Question
Graph a triangle ABC with vertices A(2, 1), B(4, 4), and C(8, 0). Rotate the triangle 180o about the origin.
Hint:
Retrieve the points from the figure and then rotate the points through 180 degrees counter clock wise to obtain the new coordinates.
The correct answer is:
* 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 That:
Graph a triangle ABC with vertices A(2, 1), B(4, 4), and C(8, 0). Rotate the triangle 180 about the origin.
>>>>Let (a, b) be a point in the plane. Then, the point after rotation through 180 degrees becomes:
= (x cos - y sin , y cos+ x sin)
= (acos180 - bsin180 , bcos180 + asin180)
= (-a, -b).
>>>similarly, the vertices of triangle A(2, 1), B(4, 4), and C(8, 0) after rotation through 180 degrees becomes A'(-2,-1); B'(-4,-4); C'(-8,0).
>>>The required graph:
Given, the vertices of a triangle ABC with vertices A(2, 1), B(4, 4), and C(8, 0).
For a rotation of 180 degrees, coordinate rule (a, b) → (-a, -b).
A(2, 1) → A’(-2, -1)
B(4, 4) → B’(-4, -4)
C(8, 0) → C’(-8, 0)
Now, graph the triangle A’B’C’.
Related Questions to study
Graph a triangle LMN with vertices L(-5, -3), M(-3, -5), and N(0, -1). Rotate the triangle 90 about the origin.
Given, the vertices of a triangle ABC with vertices L(-5, -3), M(-3, -5), and N(0, -1).
For a rotation of 90 degrees, coordinate rule (a, b) → (-b, a).
L(-5, -3) → L’(3, -5)
M(-3, -5) → M’(5, -3)
N(0, -1) → N’(1, 0)
Now, graph the triangle L’M’N’.
Graph a triangle LMN with vertices L(-5, -3), M(-3, -5), and N(0, -1). Rotate the triangle 90 about the origin.
Given, the vertices of a triangle ABC with vertices L(-5, -3), M(-3, -5), and N(0, -1).
For a rotation of 90 degrees, coordinate rule (a, b) → (-b, a).
L(-5, -3) → L’(3, -5)
M(-3, -5) → M’(5, -3)
N(0, -1) → N’(1, 0)
Now, graph the triangle L’M’N’.
Find the image coordinates of the figure w.r.t 180o rotation about the origin.
Given coordinates of the figure, A(1, 2), B(5, 2), and C(3, 5).
For a rotation of 180 degrees, coordinate rule (a, b) → (-a, -b).
A(1, 2) → A’(-1, -2)
B(5, 2) → B’(-5, -2)
C(3, 5) → C’(-3, -5)
Image coordinates of the figure are A’(-1, -2), B’(-5, -2), and C’(-3, -5).
Find the image coordinates of the figure w.r.t 180o rotation about the origin.
Given coordinates of the figure, A(1, 2), B(5, 2), and C(3, 5).
For a rotation of 180 degrees, coordinate rule (a, b) → (-a, -b).
A(1, 2) → A’(-1, -2)
B(5, 2) → B’(-5, -2)
C(3, 5) → C’(-3, -5)
Image coordinates of the figure are A’(-1, -2), B’(-5, -2), and C’(-3, -5).
Find the image vertices of the line w.r.t 90o rotation about the origin.
Coordinate of the line AB, A= (1, 2) and B= (5, 2)
For a rotation of 90, coordinate rule (a, b) → (-b, a).
A(1, 2) → A’(-2, 1)
B(5, 2) → B’(-2, 5)
Image vertices of the line AB are A’(-2, 1) and B’(-2, 5).
Find the image vertices of the line w.r.t 90o rotation about the origin.
Coordinate of the line AB, A= (1, 2) and B= (5, 2)
For a rotation of 90, coordinate rule (a, b) → (-b, a).
A(1, 2) → A’(-2, 1)
B(5, 2) → B’(-2, 5)
Image vertices of the line AB are A’(-2, 1) and B’(-2, 5).
Find the coordinate of the point and its image point w.r.t 270o rotation about the origin.
Coordinate of point V = (-3, -2)
For a rotation of 270 degrees, coordinate rule (a, b) → (b, -a).
V(-3, -2) → V’(-2, 3)
The coordinate V(-3, -2) and its image V’(-2, 3).
Find the coordinate of the point and its image point w.r.t 270o rotation about the origin.
Coordinate of point V = (-3, -2)
For a rotation of 270 degrees, coordinate rule (a, b) → (b, -a).
V(-3, -2) → V’(-2, 3)
The coordinate V(-3, -2) and its image V’(-2, 3).
When a point (9, -2) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 270o rotation.
For a rotation of 270 degrees, (a, b) → (b, -a)
(9, -2) →(-2, -9)
When a point (9, -2) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 270o rotation.
For a rotation of 270 degrees, (a, b) → (b, -a)
(9, -2) →(-2, -9)
When a point (5, -4) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 180o rotation.
For a rotation of 180 degrees, (a, b) → (-a, -b).
(5, -4) → (-5, 4)
When a point (5, -4) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 180o rotation.
For a rotation of 180 degrees, (a, b) → (-a, -b).
(5, -4) → (-5, 4)
From the below figure, identify the angle of rotation along with direction.
Angle of rotation is the angle that makes by the ray of center of rotation and the real object and the ray that joins the ray of center of rotation and the image object.
>>>Therefore, the angle of rotation is 105 degrees.
From the below figure, identify the angle of rotation along with direction.
Angle of rotation is the angle that makes by the ray of center of rotation and the real object and the ray that joins the ray of center of rotation and the image object.
>>>Therefore, the angle of rotation is 105 degrees.
When a point (a, b) is rotated 360 counterclockwise about the origin, then the coordinates of the image point will be________
Rotate the point exactly to 360 degrees to obtain the point of rotation.
When a point (a, b) is rotated 360 counterclockwise about the origin, then the coordinates of the image point will be________
Rotate the point exactly to 360 degrees to obtain the point of rotation.
When a point (a, b) is rotated 270 counterclockwise about the origin, then the coordinates of the image point will be________
Rotate the given point in the plane through 270 degrees angle of rotation to obtain the new coordinates.
When a point (a, b) is rotated 270 counterclockwise about the origin, then the coordinates of the image point will be________
Rotate the given point in the plane through 270 degrees angle of rotation to obtain the new coordinates.
When a point (a, b) is rotated 180 counterclockwise about the origin, then the coordinates of the image point will be________
Rotate the point in the plane at 180 degrees of angle of rotation to obtain the point of rotation.
When a point (a, b) is rotated 180 counterclockwise about the origin, then the coordinates of the image point will be________
Rotate the point in the plane at 180 degrees of angle of rotation to obtain the point of rotation.
When a point (a, b) is rotated 90 counterclockwise about the origin, then the coordinates of the image point will be________.
When a point (a, b) is rotated 90 counterclockwise about the origin, then the coordinates of the image point will be (-b, a).
When a point (a, b) is rotated 90 counterclockwise about the origin, then the coordinates of the image point will be________.
When a point (a, b) is rotated 90 counterclockwise about the origin, then the coordinates of the image point will be (-b, a).
Rotation can be clockwise or __________________.
Therefore, we can say that the Rotation can be done both in clockwise or counter clockwise direction.
Rotation can be clockwise or __________________.
Therefore, we can say that the Rotation can be done both in clockwise or counter clockwise direction.
Rays drawn from the center of rotation to a point and their image form the __________ of rotation.
The angle of rotation is nothing but the angle between the line that joins center of rotation to the real point and the line that joins the center of rotation and the image point.
Rays drawn from the center of rotation to a point and their image form the __________ of rotation.
The angle of rotation is nothing but the angle between the line that joins center of rotation to the real point and the line that joins the center of rotation and the image point.
It is a transformation in which a figure is rotated around a fixed point called__________.
>>>>The center of rotation is the point that rotates the given figure about a point.
It is a transformation in which a figure is rotated around a fixed point called__________.
>>>>The center of rotation is the point that rotates the given figure about a point.
Use matrix multiplication to find the image. Graph the polygon and its image.
Reflect 
in the y-axis.
The required matrix representation is:
Use matrix multiplication to find the image. Graph the polygon and its image.
Reflect in the y-axis.
The required matrix representation is:
