The problem consists of five parts, each involving different types of geometric transformations. 1. Reflect the given triangle over the x-axis and determine the coordinates of the reflected points.

1. Problem Description

The problem consists of five parts, each involving different types of geometric transformations.

1. Reflect the given triangle over the x-axis and determine the coordinates of the reflected points.

2. Describe the translation of a rectangle and express the translation as a rule.

3. Rotate the given triangle 90 degrees counterclockwise, label each point of the image, and determine the coordinates of the rotated points.

4. Determine the scale factor of a dilation, given the coordinates of a point and its image.

5. Describe a series of transformations that maps figure XYZ to figure X'Y'Z'.

2. Solution Steps

1. Reflection over the x-axis:

The rule for reflecting a point $(x, y)$ over the x-axis is $(x, -y)$.

A = (3, 2), B = (1, 4), C = (1, 2)

A' = (3, -2), B' = (1, -4), C' = (1, -2)

2. Translation:

The rectangle is translated 4 units to the left and 2 units down.

Therefore, the rule is $(x, y) \rightarrow (x - 4, y - 2)$.

3. Rotation by 90 degrees counterclockwise:

The rule for rotating a point $(x, y)$ 90 degrees counterclockwise is $(-y, x)$.

Q = (0, 0), R = (3, 0), S = (3, -3)

Q' = (0, 0), R' = (0, 3), S' = (3, 3)

4. Dilation:

Let the scale factor be $k$. If point E (12, 8) is dilated to E' (3, 2), then

$12k = 3$, so $k = \frac{3}{12} = \frac{1}{4}$

$8k = 2$, so $k = \frac{2}{8} = \frac{1}{4}$

The scale factor is $\frac{1}{4}$.

5. Series of transformations:

The figure XYZ is reflected over the y-axis and then translated down. Alternatively, it can be reflected over the x-axis and then translated to the left.

3. Final Answer

1. A' = (3, -2), B' = (1, -4), C' = (1, -2)

2. Description: Translation 4 units to the left and 2 units down. Rule: $(x, y) \rightarrow (x - 4, y - 2)$

3. Q' = (0, 0), R' = (0, 3), S' = (3, 3)

4. Scale factor = $\frac{1}{4}$

5. Reflection over the y-axis and then a downward translation.