Question
Trapezoid FGHI is rotated using the origin as the center of rotation. The preimage and image are graphed below.
On a coordinate plane, trapezoid F G H I has points (3, 4), (8, 4), (5, 2), (3, 2). Trapezoid F G H I is rotated 45 degrees clockwise to form F prime G prime H prime I prime.
Which rotation transformed FGHI to F’G’H’I’?
a 45° counterclockwise rotation
a 45° clockwise rotation
a 90° counterclockwise rotation
a 90° clockwise rotation
On a coordinate plane, trapezoid F G H I has points (3, 4), (8, 4), (5, 2), (3, 2). Trapezoid F G H I is rotated 45 degrees clockwise to form F prime G prime H prime I prime.
Which rotation transformed FGHI to F’G’H’I’?
a 45° counterclockwise rotation
a 45° clockwise rotation
a 90° counterclockwise rotation
a 90° clockwise rotation
Answers
Answered by
GPT-4o mini
To determine how trapezoid FGHI was transformed to trapezoid F'G'H'I', we need to analyze the rotation given that it's a 45-degree clockwise rotation around the origin.
The points of trapezoid FGHI are:
- F (3, 4)
- G (8, 4)
- H (5, 2)
- I (3, 2)
A rotation of 45 degrees clockwise around the origin affects the coordinates of each point \((x, y)\) according to the following formulas:
- \(x' = x \cos(-\theta) - y \sin(-\theta)\)
- \(y' = x \sin(-\theta) + y \cos(-\theta)\)
For a rotation of 45 degrees clockwise (\(\theta = 45°\)), we can use:
- \(\cos(45°) = \frac{\sqrt{2}}{2}\)
- \(\sin(45°) = \frac{\sqrt{2}}{2}\)
This leads to the following transformations:
- \(x' = x \cdot \frac{\sqrt{2}}{2} + y \cdot \frac{\sqrt{2}}{2}\)
- \(y' = -x \cdot \frac{\sqrt{2}}{2} + y \cdot \frac{\sqrt{2}}{2}\)
Now, you would apply these transformations to each point F, G, H, and I, and then compare the results with the coordinates of the new points F', G', H', and I' to confirm that they match.
Given the transformation is labeled as a 45° clockwise rotation, it is directly indicated in the problem statement. Therefore, it confirms that the transformation from trapezoid FGHI to trapezoid F'G'H'I' is indeed:
**a 45° clockwise rotation**.
The points of trapezoid FGHI are:
- F (3, 4)
- G (8, 4)
- H (5, 2)
- I (3, 2)
A rotation of 45 degrees clockwise around the origin affects the coordinates of each point \((x, y)\) according to the following formulas:
- \(x' = x \cos(-\theta) - y \sin(-\theta)\)
- \(y' = x \sin(-\theta) + y \cos(-\theta)\)
For a rotation of 45 degrees clockwise (\(\theta = 45°\)), we can use:
- \(\cos(45°) = \frac{\sqrt{2}}{2}\)
- \(\sin(45°) = \frac{\sqrt{2}}{2}\)
This leads to the following transformations:
- \(x' = x \cdot \frac{\sqrt{2}}{2} + y \cdot \frac{\sqrt{2}}{2}\)
- \(y' = -x \cdot \frac{\sqrt{2}}{2} + y \cdot \frac{\sqrt{2}}{2}\)
Now, you would apply these transformations to each point F, G, H, and I, and then compare the results with the coordinates of the new points F', G', H', and I' to confirm that they match.
Given the transformation is labeled as a 45° clockwise rotation, it is directly indicated in the problem statement. Therefore, it confirms that the transformation from trapezoid FGHI to trapezoid F'G'H'I' is indeed:
**a 45° clockwise rotation**.