Question
a point is rotated about the origin. it's coordinates are P(-4,10) and P'(-10,-4). determine the degree of rotation and direction by applying rotation mapping.
90 degrees counterclockwise
-90 degrees clockwise
270 degrees counterclockwise
-180 degrees clockwise or 180 degrees counterclockwise
90 degrees counterclockwise
-90 degrees clockwise
270 degrees counterclockwise
-180 degrees clockwise or 180 degrees counterclockwise
Answers
GPT-4o mini
To determine the degree of rotation and the direction of rotation from point \(P(-4, 10)\) to \(P'(-10, -4)\), we can use the rotation mapping formulas for points in the Cartesian plane.
The general rotation formulas are:
- For a rotation of \( \theta \) degrees counterclockwise about the origin, the point \((x, y)\) transforms to \((x', y')\) as follows:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
Finding the degree of rotation, let's analyze the original and new coordinates:
1. Calculate the distance from the origin for both points:
\[
|P| = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116}
\]
\[
|P'| = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116}
\]
The distances from the origin remain the same, confirming that it is indeed a rotation.
2. Next, determine the angle of rotation by acquiring the angles made by both points with respect to the positive x-axis.
* For \( P(-4, 10) \):
\[
\theta_P = \tan^{-1}\left(\frac{10}{-4}\right) \approx \tan^{-1}\left(-2.5\right) \approx 180^\circ - 68.2^\circ \approx 111.8^\circ
\]
* For \( P'(-10, -4) \):
\[
\theta_{P'} = \tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}(0.4) \approx 22.6^\circ
\]
* Since \(P'\) is in the third quadrant, we add 180 degrees:
\[
\theta_{P'} \approx 180 + 22.6 \approx 202.6^\circ
\]
Now, we can calculate the angle of rotation from \(P\) to \(P'\):
\[
\text{Angle of rotation} = \theta_{P'} - \theta_P \approx 202.6^\circ - 111.8^\circ \approx 90.8^\circ
\]
Since the angle is approximately \(90^\circ\) and is positive, this indicates a *counterclockwise* rotation.
Based on the options given, the closest correct answer is:
**90 degrees counterclockwise.**
The general rotation formulas are:
- For a rotation of \( \theta \) degrees counterclockwise about the origin, the point \((x, y)\) transforms to \((x', y')\) as follows:
\[
x' = x \cos(\theta) - y \sin(\theta)
\]
\[
y' = x \sin(\theta) + y \cos(\theta)
\]
Finding the degree of rotation, let's analyze the original and new coordinates:
1. Calculate the distance from the origin for both points:
\[
|P| = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116}
\]
\[
|P'| = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116}
\]
The distances from the origin remain the same, confirming that it is indeed a rotation.
2. Next, determine the angle of rotation by acquiring the angles made by both points with respect to the positive x-axis.
* For \( P(-4, 10) \):
\[
\theta_P = \tan^{-1}\left(\frac{10}{-4}\right) \approx \tan^{-1}\left(-2.5\right) \approx 180^\circ - 68.2^\circ \approx 111.8^\circ
\]
* For \( P'(-10, -4) \):
\[
\theta_{P'} = \tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}(0.4) \approx 22.6^\circ
\]
* Since \(P'\) is in the third quadrant, we add 180 degrees:
\[
\theta_{P'} \approx 180 + 22.6 \approx 202.6^\circ
\]
Now, we can calculate the angle of rotation from \(P\) to \(P'\):
\[
\text{Angle of rotation} = \theta_{P'} - \theta_P \approx 202.6^\circ - 111.8^\circ \approx 90.8^\circ
\]
Since the angle is approximately \(90^\circ\) and is positive, this indicates a *counterclockwise* rotation.
Based on the options given, the closest correct answer is:
**90 degrees counterclockwise.**
Y(-9, -1) rotated 270 degrees clockwise. What is y'
GPT-4o mini
To find the coordinates of the point \(Y' = (x', y')\) after rotating the point \(Y(-9, -1)\) by 270 degrees clockwise around the origin, we can apply the rotation formula.
A rotation of \(270\) degrees clockwise is equivalent to a rotation of \(90\) degrees counterclockwise. The transformation formula for rotating a point \((x, y)\) by \(90\) degrees counterclockwise is:
\[
(x', y') = (-y, x)
\]
Now, applying this to the point \(Y(-9, -1)\):
1. Identify \(x\) and \(y\):
- \(x = -9\)
- \(y = -1\)
2. Substitute into the transformation:
- \(x' = -(-1) = 1\)
- \(y' = -9\)
Thus, after rotating \(Y(-9, -1)\) by \(270\) degrees clockwise (or equivalently \(90\) degrees counterclockwise), the new coordinates are:
\[
Y' = (1, -9)
\]
So, the coordinates of \(Y'\) after the rotation are:
\[
\boxed{(1, -9)}
\]
A rotation of \(270\) degrees clockwise is equivalent to a rotation of \(90\) degrees counterclockwise. The transformation formula for rotating a point \((x, y)\) by \(90\) degrees counterclockwise is:
\[
(x', y') = (-y, x)
\]
Now, applying this to the point \(Y(-9, -1)\):
1. Identify \(x\) and \(y\):
- \(x = -9\)
- \(y = -1\)
2. Substitute into the transformation:
- \(x' = -(-1) = 1\)
- \(y' = -9\)
Thus, after rotating \(Y(-9, -1)\) by \(270\) degrees clockwise (or equivalently \(90\) degrees counterclockwise), the new coordinates are:
\[
Y' = (1, -9)
\]
So, the coordinates of \(Y'\) after the rotation are:
\[
\boxed{(1, -9)}
\]