02.02 Rotations

A turn, not a slide

A rotation turns a figure around a fixed point. That point is the center of rotation. Every point travels on a circle around that center, the same number of degrees.

A full turn is 360°, because a circle is 360°. Clockwise is the way clock hands move. Counterclockwise is the other way.

Canvas wants the three origin rules as functions, the same way 02.01 wanted the slide and flip rules.

Lookalike: swapping tires front-to-back is a translation. Turning a doorknob is a rotation. The word "rotate" in ordinary English is not always the geometry move.

Three rules around the origin

Unless the problem names a different center, the center is (0, 0). Plug every vertex into the same rule.

90° clockwise: (x, y) → (y, −x)

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

180° either way: (x, y) → (−x, −y)

Point P(3, 1) as a check:

90° clockwise → (1, −3)
90° counterclockwise → (−1, 3)
180° → (−3, −1)

All three land near each other. All three use the digits 3 and 1. Name the turn first, then apply the rule.

How to not mix the 90s

Both 90° rules swap. The sign is the discriminator.

Clockwise: new pair is (old y, opposite of old x). The first number of the image is the old y.

Counterclockwise: new pair is (opposite of old y, old x). The second number of the image is the old x.

A point in quadrant I, 90° clockwise, lands in quadrant IV. (3, 1) went to (1, −3). That is (y, −x), not (−y, x).

Blue PQR: P(2, 4), Q(2, 1), R(5, 1). Red is 90° clockwise: P'(4, −2), Q'(1, −2), R'(1, −5).

Same digits, different move: (y, x) is a reflection across y = x, not a 90° turn. (y, −x) is 90° clockwise. If two choices share 4 and −2, read whether the stem said rotate or reflect.

Still congruent

A rotation does not stretch or flip. Side lengths stay. Angle measures stay. A point on a side stays on that side. A midpoint stays a midpoint. Collinearity and betweenness are preserved.

Orientation is preserved too. If you named the pre-image clockwise, the image is still named clockwise. That is the difference from a reflection.

So a rotation is a rigid motion, and the image is congruent to the pre-image. The rules are functions: input (x, y), output the image pair.

If you draw the arc each vertex travels, you get concentric circles around the origin. Same center, same angle, different radii.

Sequences and 270

Do the moves in the order given. Each output is the next input.

A full turn is 360°, so 270° clockwise is the leftover 90° the other way:

270° clockwise = 90° counterclockwise → (−y, x)

270° counterclockwise = 90° clockwise → (y, −x)

Triangle XYZ with X(−4, 5). First 180°, then 270° counterclockwise.

180°: (−4, 5) → (4, −5)
270° counterclockwise is 90° clockwise: (4, −5) → (−5, −4)

Two 90° counterclockwise turns in a row are 180°. A 90° clockwise followed by 270° counterclockwise is also 180°. You can add the turns when they share a center, watching the direction.

Reason trap: after a rotation, AB ≅ A'B' because a rotation is a rigid motion, not because of the Addition Property of Equality. Same length, different toolbox.

Mapping a figure onto itself

Reflectional symmetry: a fold line that makes two matching halves. That line maps the figure onto itself.

Rotational symmetry: a turn of less than 360° around the center that makes the figure look the same. Order of rotation is how many times that happens in a full 360°, counting the original position. Angle of rotation is 360° divided by the order.

A regular octagon (equal sides, equal angles) has order 8. Angle of rotation is 45°. A regular decagon has order 10 and 36°. A regular dodecagon has order 12 and 30°.

Order 1 is not a thing. If the figure only matches after a full 360°, it has no rotational symmetry. The course writes that as order 0.

An isosceles triangle that is not equilateral has one line of symmetry (through the vertex between the two equal sides). It does not have 120° rotational symmetry. An equilateral triangle does: order 3, 120°, and three fold lines.

Two descriptions, one picture: reflect across the x-axis, then across the y-axis, is the same as a 180° rotation. Every rotation is a composition of two reflections. If the question asks for a sequence, either description can be right. If it asks for one named move, pick the rotation.

Your turn

New points, same rules. Two choices will share an ordered pair. The discriminator is the turn.