Question

geometry
determine if the following transformations result in a congruent or similar figure.
explain how you know.

1. rotation 90 degrees, clockwise, center at point c similar or congruent
2. dilation with scale factor of 1/2, center at (0,0) similar or congruent
3. reflection over the x - axis similar or congruent
4. translation 3 right and 4 down similar or congruent
5. reflection over the y axis and dilation with scale factor of 3, center at (0,0) similar or congruent

Explanation:

Step1: Recall transformation properties

• Rotation: A rotation is a rigid transformation. Rigid transformations preserve side - lengths and angles.
• Dilation: A dilation changes the size of a figure. If the scale factor \(k

eq1\), the side - lengths of the figure change. The ratio of corresponding side - lengths of the original and dilated figure is \(k\), and angles are preserved.

• Reflection: A reflection is a rigid transformation. Rigid transformations preserve side - lengths and angles.
• Translation: A translation is a rigid transformation. Rigid transformations preserve side - lengths and angles.

Step2: Analyze each transformation

1. Rotation \(90^{\circ}\) clockwise about point \(C\):

• Since rotation is a rigid transformation (preserves side - lengths \(s_1 = s_2\) and angles \(\angle_1=\angle_2\)), the figure and its image are congruent.

1. Dilation with scale factor \(\frac{1}{2}\) about \((0,0)\):

• The scale factor \(k = \frac{1}{2}

eq1\). Angles of the figure are preserved (\(\angle_1=\angle_2\)), but side - lengths change (\(s_2=\frac{1}{2}s_1\)). So the figure and its image are similar.

1. Reflection over the \(x -\)axis:

• Reflection is a rigid transformation (preserves side - lengths \(s_1 = s_2\) and angles \(\angle_1=\angle_2\)), so the figure and its image are congruent.

1. Translation \(3\) units right and \(4\) units down:

• Translation is a rigid transformation (preserves side - lengths \(s_1 = s_2\) and angles \(\angle_1=\angle_2\)), so the figure and its image are congruent.

1. Reflection over the \(y -\)axis and dilation with scale factor \(3\) about \((0,0)\):

• The reflection is a rigid transformation (preserves angles), but the dilation with scale factor \(k = 3

eq1\) changes the side - lengths (\(s_2 = 3s_1\)). Angles are preserved (\(\angle_1=\angle_2\)), so the figure and its image are similar.

Answer:

1. Congruent
2. Similar
3. Congruent
4. Congruent
5. Similar