Question

given a point (x, y), what is the new point when the point is rotated counterclockwise about the origin 270 degrees.

Answer 1

that is wrong

Answer 2

To rotate a point (x, y) counterclockwise about the origin by 270 degrees, we can use the rotation formula.

The rotation formula in a 2D coordinate system is:

x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)

For a 270-degree counterclockwise rotation, θ = 270 degrees or θ = 3π/2 radians. Plugging in the values:

x' = x * cos(3π/2) - y * sin(3π/2)
y' = x * sin(3π/2) + y * cos(3π/2)

Now let's compute the values:

x' = x * cos(3π/2) - y * sin(3π/2)
= x * 0 - y * (-1)
= y

y' = x * sin(3π/2) + y * cos(3π/2)
= x * (-1) + y * 0
= -x

So, the new point after rotating (x, y) counterclockwise by 270 degrees is (-y, -x).

Answer 3

Apologies for the incorrect response. Let's recalculate the correct rotation.

To rotate a point (x, y) counterclockwise about the origin by 270 degrees, we can use the rotation formula:

x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)

For a counterclockwise rotation of 270 degrees, we have θ = 270° or θ = 3π/2 radians. Plugging in the values:

x' = x * cos(3π/2) - y * sin(3π/2)
y' = x * sin(3π/2) + y * cos(3π/2)

Now let's compute the values:

x' = x * cos(3π/2) - y * sin(3π/2)
= x * 0 - y * (-1)
= y

y' = x * sin(3π/2) + y * cos(3π/2)
= x * (-1) + y * 0
= -x

Therefore, the new point after rotating (x, y) counterclockwise by 270 degrees is (y, -x).