Part A: Determine the two different rotations that would create the image.
To find the rotations, we need to consider standard rotation angles: 90 degrees, 180 degrees, and 270 degrees counterclockwise around the origin.
By systematically testing these standard rotations:
- When we rotate the triangle by 90 degrees counterclockwise, the vertices of the original triangle [tex]$X(-1, -3)$[/tex], [tex]$Y(-5, -5)$[/tex], [tex]$Z(-4, -2)$[/tex] transform precisely into the vertices of the image triangle [tex]$X^{\prime}(3, -1)$[/tex], [tex]$Y^{\prime}(5, -5)$[/tex], [tex]$Z^{\prime}(2, -4)$[/tex].
No other standard rotation (180 degrees or 270 degrees) matches these image coordinates accurately. Therefore, the image triangle [tex]$X^{\prime} Y^{\prime} Z^{\prime}$[/tex] is produced by a single 90-degree counterclockwise rotation.
Answer: The rotation that creates the image is 90 degrees counterclockwise.
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Part B: Explanation how you know your answer to Part A is correct.
To verify the rotation angle found, we compare the coordinates before and after the rotation.
Consider the 90-degree counterclockwise rotation:
- The formula for a point \((x, y)\) rotated 90 degrees counterclockwise is \((-y, x)\).
Let's apply this to the vertices of triangle [tex]$XYZ$[/tex].
1. Rotation of [tex]$X(-1, -3)$[/tex]:
- Rotating it 90 degrees counterclockwise gives \((-(-3), -1) = (3, -1)\), which is the coordinate of [tex]$X^{\prime}$[/tex].
2. Rotation of [tex]$Y(-5, -5)$[/tex]:
- Rotating it 90 degrees counterclockwise gives \((-(-5), -5) = (5, -5)\), which is the coordinate of [tex]$Y^{\prime}$[/tex].
3. Rotation of [tex]$Z(-4, -2)$[/tex]:
- Rotating it 90 degrees counterclockwise gives \((-(-2), -4) = (2, -4)\), which is the coordinate of [tex]$Z^{\prime}$[/tex].
Thus, we conclude that the transformation resulting in the image [tex]$X^{\prime} Y^{\prime} Z^{\prime}$[/tex] is consistent with a 90-degree counterclockwise rotation of the original triangle [tex]$XYZ$[/tex].
By performing rotations and matching the resulting points with the image coordinates, we confirm that a 90-degree counterclockwise rotation correctly maps all vertices of the original triangle to the corresponding vertices of the image triangle.
This methodical verification supports that the solution provided is accurate and reliable, explaining why the correct transformation is a 90-degree counterclockwise rotation.
