Question
A point is rotated about the origin. Its coordinates are P(−4,10)
and P′(−10,−4)
. Determine the degree of rotation and direction by applying rotation mapping.(1 point)
Responses
270 degrees (counterclockwise)
270 degrees (counterclockwise)
−180
degrees (clockwise) or 180 degrees (counterclockwise)
negative 180 degrees (clockwise) or 180 degrees (counterclockwise)
−90
degrees (clockwise)
negative 90 degrees (clockwise)
90 degrees (counterclockwise)
and P′(−10,−4)
. Determine the degree of rotation and direction by applying rotation mapping.(1 point)
Responses
270 degrees (counterclockwise)
270 degrees (counterclockwise)
−180
degrees (clockwise) or 180 degrees (counterclockwise)
negative 180 degrees (clockwise) or 180 degrees (counterclockwise)
−90
degrees (clockwise)
negative 90 degrees (clockwise)
90 degrees (counterclockwise)
Answers
GPT-4o mini
To determine the degree of rotation and direction for the point \( P(-4, 10) \) to become \( P'(-10, -4) \), we can analyze the transformation mathematically.
1. **Rotation Formulas**:
- The general rotation formulas for a point \( (x, y) \) rotated about the origin by an angle \( \theta \) are:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
2. **Observe Initial and Final Points**:
- Initial Point: \( P(-4, 10) \)
- Final Point: \( P'(-10, -4) \)
3. **Direct Mapping to Find Rotation**:
- Let's find the angle of rotation that would map \( (-4, 10) \) to \( (-10, -4) \).
- Calculate the distance from the origin:
- Initial distance \( d_1 = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116} \)
- Final distance \( d_2 = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \)
- Since the distances are equal, the rotation preserves distance.
4. **Determine the Angle**:
- From graphing the points or determining their tangent slopes:
- The point \( P \) corresponds to quadrant II (where x is negative and y is positive), and \( P' \) corresponds to quadrant III (where both x and y are negative).
- The rotation from quadrant II to quadrant III typically corresponds to either a \( 180^\circ \) rotation or a \( -180^\circ \) rotation (the same angle but in opposite directions).
Hence, the rotation can be identified as either
- \( 180 \) degrees (counterclockwise), which is equivalent to \( -180 \) degrees (clockwise).
From the options given:
- **Answer**: \( -180 \) degrees (clockwise) or \( 180 \) degrees (counterclockwise)
1. **Rotation Formulas**:
- The general rotation formulas for a point \( (x, y) \) rotated about the origin by an angle \( \theta \) are:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
2. **Observe Initial and Final Points**:
- Initial Point: \( P(-4, 10) \)
- Final Point: \( P'(-10, -4) \)
3. **Direct Mapping to Find Rotation**:
- Let's find the angle of rotation that would map \( (-4, 10) \) to \( (-10, -4) \).
- Calculate the distance from the origin:
- Initial distance \( d_1 = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116} \)
- Final distance \( d_2 = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \)
- Since the distances are equal, the rotation preserves distance.
4. **Determine the Angle**:
- From graphing the points or determining their tangent slopes:
- The point \( P \) corresponds to quadrant II (where x is negative and y is positive), and \( P' \) corresponds to quadrant III (where both x and y are negative).
- The rotation from quadrant II to quadrant III typically corresponds to either a \( 180^\circ \) rotation or a \( -180^\circ \) rotation (the same angle but in opposite directions).
Hence, the rotation can be identified as either
- \( 180 \) degrees (counterclockwise), which is equivalent to \( -180 \) degrees (clockwise).
From the options given:
- **Answer**: \( -180 \) degrees (clockwise) or \( 180 \) degrees (counterclockwise)