For every point A = (x,y) in your figure, a 180 degree counterclockwise rotation about the origin will result in a point A' = (x', y') where:
x' = x * cos(180) - y * sin(180)
y' = x * sin(180) + y * cos(180)
Happy-fun time fact: This is equivalent to using a rotation matrix from Linear Algebra!
Because a rotation is an isometry, you only have to rotate each vertex of a polygon, and then connect the respective rotated vertices to get the rotated polygon.
You can rotate a closed curve as well, but you must figure out a way to rotate the infinite number of points in the curve. We are able to do this with straight lines above due to the property of isometries, which preserves distances between points.
180 degrees.
Move it 3 times* * * * *or once in the anti-clockwise direction.
You dont, its just 90 degrees 3 times..
To rotate a figure 180 degrees clockwise about the origin you need to take all of the coordinates of the figure and change the sign of the x-coordinates to the opposite sign(positive to negative or negative to positive). You then do the same with the y-coordinates and plot the resulting coordinates to get your rotated figure.
It is (-1, 6).
180 degrees.
{1 0} {0 -1}
Move it 3 times* * * * *or once in the anti-clockwise direction.
Ex: -1,-2 Switch the numbers, so with the example it would be -2,-1. Next multiply your x coordinate by -1,so the example would be 2,-1
(-1, -4) rotated 90 degrees anticlockwise
You dont, its just 90 degrees 3 times..
To rotate a figure 90 degrees clockwise about the origin, simply swap the x and y coordinates of each point and then negate the new y-coordinate. This is equivalent to reflecting the figure over the line y = x and then over the y-axis.
To rotate a figure 180 degrees clockwise about the origin you need to take all of the coordinates of the figure and change the sign of the x-coordinates to the opposite sign(positive to negative or negative to positive). You then do the same with the y-coordinates and plot the resulting coordinates to get your rotated figure.
A counterclockwise rotation of 270 degrees about the origin is equivalent to a clockwise rotation of 90 degrees. To apply this transformation to a point (x, y), you can use the rule: (x, y) transforms to (y, -x). This means that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.
A rotation of 180 degrees counterclockwise refers to turning a point or shape around a central point (such as the origin in a coordinate plane) by half a turn. This effectively moves each point to a position that is directly opposite its starting point. For example, if a point is at coordinates (x, y), after a 180-degree counterclockwise rotation, its new coordinates will be (-x, -y). This transformation maintains the shape and size but changes its orientation.
A rotation of 270 degrees clockwise is equivalent to a rotation of 90 degrees counterclockwise. In a Cartesian coordinate system, this means that a point originally at (x, y) will move to (y, -x) after the rotation. Essentially, it shifts the point three-quarters of the way around the origin in the clockwise direction.
To rotate a figure 180 degrees clockwise, you can achieve this by first reflecting the figure over the y-axis and then reflecting it over the x-axis. This double reflection effectively rotates the figure 180 degrees clockwise around the origin.