Before we apply the similarity transformations, let's graph the original segment $\overline{C D} .$

Let's apply one transformation at time!

Rotation

When a figure is rotated

9 0^{\circ}

counterclockwise about the origin, the coordinates of the figure's endpoints change such that

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

Using this rule with the endpoints of

\overline{C D},

we can find the endpoints of

\overline{C^{'} D^{'}} .

(x, y) C (- 2, 2) D (2, 2) \to \to \to (- y, x) C^{'} (- 2, - 2) D^{'} (- 2, 2)

Now we are able to graph

\overline{C^{'} D^{'}} .

Let's do it!

Dilation

A dilation can be an enlargement, a reduction, or the same size as the preimage. Which type of dilation it is depends on the value of the scale factor $k .$

Enlargement	$k > 1$
Reduction	$0 < k < 1$
Same	$k = 1$

When the center of dilation in the coordinate plane is the origin, each coordinate of the preimage is multiplied by the scale factor

k

to find the coordinates of the image.

Preimage (x, y) \Rightarrow Image (k x, k y)

Now, let's find the endpoints of

\overline{C^{'} D^{'}}

after a dilation with a scale factor

k = \frac{1}{2} .

Dilation With Scale Factor $k = \frac{1}{2}$
Preimage	Multiply by $k$	Image
$C^{'} (- 2, - 2)$	$(\frac{1}{2} (- 2), \frac{1}{2} (- 2))$	$C^{''} (- 1, - 1)$
$D^{'} (- 2, 2)$	$(\frac{1}{2} (- 2), \frac{1}{2} (2))$	$D^{''} (- 1, 1)$

Now, we can show the transformation by plotting the newly obtained points.

Combined Transformation

Finally, we can remove the middle step and only look at the preimage and the image.

Exercise