Question
Find the vertices of each of the figures of rotation 900 counter-clockwise about the origin
- B'(0, −5), E'(−1, −2), G'(3, −2)
- B'(0, 5), E'(−1, 2), G'(3, 2)
- B'(0, −5), E'(−1, 2), G'(3, 2)
- B'(0, 5), E'(−1, 2), G'(3, −2)
Hint:
Retrieve the points from the figure and then rotate the points through 90 degrees counter clock wise to obtain the new coordinates.
The correct answer is: B'(0, −5), E'(−1, −2), G'(3, −2)
Given Data:
>>From figure, the vertices of the triangle are:
B(-5,0) and E(-2,1) and G(-2, -3).
>>>let, the point (x, y) be in the space and the Angle of Rotation becomes = 90.
>>>new coordinates are:
= (x cos - y sin , y cos + x sin)
= (x cos90 -y sin90 , y cos90 + x sin90)
= (-y, x).
* Hence, the final coordinates after rotation through 90 degrees counter clockwise are (-y, x).
>>>Similarly, for the coordinates B(-5,0) and E(-2,1) and G(-2, -3) the rotation of points through 90 degrees counter clock wise becomes:
B(0,-5) and E(3, -2) and G(3,2).
***Therefore, the coordinates of triangle B(-5,0) and E(-2,1) and G(-2, -3) after rotation through 90 degrees counter clockwise becomes B(0,-5) and E(3, -2) and G(3,2).
* 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
Find the vertices of each of the figure of rotation 900 clockwise about the origin
Given Data:
* From figure, the points are K(2, -2) and U(3, 3) and T(5, 0) are the vertices of the triangle.
>>Let (x, y) be the point in the space and angle of rotation becomes = -90 degrees.
*Then, the new coordinates are :
= (x cos - y sin , y cos + x sin)
= (x cos(-90) -y sin(-90) , y cos(-90) + x sin(-90))
= (y, -x).
>>Then, the new coordinates are (y, -x).
>>>Similarly, the rotation of triangular vertices K(2, -2) and U(3, 3) and T(5, 0) through 90 degrees clockwise becomes :
K(-2, -2) and U(3, -3) and T(0, -5).
*Therefore, the vertices K(2, -2) and U(3, 3) and T(5, 0) after rotation through 90 degrees clockwise becomes K(-2, -2) and U(3, -3) and T(0, -5).
Find the vertices of each of the figure of rotation 900 clockwise about the origin
Given Data:
* From figure, the points are K(2, -2) and U(3, 3) and T(5, 0) are the vertices of the triangle.
>>Let (x, y) be the point in the space and angle of rotation becomes = -90 degrees.
*Then, the new coordinates are :
= (x cos - y sin , y cos + x sin)
= (x cos(-90) -y sin(-90) , y cos(-90) + x sin(-90))
= (y, -x).
>>Then, the new coordinates are (y, -x).
>>>Similarly, the rotation of triangular vertices K(2, -2) and U(3, 3) and T(5, 0) through 90 degrees clockwise becomes :
K(-2, -2) and U(3, -3) and T(0, -5).
*Therefore, the vertices K(2, -2) and U(3, 3) and T(5, 0) after rotation through 90 degrees clockwise becomes K(-2, -2) and U(3, -3) and T(0, -5).
Find the vertices of each of the figure of rotation 1800 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).
Find the vertices of each of the figure of rotation 1800 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).
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 90o counter-clockwise about the origin?
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).
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 90o counter-clockwise about the origin?
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).
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?
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).
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?
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 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).