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Unit 3: Rigid Transformations and Symmetry

Sliding, flipping, and turning a shape without changing its size, and the surprising thing that happens when you chain two moves together.

7 topics · 27 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Translations

  1. How it worksHow does a translation vector move every single point of a figure by the same distance and direction?
  2. Worked exampleFind the image of the point (2, -3) after a translation of (5, 4).
  3. Common mistakeWhy doesn't a translation ever rotate or flip the shape it moves?
  4. Going furtherWhat happens when you apply the same translation vector twice in a row to a triangle?
Your turn1 of 2
Point (4, 3) is translated by the vector (8, -4). Find the sum of the coordinates of the image point.

Reflections

  1. How it worksHow does reflecting a point over a line send it to the same distance on the opposite side?
  2. Worked exampleFind the image of the point (4, 1) reflected over the y-axis.
  3. Common mistakeWhy isn't a reflected shape considered a different shape from the original?
  4. Going furtherWhat happens to a shape's orientation, clockwise or counterclockwise, after one reflection?
Your turn1 of 2
Point (4, 3) is reflected over the y-axis. Find the sum of the coordinates of its image.

Rotations

  1. How it worksHow does a center of rotation and an angle of turn together determine where every point of a figure lands?
  2. Worked exampleFind the image of the point (3, 2) rotated 90 degrees counterclockwise about the origin.
  3. Common mistakeWhy doesn't rotating a shape 180 degrees give the same result as reflecting it?
  4. Going furtherWhat happens to a figure after four rotations of 90 degrees each around the same center?
Your turn1 of 2
Point (4, 3) is rotated 90 degrees counterclockwise about the origin. Find the sum of the coordinates of the image point.

Composing Two Reflections

  1. How it worksHow do two reflections over intersecting lines combine into one rotation about their intersection point?
  2. Worked exampleFind the angle of the single rotation equivalent to reflecting first over a line at 10 degrees and then over a line at 55 degrees.
  3. Common mistakeWhy isn't the result of two reflections just a bigger reflection?
  4. Going furtherWhat happens when the two lines you reflect over are parallel instead of intersecting?
Your turn1 of 2
Reflecting first over a line through the origin at 42 degrees, then over a second line through the origin at 58 degrees, is equivalent to a single rotation about the origin. Find the angle of that rotation, in degrees.

Glide Reflections

  1. How it worksHow does a glide reflection combine a translation parallel to a line with a reflection over that same line?
  2. Worked exampleFind the image of the point (2, 3) under a glide reflection that translates by (4, 0) and reflects over the x-axis.
  3. Common mistakeWhy isn't a glide reflection the same transformation as just a translation followed by any reflection?
  4. Ties togetherHow does a glide reflection combine the translations and reflections you already learned into one single move?
Your turn1 of 2
A glide reflection translates a point by (8, 0) and then reflects it over the x-axis. Find the sum of the coordinates of the image of the point (4, 3).

Line and Rotational Symmetry

  1. How it worksHow do you count the number of lines of symmetry hiding inside a regular polygon?
  2. Worked exampleFind the smallest angle of rotational symmetry for a regular pentagon.
  3. Worked exampleFind the number of lines of symmetry in a regular nonagon.
  4. Common mistakeWhy doesn't every shape with matching left and right sides also have rotational symmetry?
Your turn1 of 2
Find the smallest angle of rotational symmetry, in degrees, for a regular polygon with 18 sides.

Congruence Defined by Rigid Motions

  1. How it worksHow does a sequence of translations, rotations, and reflections carry one triangle exactly onto another?
  2. Common mistakeWhy isn't 'congruent' the same word as 'equal' when you're talking about two triangles?
  3. Ties togetherHow does defining congruence through rigid motions connect back to the translations, reflections, and rotations you just learned?
Your turn1 of 2
Why isn't 'congruent' the same word as 'equal' when talking about two triangles?