A quadrilateral with vertices G(-10,1), E(-6,-4), O(2,0), and M(0,4) is rotated about the origin 270 degrees (counterclockwise). apply rotation mapping rules to find image of M.
To rotate a point (x, y) counterclockwise about the origin by a certain angle, we can use the following rotation mapping rules:
x' = x * cosθ - y * sinθ
y' = x * sinθ + y * cosθ
where (x', y') represents the coordinates of the image point after rotation, and θ represents the angle of rotation.
In this case, we need to rotate the point M(0, 4) about the origin 270 degrees counterclockwise. Let's apply the rotation mapping rules:
x' = 0 * cos270 - 4 * sin270
= 0 - (-4)
= 4
y' = 0 * sin270 + 4 * cos270
= 0 + 4
= 4
Therefore, the image of M(0, 4) after rotating 270 degrees counterclockwise about the origin is M'(4, 4).
To apply rotation mapping rules to find the image of M after a 270-degree counterclockwise rotation about the origin, you can follow these steps:
1. Calculate the new coordinates of the point M(x, y) after the rotation.
- For a counterclockwise rotation of 270 degrees, the rotation matrix is:
[ cos(270) -sin(270) ]
[ sin(270) cos(270) ]
The sine and cosine values for 270 degrees can be calculated as follows:
sin(270) = -1
cos(270) = 0
2. Substitute the coordinates of point M(0, 4) into the rotation matrix.
By applying the rotation matrix to the coordinates (0, 4), you get:
[ 0 ] [ cos(270) -sin(270) ] [ 0 ] [ 0 ]
[ 4 ] = [ sin(270) cos(270) ] * [ 4 ] = [ -4 ]
3. Therefore, the image of point M(0, 4) after a 270-degree counterclockwise rotation about the origin is (-4, 0).
To find the image of point M (-10,1) after rotating it counterclockwise by 270 degrees about the origin, we can use the rotation mapping rules.
The rotation mapping rule for counterclockwise rotation by θ degrees about the origin is given by:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, θ = 270 degrees.
Plugging the values of point M into the rotation mapping rules, we have:
x' = 0 * cos(270) - 4 * sin(270)
y' = 0 * sin(270) + 4 * cos(270)
To find the values of the trigonometric functions for 270 degrees, we convert it into radians.
270 degrees = (270 * π) / 180 = (3/2) * π
Using the trigonometric functions for (3/2) * π, we have:
cos(270) = cos((3/2) * π) = 0
sin(270) = sin((3/2) * π) = -1
Plugging these values back into the rotation mapping rules, we get:
x' = 0 * 0 - 4 * (-1) = 0 + 4 = 4
y' = 0 * (-1) + 4 * 0 = 0 + 0 = 0
Therefore, after rotating point M (-10,1) counterclockwise by 270 degrees about the origin, its image is point M' (4,0).