Question
What will be the coordinates of the image of ABC with points A (-3,4) B (0, 1), C (-5, 2) after it is rotated 180o about the origin?
- (-4, -3), (-1, 0), (-2, -5)
- (3, -4), (0, -1), (5, -2)
- (-3, 4), (0, 1), (-5, 2)
- (4, 3), (1, 0), (2, 5)
Hint:
We know that the point that rotated through 180 degrees will changes sign of the coordinates.
The correct answer is: (3, -4), (0, -1), (5, -2)
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).
Related Questions to study
In what quadrant will an image be if the figure is in quadrant I and is rotated 90° counter clockwise?
* 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 what quadrant will an image be if the figure is in quadrant I and is rotated 90° counter clockwise?
* 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 what quadrant will an image be if the figure is in quadrant II and is rotated 180° clockwise?
* 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 what quadrant will an image be if the figure is in quadrant II and is rotated 180° clockwise?
* 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 what quadrant will an image be if the figure is in quadrant III and is rotated 90° clockwise?
* 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 what quadrant will an image be if the figure is in quadrant III and is rotated 90° clockwise?
* 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).
If the point (-4, -6) rotates 180° counter-clockwise, then the point will be
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).
If the point (-4, -6) rotates 180° counter-clockwise, then the point will be
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).
If the point (4, 6) rotates 90° counter-clockwise, then the point will be
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).
If the point (4, 6) rotates 90° counter-clockwise, then the point will be
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).
If the point (2, 3) rotates 90°clockwise, then the point will be
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).
If the point (2, 3) rotates 90°clockwise, then the point will be
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).
In rotation, the point (x, y) after moving 180° both clockwise and anti-clockwise will be
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).
In rotation, the point (x, y) after moving 180° both clockwise and anti-clockwise will be
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).
In rotation, the point (x, y) after moving 270° clockwise and 90° counter-clockwise will be
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).
In rotation, the point (x, y) after moving 270° clockwise and 90° counter-clockwise will be
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).
In rotation, the point (x, y) after moving 270° clockwise will be
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)
In rotation, the point (x, y) after moving 270° clockwise will be
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)
In rotation, the point (x, y) after moving 180°counter-clockwise will be
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).
In rotation, the point (x, y) after moving 180°counter-clockwise will be
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).
In rotation, the point (x, y) after moving 180° clockwise will be
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).
In rotation, the point (x, y) after moving 180° clockwise will be
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).
In rotation, the point (x, y) after moving 90° counter-clockwise will be
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).
In rotation, the point (x, y) after moving 90° counter-clockwise will be
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).
In rotation, the point (x, y) after moving 90° clockwise will be
Given That:
In rotation, the point (x, y) after moving 90° clockwise will be
>>point (x, y) will present in the 1st Quadrant.
>>Since, to rotate the point in clockwise denotes backward movement of point.
>> hence, the point is rotated to fourth Quadrant and the point changes to (y, -x).
>>>Final point obtained on rotation of (x, y) by 90 degrees clockwise is (y, -x).
In rotation, the point (x, y) after moving 90° clockwise will be
Given That:
In rotation, the point (x, y) after moving 90° clockwise will be
>>point (x, y) will present in the 1st Quadrant.
>>Since, to rotate the point in clockwise denotes backward movement of point.
>> hence, the point is rotated to fourth Quadrant and the point changes to (y, -x).
>>>Final point obtained on rotation of (x, y) by 90 degrees clockwise is (y, -x).
In rotation, images after transformation will be
Hence, we can say that the image or object transformation in rotation will be equal.
In rotation, images after transformation will be
Hence, we can say that the image or object transformation in rotation will be equal.
Rotation is transformation that
Hence, we can say that rotation is the transformation that rotates about a point.
Rotation is transformation that
Hence, we can say that rotation is the transformation that rotates about a point.
