Question

1. reflect across y=x.

translate left 4 and up 2.
rotate 90° ccw about the origin.
coordinate notation:

Explanation:

To solve the problem of transforming the triangle with vertices \( U \), \( G \), and \( W \) through reflection, translation, and rotation, we first need to determine the coordinates of the original vertices. From the graph:

• \( U(-4, 5) \)
• \( G(-5, 3) \)
• \( W(-2, 3) \)

Step 1: Reflect across \( y = x \)

The rule for reflecting a point \( (x, y) \) across \( y = x \) is \( (x, y) \to (y, x) \).

• For \( U(-4, 5) \): \( (-4, 5) \to (5, -4) \)
• For \( G(-5, 3) \): \( (-5, 3) \to (3, -5) \)
• For \( W(-2, 3) \): \( (-2, 3) \to (3, -2) \)

Step 2: Translate left 4 and up 2

The rule for translating a point \( (x, y) \) left 4 (subtract 4 from \( x \)) and up 2 (add 2 to \( y \)) is \( (x, y) \to (x - 4, y + 2) \).

• For \( U(5, -4) \): \( (5 - 4, -4 + 2) = (1, -2) \)
• For \( G(3, -5) \): \( (3 - 4, -5 + 2) = (-1, -3) \)
• For \( W(3, -2) \): \( (3 - 4, -2 + 2) = (-1, 0) \)

Step 3: Rotate \( 90^\circ \) CCW about the origin

The rule for rotating a point \( (x, y) \) \( 90^\circ \) counterclockwise about the origin is \( (x, y) \to (-y, x) \).

• For \( U(1, -2) \): \( (-(-2), 1) = (2, 1) \)
• For \( G(-1, -3) \): \( (-(-3), -1) = (3, -1) \)
• For \( W(-1, 0) \): \( (-0, -1) = (0, -1) \)

Final Coordinates

After all transformations, the coordinates of the vertices are:

• \( U' = (2, 1) \)
• \( G' = (3, -1) \)
• \( W' = (0, -1) \)

(Note: If the original coordinates were misidentified, adjust the steps accordingly. The key is to apply each transformation rule sequentially.)

Answer:

The final coordinates after reflection, translation, and rotation are \( U'(2, 1) \), \( G'(3, -1) \), and \( W'(0, -1) \).