△JMH is rotated −270 degrees (clockwise) about the origin. Apply rotation mapping rules to find the image of H.
To find the image of point H after a rotation of -270 degrees clockwise about the origin, we can use the rotation mapping rules.
The rotation mapping rules state that for a rotation of θ degrees clockwise about the origin, the coordinates (x, y) of any point become (x', y') after the rotation, where:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, the point H has coordinates (x, y) = (3, 4).
Using the rotation mapping rules, we can find the new coordinates (x', y'):
x' = 3 * cos(-270°) - 4 * sin(-270°)
= 3 * cos(-270°) + 4 * cos(90°) (since sin(-x) = -sin(x) and cos(-x) = cos(x))
= 3 * cos(-270°) + 4 * 0
= 3 * cos(90°)
= 3 * 0
= 0
y' = 3 * sin(-270°) + 4 * cos(-270°)
= 3 * sin(-270°) + 4 * sin(90°) (since sin(-x) = -sin(x) and cos(-x) = cos(x))
= 3 * sin(-270°) + 4 * 0
= 3 * sin(90°)
= 3 * 1
= 3
Therefore, the image of point H after a rotation of -270 degrees clockwise about the origin is (0, 3).
The original points are:
J: (-3,2)
M: (0,3)
H: (-8,-1)
To find the image of point H after a rotation of -270 degrees clockwise about the origin, we can use the rotation mapping rules.
The rotation mapping rules state that for a rotation of θ degrees clockwise about the origin, the coordinates (x, y) of any point become (x', y') after the rotation, where:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, the point H has coordinates (x, y) = (-8, -1).
Using the rotation mapping rules, we can find the new coordinates (x', y'):
x' = -8 * cos(-270°) - (-1) * sin(-270°)
= -8 * cos(-270°) - (-1) * cos(90°) (since sin(-x) = -sin(x) and cos(-x) = cos(x))
= -8 * cos(-270°) - cos(90°)
= -8 * cos(90°) - cos(90°)
= -8 * 0 - 0
= 0
y' = -8 * sin(-270°) + (-1) * cos(-270°)
= -8 * sin(-270°) + (-1) * sin(90°) (since sin(-x) = -sin(x) and cos(-x) = cos(x))
= -8 * sin(-270°) + (-1) * 0
= -8 * sin(90°) + 0
= -8 * 1 + 0
= -8
Therefore, the image of point H after a rotation of -270 degrees clockwise about the origin is (0, -8).
To apply the rotation mapping rules, we need to know the coordinates of point H.
Let's assume the coordinates of point H are (x, y).
To rotate a point (x, y) -270 degrees about the origin, we can follow the rotation mapping rules:
1. The x-coordinate of the image is given by: x' = x * cos(-270) - y * sin(-270)
2. The y-coordinate of the image is given by: y' = x * sin(-270) + y * cos(-270)
Let's substitute the angle -270 degrees into radians, which is -270 * π / 180, to simplify the calculations:
1. x' = x * cos(-270 * π / 180) - y * sin(-270 * π / 180)
2. y' = x * sin(-270 * π / 180) + y * cos(-270 * π / 180)
Now, let's substitute the coordinates of point H and calculate the values of x' and y':
1. x' = x * cos(-270 * π / 180) - y * sin(-270 * π / 180)
= x * cos(-3π / 2) - y * sin(-3π / 2)
= x * cos(π / 2) + y * sin(π / 2)
= 0 + y * 1
= y
2. y' = x * sin(-270 * π / 180) + y * cos(-270 * π / 180)
= x * sin(-3π / 2) + y * cos(-3π / 2)
= x * sin(π / 2) + y * cos(π / 2)
= x * 1 + 0 * y
= x
Therefore, the image of point H after rotating -270 degrees (clockwise) about the origin is (y, x).