Mathematics
Grade-8
Easy

Question

In which rotation movement does (x, y)      (-x, -y)

  1. CCW 90° and CW 270°
  2. CW 90°
  3. CW180° and   CCW180°
  4. CCW 270°

hintHint:

TO find the Angle of Rotation, we need compare both the point and it's rotation point.

The correct answer is: CW180° and   CCW180°


    Given Data:
    In which rotation movement does (x, y)      (-x, -y)
    ***we were asked to find the Angle of Rotation of a point (x, y) to rotate it to (-x, -y).
    >>>The rotated coordinates are:
    (x', y') =  (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
    >>>From the data given  (x', y') = (-x, -y)
    * Hence, By comparing the above equation's we get:
    -x = x cosalpha - y sinalpha and  -y = y cosalpha + x sinalpha. Then
    * By solving the above equation's we get:
    (y cross times- x) = (y cross times x ) cosalpha - y2 sinalpha
    (-y cross times x) = (y cross times x ) cosalpha + x2sinalpha
    ___________________________________
    0   = 0 + (x2+y2)sinalpha

    sinalpha=0
    alpha = 180 degrees or -180 degrees.
    ***Hence, the Angle of Rotation to rotate the point (x, y) to (-x, -y) is counter clockwise 180 degrees and clockwise 180 degrees.


    * In Mathematics, rotation means the Circular movement of an object around one fixed point.

    * In rotation, the image after transformation remains constant.

    * Hence, it is called as a rigid transformation.

    * No Change in shape and size.

    * The Shape rotates counter- clockwise when the degrees is positive and rotates clockwise when degrees is negative.

    *The Rotation of a point (x, y) about origin and through angle alpha, then:
    New coordinates of a point (x, y) after it's rotation becomes (x cosalpha - y sinalpha , y cosalpha + x sinalpha).

    Related Questions to study

    Grade-8
    Mathematics

    In which rotation movement does (x, y)      (-y, x)

    Given Data:
    In which rotation movement does (x, y)      (-y, x)
    >>>We were asked to find the angle of rotation of a point to rotate a point from (x, y) to (-y, x).
    *** Rotated coordinates are:
    (x', y') =  (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
    >>Here, the rotated points are :
                  (x', y') = (-y, x).
    * Hence, By comparing the above equation's we get:
                 -y =  x cosalpha - y sinalpha ; and x = y cosalpha + x sinalpha

    Hence, By solving the above equation's we get:

                (x cross times -y) = x2cosalpha - (x cross times y)sinalpha

    and      (y cross times x) = y2 cosalpha + (x cross times y)sinalpha
              ________________________________
                          0 = ( x2 + y2)cosalpha 

    * Hence, cosalpha =0 leads to 90 degrees or -270 degrees.
    >>>>Therefore, the Angle of Rotation is counter clockwise 90 degrees and clockwise 270 degrees.

    In which rotation movement does (x, y)      (-y, x)

    MathematicsGrade-8

    Given Data:
    In which rotation movement does (x, y)      (-y, x)
    >>>We were asked to find the angle of rotation of a point to rotate a point from (x, y) to (-y, x).
    *** Rotated coordinates are:
    (x', y') =  (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
    >>Here, the rotated points are :
                  (x', y') = (-y, x).
    * Hence, By comparing the above equation's we get:
                 -y =  x cosalpha - y sinalpha ; and x = y cosalpha + x sinalpha

    Hence, By solving the above equation's we get:

                (x cross times -y) = x2cosalpha - (x cross times y)sinalpha

    and      (y cross times x) = y2 cosalpha + (x cross times y)sinalpha
              ________________________________
                          0 = ( x2 + y2)cosalpha 

    * Hence, cosalpha =0 leads to 90 degrees or -270 degrees.
    >>>>Therefore, the Angle of Rotation is counter clockwise 90 degrees and clockwise 270 degrees.

    Grade-8
    Mathematics

    In rotation of clockwise movement maps (x , y) (y,-x)

    Given Data:
    The point (x, y) is transformed to (x , y) (y,-x) in clockwise direction.
    >>> we were asked to find the Angle of Rotation.
    >>>The coordinates of a point (x, y) after rotation through 90 degrees in clockwise direction are:
    (x', y') = (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
    >>>we were given that (x', y') = (y, -x)
    >>> (y, -x) = (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
        Hence, y = x cosalpha - y sinalpha and -x = y cosalpha + x sinalpha
    By solving the above equation's we get:
                  (x cross times y)  = x2cosalpha - (x cross times y) sinalpha
    and        (y cross times -x) = y2cosalpha + (x cross times y) sinalpha

    __________________________________
    0 = (x2+y2)cosalpha
    *This implies cosalpha=0, then:
    alpha = 90 degrees.
    >>>Therefore, the angle of rotation is 90 degrees.

    In rotation of clockwise movement maps (x , y) (y,-x)

    MathematicsGrade-8

    Given Data:
    The point (x, y) is transformed to (x , y) (y,-x) in clockwise direction.
    >>> we were asked to find the Angle of Rotation.
    >>>The coordinates of a point (x, y) after rotation through 90 degrees in clockwise direction are:
    (x', y') = (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
    >>>we were given that (x', y') = (y, -x)
    >>> (y, -x) = (x cosalpha - y sinalpha , y cosalpha + x sinalpha)
        Hence, y = x cosalpha - y sinalpha and -x = y cosalpha + x sinalpha
    By solving the above equation's we get:
                  (x cross times y)  = x2cosalpha - (x cross times y) sinalpha
    and        (y cross times -x) = y2cosalpha + (x cross times y) sinalpha

    __________________________________
    0 = (x2+y2)cosalpha
    *This implies cosalpha=0, then:
    alpha = 90 degrees.
    >>>Therefore, the angle of rotation is 90 degrees.

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