Rotations and Dilations - Transformations, Congruence, and Similarity (Pre-Algebra)

$P A$		$P A^{'}$
$P B$		$P B^{'}$
$P C$		$P C^{'}$
$P D$		$P D^{'}$

P A

P A^{'}

P B

P B^{'}

P C

P C^{'}

P D

P D^{'}

$m ∠ A P A^{'}$
$m ∠ B P B^{'}$
$m ∠ C P C^{'}$

m ∠ A P A^{'}

m ∠ B P B^{'}

m ∠ C P C^{'}

Zosia's goal is to find the image of the triangle after a $18 0^{\circ}$ clockwise rotation about its fixed vertex. Labeling the vertices of the triangle will make this rotation process simpler.

The center of rotation is $O$ and the angle of rotation is $18 0^{\circ} .$ Perform the rotation by rotating one point at a time. To rotate $N,$ place the center of the protractor on $O$ and align it with $\overline{N O} .$ Use the protractor to draw a ray that starts from $O$ and makes a $18 0^{\circ}$ angle with $\overline{N O} .$

Then, mark a point $N^{'}$ on this ray so that $\overline{O N^{'}}$ is the same length as $\overline{O N} .$ This is the image of $N$ after the rotation. Since $\overline{O N}$ is the radius of the inner circle, $N^{'}$ should also be on that circle.

Repeat the same process for the vertex $E$ to find $E^{'} .$ Since $O$ is the center of rotation, $O^{'}$ will be in the same position as $O .$

Finally connect $N^{'},$ $E^{'},$ and $O^{'}$ to draw the image of $N E O$ after the $18 0^{\circ}$ rotation.

As shown, the image $N^{'}$ corresponds to P and $E^{'}$ corresponds to T. Therefore, the clue Vincenzo and Zosia is looking for is NEPT.

Counterclockwise Rotations Around the Origin
Angle of Rotation	Rule
$9 0^{\circ}$	$(x, y) \to (- y, x)$
$18 0^{\circ}$	$(x, y) \to (- x, - y)$
$27 0^{\circ}$	$(x, y) \to (y, - x)$

Counterclockwise Rotations Around the Origin

Angle of Rotation

Rule

9 0^{\circ}

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

18 0^{\circ}

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

27 0^{\circ}

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

Counterclockwise Rotations About the Origin
$9 0^{\circ}$ Rotation	$18 0^{\circ}$ Rotation	$27 0^{\circ}$ Rotation
$Preimage (x, y) \to Image (- y, x)$	$Preimage (x, y) \to Image (- x, - y)$	$Preimage (x, y) \to Image (y, - x)$

Counterclockwise Rotations About the Origin

9 0^{\circ}

Rotation

18 0^{\circ}

Rotation

27 0^{\circ}

Rotation

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

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

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

Dilation With Scale Factor $k = 3$
Preimage	Multiply by $k$	Image
$A (0, 2)$	$(3 \cdot 0, 3 \cdot 2)$	$A^{'} (0, 6)$
$B (3, 1)$	$(3 \cdot 3, 3 \cdot 1)$	$B^{'} (9, 3)$
$C (2, - 1)$	$(3 \cdot 2, 3 \cdot (- 1))$	$C^{'} (6, - 3)$

Dilation With Scale Factor

k = 3

Preimage

Multiply by

k

Image

A (0, 2)

(3 \cdot 0, 3 \cdot 2)

A^{'} (0, 6)

B (3, 1)

(3 \cdot 3, 3 \cdot 1)

B^{'} (9, 3)

C (2, - 1)

(3 \cdot 2, 3 \cdot (- 1))

C^{'} (6, - 3)

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Catch-Up and Review

Rotating Points About a Fixed Point

Rotating a Triangle About a Fixed Point

Rotation

Rotation of a Point Around Another Point

Cyrptawheel

Hint

Solution

Extra

Practice Rotating Triangles

Rotating a Point Around the Origin at $9 0^{\circ},$ $18 0^{\circ},$ and $27 0^{\circ}$

A Gateway to a New Realm

Answer

Hint

Solution

Extra

Exploring Dilations

Dilating a Drawing

How to Dilate?

Dilation

Coordinate Rule for Dilations

Practicing Dilatius's Teachings

Answer

Hint

Solution

Checking Our Answer

Dilatius's Final Lesson

Hint

Solution

Checking Our Answer

Comparing Rotations and Dilations

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