A quadrilateral with the 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 the image of M.
To rotate a point (x, y) counterclockwise around the origin by θ degrees, the new coordinates (x', y') can be found using the following formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, we are rotating the point M (0, 4) by 270 degrees counterclockwise. Therefore, θ = 270 degrees.
Applying the rotation formulas, we have:
x' = 0 * cos(270) - 4 * sin(270)
y' = 0 * sin(270) + 4 * cos(270)
Simplifying, cos(270) = 0 and sin(270) = -1, so we get:
x' = 0 - 4 * (-1) = 0 + 4 = 4
y' = 0 + 4 * (0) = 0 + 0 = 0
Therefore, the image of M (0, 4) after rotating it 270 degrees counterclockwise is M' (4, 0).
To find the image of point M(0,4) after rotating the quadrilateral counterclockwise by 270 degrees about the origin, we can apply the rotation mapping rules.
The rotation mapping rule is given by:
(x', y') = (x * cosθ - y * sinθ, x * sinθ + y * cosθ),
where (x, y) are the coordinates of the original point, (x', y') are the coordinates of the rotated point, and θ is the angle of rotation.
In this case, the angle of rotation is 270 degrees. Let's apply the rotation mapping rule:
(x', y') = (0 * cos270° - 4 * sin270°, 0 * sin270° + 4 * cos270°)
To simplify, let's convert the angles to radians:
(x', y') = (0 * cos(270° * π/180) - 4 * sin(270° * π/180), 0 * sin(270° * π/180) + 4 * cos(270° * π/180))
Calculating the values:
(x', y') = (0 * cos(270 * π/180) - 4 * sin(270 * π/180), 0 * sin(270 * π/180) + 4 * cos(270 * π/180))
= (0 * (-1) - 4 * 0, 0 * 0 + 4 * (-1))
= (0, -4)
Therefore, the image of point M(0,4) after rotating the quadrilateral counterclockwise by 270 degrees about the origin is M'(0, -4).
To find the image of point M(-10,1) after rotating 270 degrees counterclockwise about the origin, we can apply the rotation mapping rules.
First, let's understand the rotation mapping rules:
1. To rotate a point (x, y) counterclockwise by θ degrees about the origin, you need to apply the following formula:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Now, let's apply these rules to find the image of point M(-10,1) after a 270-degree counterclockwise rotation:
x' = -10 * cos(270°) - 1 * sin(270°)
= -10 * 0 - 1 * (-1)
= 0 - (-1)
= 1
y' = -10 * sin(270°) + 1 * cos(270°)
= -10 * (-1) + 1 * 0
= 10 + 0
= 10
Therefore, after rotating M(-10,1) by 270 degrees counterclockwise about the origin, its image point is M'(1, 10).