Transformations and Congruence

Transformations

  • A transformation takes a pre-image (original figure) to its image (final figure). Use prime notation (e.g., A′A′) to denote the image of a point.

  • Rigid Transformations (Isometries): Transformations that preserve length and angle measures. These include:

    • Translations (shifts)

    • Reflections (flips)

    • Rotations (turns)

Translations

  • A translation shifts every point of a figure the same distance in the same direction.

  • Translations preserve length and angle measures (rigid motion).

  • Translation Rule: (x,y)→(x+a,y+b)(x,y)→(x+a,y+b), where a is the horizontal movement and b is the vertical movement.

    • a>0: right

    • a<0: left

    • b>0: up

    • b<0: down

Reflections

  • A reflection flips a figure over a line (line of reflection).

  • Reflections preserve length and angle measures (rigid motion).

  • Reflection Rules:

    • Reflection across the x-axis: (x,y)→(x,−y)(x,y)→(x,−y)

    • Reflection across the y-axis: (x,y)→(−x,y)(x,y)→(−x,y)

    • Reflection across the line y=x: (x,y)→(y,x)(x,y)→(y,x)

  • To reflect a point, count units to the line of reflection and then move the same amount past the line.

Rotations

  • A rotation turns a figure around a point (center of rotation).

  • Rotations preserve length and angle measures (rigid motion).

  • Rotation Rules (counterclockwise about the origin):

    • 90° CCW (counterclockwise): (x,y)→(−y,x)(x,y)→(−y,x)

    • 180° CCW: (x,y)→(−x,−y)(x,y)→(−x,−y)

    • 270° CCW: (x,y)→(y,−x)(x,y)→(y,−x)

  • Clockwise rotations are equivalent to counterclockwise rotations: 90° CW = 270° CCW, 180° CW = 180° CCW, 270° CW = 90° CCW.

Compositions of Transformations

  • A composition of transformations is a sequence of two or more transformations.

  • Order matters! The image of the first transformation becomes the pre-image for the second transformation.

  • Example: △ABC→△A′B′C′→△A′′B′′C′′△ABC→△A′B′C′→△A′′B′′C′′

Dilations

  • Dilations enlarge or reduce a figure from a center of dilation by a scale factor.

  • Angles and ratios of sides stay the same; lines stay parallel.

  • The center of dilation is the only point that does not move (if the center is the origin).

  • If the scale factor is greater than 1, the dilation is an enlargement.

  • If the scale factor is between 0 and 1, the dilation is a reduction.

Congruence

  • Two figures are congruent if you can get from one figure to the other with rigid transformations (isometries).

  • Corresponding sides and angles of congruent figures are congruent.

  • Congruence Statement: A congruence statement lists the corresponding vertices in the correct order. For example, △ABC≅△DEF△ABC≅△DEF means:

    • ∠A≅∠D

    • ∠B≅∠E

    • ∠C≅∠F

    • AB≅DE

    • BC≅EF

    • AC≅DF

Coordinate Connection: Transformations of Equations

  • Reflection over the y-axis: Replace x with (−x) in the equation.

  • Reflection over the x-axis: Replace y with (−y) in the equation.

  • Vertical Shift:

    • To move the graph up, add to the original equation: y=original equation+ky=original equation+k (where k>0).

    • To move the graph down, subtract from the original equation: y=original equation −k(where k>0).

  • Horizontal Shift:

    • To move the graph to the right, replace x with (x−h) in the equation (where h>0).

    • To move the graph to the left, replace x with (x+h) in the equation (where h>0).

Polygons

  • Sum of Interior Angles: s=(n−2)×180∘s=(n−2)×180∘, where n is the number of sides.

  • Interior Angle of a Regular Polygon: i= s/n​

  • Exterior Angle of a Regular Polygon: e=180∘−i

  • Number of Diagonals: n(n−3)/2

  • Lines of Symmetry: A regular polygon has the same number of lines of symmetry as it has sides.

  • Degrees of Rotational Symmetry: 360/n ​, where n is the number of sides.

Example Problem

Problem: Given A(1,2)A(1,2), perform the following transformations in sequence: 1) Reflect over the y-axis, 2) Translate according to the rule (x,y)→(x+3,y−1)(x,y)→(x+3,y−1). What are the final coordinates of A′′A′′?

Solution:

  1. Reflection over the y-axis: A(1,2)→A′(−1,2)A(1,2)→A′(−1,2)

  2. Translation: A′(−1,2)→A′′(−1+3,2−1)→A′′(2,1)A′(−1,2)→A′′(−1+3,2−1)→A′′(2,1)

Therefore, the final coordinates of A′′A′′ are (2,1)(2,1).