Triangle XYZ is rotated 90° counterclockwise about the origin to produce AXYZ. What are the coordinates of AX'YZ'? 4 2 -4 -2 O 4 -2 -4 O A. X(1, 1), Y(-3, 2), Z'(-2, 5) В. X-1,-1), Y(3, -2), Z(2, -5) С. X-1, -1), Y-2, 3), Z(-5, 2) оD. X(0, 0), Y(1, 4), Z(4, 3) N

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
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Question 3

### Rotation of Triangle XYZ

#### Problem Statement:
Triangle XYZ is rotated 90° counterclockwise about the origin to produce triangle \(ΔX'Y'Z'\). What are the coordinates of \(ΔX'Y'Z'\)?

#### Diagram:
The given diagram shows triangle XYZ on the coordinate plane. The points of triangle XYZ are plotted and connected to form the triangle. The coordinate grid is labeled with the x-axis and y-axis, both extending from -4 to 4.

Triangle XYZ operates over individual quadrants with the following vertices coordinates identified:
- \(X(2, -3)\)
- \(Y(4, 2)\)
- \(Z(3, 5)\)

#### Answer Choices:
a. \(X'(1, 1), Y'(-3, 2), Z'(-2, 5)\)  
b. \(X'(-1, -1), Y'(3, -2), Z'(2, -5)\)  
c. \(X'(-1, -1), Y'(-2, 3), Z'(-5, 2)\)  
d. \(X'(0, 0), Y'(1, 4), Z'(4, 3)\)

#### Diagram Explanation:
The graph illustrates the following:

1. **Original Triangle (XY Coordinates):**
   - Point \(X\) is situated at (2, -3)
   - Point \(Y\) is situated at (4, 2)
   - Point \(Z\) is situated at (3, 5) 

2. **Rotation:**
   The rotation of 90° counterclockwise about the origin changes coordinates \((x, y)\) to \((-y, x)\).

#### Solution:
To find the new coordinates \(X'Y'Z'\) after the 90° counterclockwise rotation:
- For Point \(X(2, -3)\), new coordinates are \((-(-3), 2) = (3, 2)\)
- For Point \(Y(4, 2)\), new coordinates are \((-2, 4) = (-2, 4)\)
- For Point \(Z(3, 5)\), new coordinates are \((-5, 3) = (-5, 3)\)

Thus, the coordinates of triangle \(ΔX'Y'Z'\) after rotation are:
- \(X
Transcribed Image Text:### Rotation of Triangle XYZ #### Problem Statement: Triangle XYZ is rotated 90° counterclockwise about the origin to produce triangle \(ΔX'Y'Z'\). What are the coordinates of \(ΔX'Y'Z'\)? #### Diagram: The given diagram shows triangle XYZ on the coordinate plane. The points of triangle XYZ are plotted and connected to form the triangle. The coordinate grid is labeled with the x-axis and y-axis, both extending from -4 to 4. Triangle XYZ operates over individual quadrants with the following vertices coordinates identified: - \(X(2, -3)\) - \(Y(4, 2)\) - \(Z(3, 5)\) #### Answer Choices: a. \(X'(1, 1), Y'(-3, 2), Z'(-2, 5)\) b. \(X'(-1, -1), Y'(3, -2), Z'(2, -5)\) c. \(X'(-1, -1), Y'(-2, 3), Z'(-5, 2)\) d. \(X'(0, 0), Y'(1, 4), Z'(4, 3)\) #### Diagram Explanation: The graph illustrates the following: 1. **Original Triangle (XY Coordinates):** - Point \(X\) is situated at (2, -3) - Point \(Y\) is situated at (4, 2) - Point \(Z\) is situated at (3, 5) 2. **Rotation:** The rotation of 90° counterclockwise about the origin changes coordinates \((x, y)\) to \((-y, x)\). #### Solution: To find the new coordinates \(X'Y'Z'\) after the 90° counterclockwise rotation: - For Point \(X(2, -3)\), new coordinates are \((-(-3), 2) = (3, 2)\) - For Point \(Y(4, 2)\), new coordinates are \((-2, 4) = (-2, 4)\) - For Point \(Z(3, 5)\), new coordinates are \((-5, 3) = (-5, 3)\) Thus, the coordinates of triangle \(ΔX'Y'Z'\) after rotation are: - \(X
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