What is the image of the point (0,5) after a rotation of 180 degrees
counterclockwise about the origin?
To find the image of the point (0,5) after a rotation of 180 degrees counterclockwise about the origin, we can use the formula for rotating a point (x, y) counterclockwise by an angle θ about the origin.
The formula is:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, θ = 180 degrees, x = 0, and y = 5.
Using the formula, we get:
x' = 0*cos(180) - 5*sin(180) = 0 - 0 = 0
y' = 0*sin(180) + 5*cos(180) = 0 + (-5) = -5
Therefore, the image of the point (0,5) after a rotation of 180 degrees counterclockwise about the origin is (0, -5).
What is the image of the point (−7,−3) after a rotation of 90 degrees
counterclockwise about the origin?
To find the image of the point (-7, -3) after a rotation of 90 degrees counterclockwise about the origin, we can use the same formula as before:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, θ = 90 degrees, x = -7, and y = -3.
Using the formula, we get:
x' = (-7)*cos(90) - (-3)*sin(90) = -7*0 - (-3)*1 = -3
y' = (-7)*sin(90) + (-3)*cos(90) = -7*1 + (-3)*0 = -7
Therefore, the image of the point (-7, -3) after a rotation of 90 degrees counterclockwise about the origin is (-3, -7).
What is the image of the point (−1,4) after a rotation of 270 degrees
counterclockwise about the origin?
To find the image of the point (-1, 4) after a rotation of 270 degrees counterclockwise about the origin, we can use the same formula as before:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, θ = 270 degrees, x = -1, and y = 4.
Using the formula, we get:
x' = (-1)*cos(270) - 4*sin(270) = 1*0 - 4*(-1) = 4
y' = (-1)*sin(270) + 4*cos(270) = (-1)*(-1) + 4*0 = 1
Therefore, the image of the point (-1, 4) after a rotation of 270 degrees counterclockwise about the origin is (4, 1).
What is the image of the point (−6,−3) after a rotation of 180 degrees
counterclockwise about the origin?
To find the image of the point (-6, -3) after a rotation of 180 degrees counterclockwise about the origin, we can use the same formula as before:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, θ = 180 degrees, x = -6, and y = -3.
Using the formula, we get:
x' = (-6)*cos(180) - (-3)*sin(180) = -6*(-1) - (-3)*0 = -6
y' = (-6)*sin(180) + (-3)*cos(180) = (-6)*0 + (-3)*(-1) = 3
Therefore, the image of the point (-6, -3) after a rotation of 180 degrees counterclockwise about the origin is (-6, 3).