Find the vertices of each of the figure of rotation 90⁰ clockwise about the origin

Given Data:

>>From figure, the coordinates of the points w, u, x are.
>>> W(-4, -3) and U(4, 0) and X(-3, -2) are the required coordinates of a triangle.
>>Let, (x, y) be the point in the space and are rotated through 180 degrees.
Then, the new coordinates are:
=  (x cos - y sin , y cos + x sin)
= (x cos180 -y sin180 , y cos180 + xsin180)
= (-x , -y)
* Then, the new coordinates of (x, y) after rotation through 180 degrees is (-x, -y).
>>>Similarly, for  W(-4, -3) and U(4, 0) and X(-3, -2) the new coordinates are:
W(4, 3) and U(-4, 0) and (3, 2).
>>>Hence, the rotation of points W(-4, -3) and U(4, 0) and X(-3, -2) through 180 degrees becomes  W(4, 3) and U(-4, 0) and (3, 2).

Given That:
                    A (-3,4) B (0, 1), C (-5, 2) are the vertices of a triangle.
>>>The vertices of a triangle are rotated through 90 degrees counter clockwise direction.
>>>Let, the point on the space be (x, y). Then Angle of Rotation  becomes 90 degrees.
>>>Hence, new coordinates are:
                                 = (x cos - y sin , y cos + x sin)
                                 = (x cos90 - y sin90 , y cos90 + x sin90)
                                 = (-y , x).
>>>Similarly, the rotation of the points  A (-3,4) B (0, 1), C (-5, 2)  becomes:
                A(-4,-3) and B(-1,0) and C(-2, -5).
***Therefore, the triangle vertices A (-3,4) B (0, 1), C (-5, 2)  after rotation through 90 degrees counter clockwise are   A(-4,-3) and B(-1,0) and C(-2, -5).

Given That:
                  A (-3,4), B(0, 1), C (-5, 2) are the points of triangle.
>>we re asked to rotate the vertices of triangle by 180 degrees.
>>>let, the point be (x, y) then the angle of rotation be 180 degrees. Then:
>>New Coordinates are:
= (x cos - y sin , y cos + x sin).
                             = (x cos180 - y sin180 , y cos180 + x sin180)
                             = (-x , -y).
>>>Hence, any point that rotated by 180 degrees will just change the sign of the coordinates.
>>>Similarly, for triangle coordinates  A (-3,4), B(0, 1), C (-5, 2) the rotation through 180 degrees about origin becomes : A(3, -4) and B(0, -1) and C(5, -2).

* 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 cos - y sin , y cos + x sin).
 

* 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 cos - y sin , y cos + x sin).

 

* 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 cos - y sin , y cos + x sin).

 

Given Data:
If the point (-4, -6) rotates 180° counter-clockwise, then the point will be
>>>Since it is counter clockwise the angle of rotation becomes = 180 degrees.
>>Point (x, y) = (-4, -6)
>>>New Coordinates are:
= (x cos - y sin , y cos + x sin).
= (-4cos180 + 6sin180 , -6cos180 -6sin180)
= (4 , 6).
***Therefore, the rotation of the point (-4, -6 ) through 180 degrees counter clockwise becomes (4, 6).

Given Data:
If the point (4, 6) rotates 90° counter-clockwise, then the point will be
>>>Since, it is counter clockwise rotation angle of rotation becomes = 90 degrees.
>>point (x, y)= (4, 6)
*New Coordinates are:
= (x cos - y sin , y cos + x sin).
= (4cos90 - 6sin90 , 6cos90 + 4sin90)
= (-6 , 4).
***The rotation of the point (4, 6 ) through 90 degrees counter clockwise becomes (-6, 4).

Given data:
If the point (2, 3) rotates 90°clockwise, then the point will be
>>>Since, it is clockwise rotation angle of rotation  becomes negative that is  = -90.
>>And the point (x, y) = (2, 3).
>>>Hence, the new coordinates are:
= (x cos - y sin , y cos + x sin).
                             = (2 cos(-90) - 3 sin(-90) , 3 cos(-90) + 2 sin(-90))
                             = (3 ,-2).
**Hence, the rotation of the point (2, 3) through 90 degrees clockwise is (3, -2).

Given Data:
                    In rotation, the point (x, y) after moving 180° both clockwise and anti-clockwise will be
>>First step is to rotate the point (x, y) through 180 degrees clockwise.
* Hence, the angle of rotation becomes = -180
The new coordinates are:
                                = (x cos - y sin , y cos + x sin)
                                = (x cos(-180) - y sin(-180) , y cos(-180) + x sin(-180))
                                = (-x, -y).
***Therefore, the rotation of a point (x, y) through 180 degrees clockwise gives (-x, -y).
>>>similarly, the rotation of the point (x, y) through 180 degrees counter clockwise gives:
*The new Coordinates are:
              =  (x cos - y sin , y cos + x sin)
              = (x cos(180) -y sin(180) , y cos(180) + x sin(180))
              = (-x , -y).
***Therefore, the rotation of the point (x, y) through 180 degrees counter clockwise gives (-x, -y)
>>>Hence, we can say that the rotation of the point (x, y) through any direction yields (-x, -y).

Given Data:
In rotation, the point (x, y) after moving 270° clockwise and 90° counter-clockwise will be
>>>The first step is to rotate a point (x, y) through 270 degrees clockwise
* Angle of rotation  becomes -270
* Then, the coordinates after rotation :
                      = (x cos - y sin , y cos + x sin)
                      = (x cos(-270) - y sin(-270) , y cos(-270) + x sin(-270))
                      = (-y, x).
>>>Second step is to rotate (-y, x) through 90 degrees counter clockwise 
* Angle of rotation  becomes 90 degrees.
* Then , the new coordinates after rotation becomes:
                             =  (x cos - y sin , y cos + x sin)
                             = (-y cos(90) - x sin(90) , x cos(90) -y sin(90))
                             = (-x , -y).
***Therefore the rotation of a point (x, y) through 270 degrees clockwise and 90 degrees counter clockwise becomes (-x, -y).

Given Data:
                      In rotation, the point (x, y) after moving 270° clockwise will be
>>>since, it is clockwise rotation the angle of rotation  becomes negative.
>>= -270.
>>>New coordinates of a point after rotation:
                             = (x cos - y sin , y cos + x sin)
= (x cos(-270) - y sin(-270) , y sin(-270) + x sin(-270))
= (-y , x).
***Hence, the rotation of the point (x, y) through 270 degrees clockwise becomes (-y, x)

Given That:
In rotation, the point (x, y) after moving 180°counter-clockwise will be
>>>Since, it is counter clockwise rotation the angle of rotation remains positive.
>>>Therefore, the angle of rotation  becomes 180 degrees.
>>>The new coordinates becomes:
=  (x cos - y sin , y cos + x sin)
                     = (x cos(180) -y sin(180) , y cos(180) + x sin(180))
= (-x , -y).
***Hence, the rotation of the point (x, y) through 180 degrees counter clockwise becomes (-x, -y).

Given Data:
In rotation, the point (x, y) after moving 180° clockwise will be
>>> Since, the clockwise rotation denotes negative degrees the angle of rotation  becomes -180 degrees.
>>>New Coordinates of a point (x, y) are:
=  (x cos - y sin , y cos + x sin)
= (x cos(-180) - y sin(-180) , y cos(-180) + x sin(-180))
= (-x , -y).
**Therefore, the rotation of the point (x, y) through 180 degrees clockwise gives (-x , -y).

Given That:
In rotation, the point (x, y) after moving 90° counter-clockwise will be
* We were asked to rotate the point (x, y) through 90 degrees counter clockwise.
>Hence, the point after rotation becomes:
= (x cos - y sin , y cos  + x sin)
= (x cos(90) - y sin(90) , y cos(90) + x sin(90))
= (-y , x ).
>>>Hence, the rotation of the point (x, y) through 90 degrees counter clockwise becomes (-y, x).