Geometric Transformations

Geometry is the math of shape, size, and position. You use it to find lengths, angles, areas, and volumes.

Geometric Transformations: the central idea

A transformation maps every point of a figure according to one rule. Rigid motions preserve distance and angle; dilations preserve shape but change scale.

Words you need

Pre-image

The pre-image is the original figure before any rule is applied, and its points are usually labelled with plain letters such as $A$, $B$, and $C$.

Image

The image is the figure you get after the transformation rule has been applied to every point, and its points are usually labelled with primes such as $A'$, $B'$, and $C'$.

Rigid motion

A rigid motion, also called an isometry, is a transformation that leaves every length and every angle unchanged, so translations, reflections, and rotations are rigid motions while dilations are not.

Line of reflection

The line of reflection is the mirror line, and it is the perpendicular bisector of the segment joining any point to its image, meaning both points are the same distance from it on opposite sides.

Scale factor

The scale factor $k$ is the number a dilation multiplies every distance from the center by, so $k>1$ enlarges the figure, $0<k<1$ shrinks it, and $k=1$ leaves it alone.

Orientation

Orientation is the order the vertices run in, clockwise or counterclockwise, and a reflection reverses it while translations, rotations, and positive dilations preserve it.

What to know before this lesson

Be able to plot ordered pairs and distinguish distance, angle, orientation, and size.

If one of those prerequisites is uncertain, use the Geometry subject guide to locate the earlier concept before memorizing a procedure.

Geometric transformations: a worked example

Every step, with the arithmetic

1. Step 1 - Set upDilate triangle $A(2,1)$, $B(6,1)$, $C(2,4)$ by scale factor $3$ about the origin, then translate the result by $(-1,5)$.
2. Step 2 - Write the dilation rule$(x,y)\mapsto(3x,3y)$
3. Step 3 - Apply it to every vertex$A(2,1)\mapsto(6,3)$, $B(6,1)\mapsto(18,3)$, $C(2,4)\mapsto(6,12)$
4. Step 4 - Check a length$AB$ was $6-2=4$ units long. The image side runs $18-6=12$ units, and $12=3\times4$.
5. Step 5 - Check an angleThe corner at $A$ was a right angle between a horizontal and a vertical side, and the image at $(6,3)$ still has a horizontal and a vertical side, so it is still $90^\circ$.
6. Step 6 - Write the translation rule and apply it$(x,y)\mapsto(x-1,y+5)$ sends $(6,3)\mapsto(5,8)$, $(18,3)\mapsto(17,8)$, and $(6,12)\mapsto(5,17)$.
7. Step 7 - Compare areasOriginal area $=\tfrac12(4)(3)=6$. Final area $=\tfrac12(12)(9)=54$, and $54=3^2\times6$. The translation moved the figure but changed nothing about its size.

Reflection across the $y$-axis uses $(x,y)\mapsto(-x,y)$, so $(4,-2)$ maps to $(-4,-2)$.

Coordinate rules for the transformations you meet most, tested on the point (4, -2)

The step-by-step method for geometric transformations

1. Name the transformation and its center, axis, vector, angle, or scale factor.
2. Apply the coordinate rule to every vertex in the same order.
3. Compare corresponding lengths, angles, orientation, and location to verify the claimed properties.

How to check your answer

Both points are four units from the $y$-axis and share the same $y$-coordinate, so the axis is their perpendicular bisector.

Test triangle inequalities, angle sums, units, scale, and whether the result is compatible with the diagram without assuming the drawing is exact.

A mistake that changes the mathematics

A rotation rule depends on both direction and angle. Confusing clockwise with counterclockwise changes every image coordinate.

Where you will actually use this

Video games and animation

Moving a character across the screen is a translation, spinning a wheel is a rotation, and zooming the camera is a dilation. Game engines store these as matrices so hundreds of points can be moved with one calculation per frame.

Pattern and textile design

Wallpaper, tile floors, and fabric prints are built by taking one small motif and repeating it with translations, reflections, and rotations. Choosing which combination to use is what gives a pattern its style.

Manufacturing and CAD

An engineer draws half of a symmetric part and reflects it to get the other half, guaranteeing the two sides match exactly. Scaling a drawing from a $1:10$ model to full size is a dilation with scale factor $10$.

How geometric transformations connects to the rest of geometry

• Function transformations — Shifting, flipping, and stretching a graph is the same set of moves applied to a curve, which is why $y=f(x-3)+2$ behaves exactly like a translation by $(3,2)$.
• Matrix operations — Every rotation, reflection, and dilation about the origin can be written as a $2\times2$ matrix, and applying two transformations in a row is the same as multiplying their matrices.
• Inverse functions — The graph of an inverse function is the reflection of the original graph over the line $y=x$, which uses the rule $(x,y)\mapsto(y,x)$.

Try a transfer problem

Rotate the triangle with vertices $(1,1),(4,1),(1,3)$ by $90^\circ$ counterclockwise about the origin and compare side lengths.

Show the worked answer

A $90^\circ$ counterclockwise rotation about the origin uses the rule $(x,y)\mapsto(-y,x)$. Applying it vertex by vertex: $(1,1)\mapsto(-1,1)$, $(4,1)\mapsto(-1,4)$, and $(1,3)\mapsto(-3,1)$. Now compare the sides. The original horizontal side from $(1,1)$ to $(4,1)$ is $3$ units; its image runs from $(-1,1)$ to $(-1,4)$, which is also $3$ units, though it is now vertical. The original vertical side from $(1,1)$ to $(1,3)$ is $2$ units; its image runs from $(-1,1)$ to $(-3,1)$, also $2$ units. The slanted side measures $\sqrt{3^2+2^2}=\sqrt{13}$ both before and after. Every length matches, which is exactly what it means for a rotation to be a rigid motion.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to geometry practice questions.

Questions about geometric transformations

What is the difference between a rigid motion and a dilation?

A rigid motion keeps every length and angle the same, so the image is congruent to the pre-image and you could slide one on top of the other. A dilation keeps the angles but multiplies the lengths by the scale factor, so the image is similar, the same shape at a different size.

Does a rotation change orientation?

No. If the vertices of the pre-image read counterclockwise, they still read counterclockwise after any rotation or translation. Only a reflection reverses that order, which is why a reflected figure looks like a mirror copy that cannot be slid back on top of the original.

What happens to area under a dilation?

Area is multiplied by $k^2$, not by $k$. A triangle of area $6$ dilated by scale factor $3$ has area $6\times3^2=54$. Volume in three dimensions would scale by $k^3$.

How do I rotate about a point that is not the origin?

Do it in three moves: translate so the center lands on the origin, rotate with the usual rule, then translate back. To turn $(5,3)$ by $90^\circ$ counterclockwise about $(2,1)$, subtract the center to get $(3,2)$, apply $(x,y)\mapsto(-y,x)$ to get $(-2,3)$, then add the center back to get $(0,4)$.