The three isometric transformations you meet most often are translations, rotations, and reflections.
Definition
Isometric transformation
A transformation that preserves distances (and therefore preserves shape and size). Translations, rotations, and reflections are isometries.
Translation Moves Every Point By The Same Vector
A translation slides a figure without turning or flipping it.
It changes position but not shape, size, direction, or orientation.
A translation is described by a displacement vector $$\begin{pmatrix}a\\b\end{pmatrix},$$ meaning "$a$ units horizontally and $b$ units vertically."
In coordinate form: $$T(x,y)=(x+a,\,y+b)$$
Example question
Translate a point $P(1,-3)$ by $\begin{pmatrix}4\\2\end{pmatrix}$.
Solution
$$T(1,-3)=(1+4,-3+2)=(5,-1)$$
Rotation Turns A Figure About A Fixed Center
A rotation turns a shape around a center of rotation by a given angle and direction (clockwise or anticlockwise).
It changes position and direction, but not size or shape.
Common special cases about the origin $(0,0)$ are:
$90^\circ$ anticlockwise: $(x,y)\mapsto(-y,x)$
$90^\circ$ clockwise: $(x,y)\mapsto(y,-x)$
$180^\circ$: $(x,y)\mapsto(-x,-y)$
For rotations about a point $C(h,k)$ (not the origin), a reliable method is:
Translate so $C$ becomes the origin.
Apply the origin-rotation rule.
Translate back.
Hint
For "$90^\circ$ clockwise about $(h,k)$":
Start with vector from the center: $(x-h,\,y-k)$
Rotate it to $(y-k,\,-(x-h))$
Add the center back: $(h+y-k,\,k-(x-h))$
Reflection Flips A Figure Over A Mirror Line
A reflection creates a mirror image across a mirror line.
It changes position and orientation (it flips the figure), but does not change size or shape.
Key properties of reflections:
The mirror line is the perpendicular bisector of the segment joining a point and its image.
Points on the mirror line stay fixed.
Common reflection rules:
In the $x$-axis: $(x,y)\mapsto(x,-y)$
In the $y$-axis: $(x,y)\mapsto(-x,y)$
In the line $y=x$: $(x,y)\mapsto(y,x)$
For a slanted mirror line like $y=x-3$, you usually use geometric reasoning (equal perpendicular distances) or coordinate methods.
Common Mistake
Reflections reverse orientation.
If a polygon's vertices go clockwise in the pre-image, they will go anticlockwise in the image.
Dilations Change Size But Preserve Shape
A dilation changes the size of a figure but keeps it the same shape.
Because distances change, a dilation is not an isometric transformation.
A dilation is described by:
a center of dilation (often a point $O$), and
a scale factor $k$.
Definition
Dilation
A transformation that multiplies all distances from a center point by the same scale factor, producing a similar image.
Enlargement And Reduction Depend On The Scale Factor
If $k>1$, the image is an enlargement.
If $0<k<1$, the image is a reduction.
If the dilation is centered at the origin, the coordinate rule is: $$T(x,y)=(kx,\,ky).$$
For a center $C(h,k)$, conceptually each point moves along the ray from $C$ through the point.
Definition
Scale factor
The multiplier that compares corresponding lengths in an image and its pre-image under dilation.
Negative Scale Factors Reverse Direction Through The Center
If the scale factor is negative, the image appears on the opposite side of the center of dilation.
$k=-1$ gives a half-turn effect (similar to a $180^\circ$ rotation about the center), with no change in size.
$|k|>1$ makes a larger image on the opposite side.
$|k|<1$ makes a smaller image on the opposite side.
Similarity And Congruence Describe What Is Preserved
Transformations help you classify relationships between figures.
Definition
Congruent figures
Two figures are congruent if they have the same shape and the same size.
Definition
Similar figures
Two figures are similar if corresponding angles are equal and corresponding lengths are in a constant ratio.
After an isometric transformation, the image is congruent to the pre-image.
After a dilation, the image is similar to the pre-image (same angles, lengths scaled by $k$), but not usually congruent.
Note
A dilation preserves angle measures, but it does not preserve lengths unless $|k|=1$.
Describing A Transformation Precisely
In problems you are often asked to "describe the transformation."
Each type has specific information that must be given.
What You Must State
Translation: the displacement vector $\begin{pmatrix}a\\b\end{pmatrix}$.
Rotation: the center, the angle, and the direction.
Reflection: the mirror line.
Dilation: the center of dilation and the scale factor.
Exam technique
When describing a transformation between two figures on a grid:
Match corresponding vertices (often by labels or relative position).
Check whether distances are preserved (if yes, it is isometric).
If size changes but shape stays the same, it is a dilation.
State the required descriptors (vector, center and angle, mirror line, or center and scale factor).
Example
Suppose triangle $A$ maps onto triangle $B$ by a translation of $\begin{pmatrix}4\\2\end{pmatrix}$.
That means every vertex moves the same way:
If $A_1(x_1,y_1)$, then $B_1(x_1+4,\,y_1+2)$.
If $A_2(x_2,y_2)$, then $B_2(x_2+4,\,y_2+2)$.
So the transformation function can be written as: $$T(x,y)=(x+4,\,y+2)$$
Tip
To test a suspected translation, pick one point and calculate the horizontal and vertical change to its image.
Then verify that a second point has the same change.
Active recall
Which transformations always produce a congruent image?
A figure is reduced with scale factor $\tfrac12$. What happens to its side lengths and angles?
What extra information do you need to fully describe a rotation that you do not need for a translation?