Question

triangle xyz is the image of triangle xyz after the following sequence of transformations: rotation 270° clockwise about the origin translation 2 units to the right and 1 unit up graph triangle xyz.

Explanation:

Step1: Identify coordinates of \( X' \), \( Y' \), \( Z' \)

From the graph:

• \( X' \): \( (3, -2) \)
• \( Z' \): \( (4, -8) \)
• \( Y' \): \( (8, -8) \)

Step2: Reverse translation (2 right, 1 up → 2 left, 1 down)

To reverse translation, subtract 2 from \( x \)-coordinate and 1 from \( y \)-coordinate.

• For \( X' \): \( (3 - 2, -2 - 1) = (1, -3) \)
• For \( Z' \): \( (4 - 2, -8 - 1) = (2, -9) \)
• For \( Y' \): \( (8 - 2, -8 - 1) = (6, -9) \)

Step3: Reverse 270° clockwise rotation (equivalent to 90° clockwise rotation or use rotation formula: \( (x, y) \to (y, -x) \) for 270° clockwise; reverse: \( (x, y) \to (-y, x) \))

Reverse 270° clockwise rotation (formula: \( (x, y) \to (-y, x) \)):

• For \( X'' = (1, -3) \): \( (-(-3), 1) = (3, 1) \)
• For \( Z'' = (2, -9) \): \( (-(-9), 2) = (9, 2) \)
• For \( Y'' = (6, -9) \): \( (-(-9), 6) = (9, 6) \)

These \( (3, 1) \), \( (9, 2) \), \( (9, 6) \) are coordinates of \( X \), \( Z \), \( Y \) (or \( X \), \( Y \), \( Z \) as per triangle).

Step4: Graph \( XYZ \)

Plot points \( X(3, 1) \), \( Y(9, 6) \), \( Z(9, 2) \) and connect them to form triangle \( XYZ \).

Answer:

To graph \( \triangle XYZ \), plot the points \( X(3, 1) \), \( Y(9, 6) \), and \( Z(9, 2) \) on the coordinate plane and connect them. (Note: The process involves reversing transformations to find original coordinates.)