Rotations about the origin

Warm-up

Three quick questions from earlier lessons.

Discover

Rotate the point $(1, 0)$ by $90^\circ$ counterclockwise about the origin. Where does it go?

$(0, 1)$
$(0, -1)$

Move P and change the angle. Compare OP and OQ.

They are always equal
OQ grows with the angle

Key idea

Rotations about the origin by quarter turns.

About the origin:

90^\circ

counterclockwise:

(x, y) \to (-y, x)

;

180^\circ

:

(x, y) \to (-x, -y)

;

90^\circ

clockwise (

270^\circ

counterclockwise):

(x, y) \to (y, -x)

.

Worked example

Rotate $C(3, 2)$ .

Step 1Triangle ABC.
Step 2Turn it about O: every point keeps its distance to O.
Step 3A quarter turn (90° counterclockwise): (x, y) → (−y, x).

90^\circ

ccw:

(-2, 3)

(−y, x).

180^\circ

:

(-3, -2)

(−x, −y).

90^\circ

cw:

(2, -3)

(y, −x).

Your turn, guided

Your turn: the y-coordinate of $(5, 1)$ after $90^\circ$ counterclockwise.

5

(−y, x) = (−1, 5).

Practice ★

Rotate $A(-2, -1)$ by $180^\circ$ about the origin. Tap its image.
Rotate $A(3, -2)$ by $180^\circ$ about the origin. Tap its image.

Rotate $A(1, 3)$ by $90^\circ$ counterclockwise about the origin. Tap its image.
Rotate $A(1, 2)$ by $90^\circ$ counterclockwise about the origin. Tap its image.

Watch out!

Practice ★★

A figure is rotated 7 times by $90^\circ$ in the same direction. What single rotation, between $0^\circ$ and $360^\circ$ , gives the same result? (degrees)

Boss problem. $A = (4, 6)$ is reflected across the $x$ -axis, then rotated $90^\circ$ counterclockwise about the origin, then translated 5 units right.

Rotate triangle ABC by 90° counterclockwise about the origin. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

Rotate $A(1, 3)$ by $90^\circ$ counterclockwise about the origin. Tap its image.
Rotate $A(1, 2)$ by $90^\circ$ counterclockwise about the origin. Tap its image.
Rotate $A(2, 6)$ by $90^\circ$ counterclockwise about the origin. Tap its image.

Summary card

Rotations about the origin

ccw 90°: $(-y, x)$ ; 180°: $(-x, -y)$ ; cw 90°: $(y, -x)$ .

Watch out: check the quadrant the image lands in.

About this lesson