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

3 answers

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.

Y(-9, -1) rotated 270 degrees clockwise. What is y'

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:

\[ \boxed{(1, -9)} \]