Slide a chess piece across the board, flip a pancake, spin a Ferris wheel: in each case an object changes position, yet it keeps exactly the same size and shape. In this chapter you will describe these moves with coordinate rules, combine them, and use them to decide when two figures are congruent.
1. Transformations and rigid motions
A transformation assigns to every point of the plane a new point. The starting figure is the preimage; the figure you get is the image. We write \(A\) for a point of the preimage and \(A′\) (read “A prime”) for its image.
A rigid motion (also called an isometry) is a transformation that preserves distance and angle measure. The image has exactly the same size and shape as the preimage; only its position or its orientation may change.
Three basic rigid motions are studied here: translations (slides), reflections (flips) and rotations (turns). A dilation, which enlarges or shrinks a figure, is not a rigid motion because it changes distances.
2. Translations
A translation slides every point the same distance in the same direction. It is described by a vector \(\langle a, b\rangle\): move \(a\) units horizontally and \(b\) units vertically.
The translation by \(\langle a, b\rangle\) maps \((x, y)\) to \((x + a,\; y + b)\).
Translate \(A(-5, 3)\), \(B(-2, 3)\), \(C(-4, 1)\) by \(\langle 6, -4\rangle\) (6 right, 4 down).
\(A′ = (-5 + 6,\; 3 - 4) = (1, -1)\), \(B′ = (4, -1)\) and \(C′ = (2, -3)\).
Check: \(AB = 3\) and \(A′B′ = 3\). The sliding keeps every length.
3. Reflections
A reflection in a line \(\ell\) (the line of reflection) maps each point \(P\) to a point \(P′\) such that \(\ell\) is the perpendicular bisector of \(\overline{PP′}\). Points on \(\ell\) do not move.
In the x-axis: \((x, y)\to(x, -y)\). In the y-axis: \((x, y)\to(-x, y)\). In the line \(y = x\): \((x, y)\to(y, x)\). In the line \(y = -x\): \((x, y)\to(-y, -x)\).
Reflect \(A(1, 4)\), \(B(5, 2)\), \(C(2, 1)\) in the y-axis.
Change the sign of each x-coordinate: \(A′(-1, 4)\), \(B′(-5, 2)\), \(C′(-2, 1)\). Each segment \(\overline{AA′}\) is horizontal and is cut in half by the y-axis.
A reflection reverses the orientation: if you read \(A, B, C\) counterclockwise on the preimage, you read \(A′, B′, C′\) clockwise on the image. Translations and rotations keep the orientation.
4. Rotations
A rotation about a point \(O\) (the center) by an angle \(\theta\) turns every point around \(O\) by \(\theta\), keeping its distance to \(O\). A positive angle means counterclockwise.
\(90^\circ\) counterclockwise: \((x, y)\to(-y, x)\). \(180^\circ\): \((x, y)\to(-x, -y)\). \(270^\circ\) counterclockwise (the same as \(90^\circ\) clockwise): \((x, y)\to(y, -x)\).
Rotate \(A(2, 1)\), \(B(5, 1)\), \(C(2, 3)\) by \(90^\circ\) counterclockwise about the origin.
Apply \((x, y)\to(-y, x)\): \(A′(-1, 2)\), \(B′(-1, 5)\), \(C′(-3, 2)\). The side \(\overline{AB}\), horizontal and 3 units long, becomes the vertical side \(\overline{A′B′}\), also 3 units long.
- Subtract the center: \((x - h,\; y - k)\).
- Apply the origin rule for the angle.
- Add the center back.
5. Compositions of transformations
A composition applies one transformation, then another to the result. The order matters: reflecting and then sliding usually does not give the same image as sliding and then reflecting.
Reflect \(A(-5, 1)\), \(B(-2, 1)\), \(C(-4, 3)\) in the x-axis, then translate by \(\langle 7, 0\rangle\).
Reflection: \(A′(-5, -1)\), \(B′(-2, -1)\), \(C′(-4, -3)\). Translation: \(A″(2, -1)\), \(B″(5, -1)\), \(C″(3, -3)\).
Reflecting in two parallel lines that are \(d\) units apart is a translation by \(2d\), perpendicular to the lines. Reflecting in two lines that intersect at an angle \(\theta\) is a rotation by \(2\theta\) about the intersection point.
When you compose several moves, write the new coordinates after every step instead of trying to do everything in your head. One slip in the middle spoils the final answer.
6. Congruence through rigid motions
Two figures are congruent (\(\cong\)) if a rigid motion, or a sequence of rigid motions, maps one exactly onto the other. Then all corresponding sides and corresponding angles are congruent.
Use the triangles of Example 1. \(AC^2 = 1^2 + 2^2 = 5\) and \(A′C′^2 = 1^2 + 2^2 = 5\). \(BC^2 = 2^2 + 2^2 = 8\) and \(B′C′^2 = 2^2 + 2^2 = 8\). Together with \(AB = A′B′ = 3\), the three pairs of sides match, so \(\triangle ABC\cong\triangle A′B′C′\), as expected for a translation.
To prove two triangles congruent without drawing, show that they have three pairs of congruent sides (SSS), or use SAS or ASA. Each of these criteria guarantees that a sequence of rigid motions exists.
7. Symmetry of figures
A figure has line symmetry if a reflection in some line maps it onto itself. It has rotational symmetry if a rotation of less than \(360^\circ\) about its center maps it onto itself.
A regular polygon with \(n\) sides has \(n\) lines of symmetry and rotational symmetry with smallest angle \(\dfrac{360^\circ}{n}\). The equilateral triangle has \(120^\circ\), the square \(90^\circ\), the regular hexagon \(60^\circ\).
8. Summary of coordinate rules
| Transformation | Rule for \((x, y)\) |
|---|---|
| Translation by \(\langle a, b\rangle\) | \((x + a,\; y + b)\) |
| Reflection in the x-axis | \((x, -y)\) |
| Reflection in the y-axis | \((-x, y)\) |
| Reflection in \(y = x\) | \((y, x)\) |
| Rotation \(90^\circ\) counterclockwise about the origin | \((-y, x)\) |
| Rotation \(180^\circ\) about the origin | \((-x, -y)\) |
| Rotation \(270^\circ\) counterclockwise about the origin | \((y, -x)\) |
Key takeaways
- A rigid motion keeps all distances and angle measures: preimage and image are congruent.
- Translation: \((x, y)\to(x + a, y + b)\). Reflection: the line of reflection is the perpendicular bisector of each \(\overline{PP′}\).
- Rotation: positive angles turn counterclockwise; about the origin, \(90^\circ\) gives \((-y, x)\) and \(180^\circ\) gives \((-x, -y)\).
- The order of a composition matters. Two reflections give a translation (parallel lines) or a rotation (intersecting lines).
- Two figures are congruent when a sequence of rigid motions maps one onto the other.
- A regular \(n\)-gon has \(n\) lines of symmetry and a smallest rotation angle of \(360^\circ \div n\).
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