Question

determine which of the following transformations preserve congruence. select all that apply.
a. $(x,y)\to(-y,x)$
b. $(x,y)\to(-x,-y)$
c. $(x,y)\to(\frac{4}{5}x,\frac{4}{5}y)$
d. $(x,y)\to(4x,4y)$
e. $(x,y)\to(x + 3,y + 3)$

Explanation:

Step1: Recall congruence - preserving transformations

Rigid motions (translations, rotations, reflections) preserve congruence. Translations move points by a fixed distance in a given direction, rotations turn points around a fixed point, and reflections flip points across a line. Dilations change the size of a figure and do not preserve congruence.

Step2: Analyze Option A

The transformation $(x,y)\to(-y,x)$ is a $90^{\circ}$ - counter - clockwise rotation about the origin. Rotations are rigid motions and preserve congruence.

Step3: Analyze Option B

The transformation $(x,y)\to(-x,-y)$ is a $180^{\circ}$ rotation about the origin. Rotations are rigid motions and preserve congruence.

Step4: Analyze Option C

The transformation $(x,y)\to(\frac{4}{5}x,\frac{4}{5}y)$ is a dilation with a scale factor of $\frac{4}{5}$. Dilations change the size of the figure and do not preserve congruence.

Step5: Analyze Option D

The transformation $(x,y)\to(4x,4y)$ is a dilation with a scale factor of 4. Dilations change the size of the figure and do not preserve congruence.

Step6: Analyze Option E

The transformation $(x,y)\to(x + 3,y + 3)$ is a translation. Translations are rigid motions and preserve congruence.

Answer:

A. $(x,y)\to(-y,x)$
B. $(x,y)\to(-x,-y)$
E. $(x,y)\to(x + 3,y + 3)$