Given a point (x, y), what is the new point when the point is rotated counterclockwise about the origin 270 degrees?
Given a point on a geometric figure (1, 2), what is the new point when the point is rotated clockwise about the origin 90 degrees.
To find the new point after rotating (1, 2) clockwise about the origin 90 degrees, we can use the rotation formulas given earlier:
x' = x*cos(θ) + y*sin(θ)
y' = -x*sin(θ) + y*cos(θ)
In this case, since we're rotating clockwise 90 degrees, we have θ = -90 degrees.
Converting θ to radians: -90 degrees * (π/180) radians/degree = -π/2 radians
Substituting the values into the formulas:
x' = 1*cos(-π/2) + 2*sin(-π/2)
= 1*0 + 2*(-1)
= -2
y' = -1*sin(-π/2) + 2*cos(-π/2)
= -1*(-1) + 2*0
= 1
Therefore, the new point after rotating (1, 2) clockwise about the origin 90 degrees is (-2, 1).
Given a point on a line segment (−4, 5), what is the new point when the point is rotated counterclockwise about the origin 270 degrees?
To determine the new point after rotating (-4, 5) counterclockwise about the origin 270 degrees, we will use the rotation formulas mentioned earlier:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, since we are rotating counterclockwise by 270 degrees, θ = 270°.
Converting θ to radians: 270° * (π/180) radians/degree = 3π/2 radians
Substituting the values into the formulas:
x' = (-4)*cos(3π/2) - 5*sin(3π/2)
= (-4)*0 - 5*(-1)
= 5
y' = (-4)*sin(3π/2) + 5*cos(3π/2)
= (-4)*(-1) + 5*0
= 4
Hence, the new point after rotating (-4, 5) counterclockwise about the origin 270 degrees is (5, 4).
To find the new point after rotating (x, y) counterclockwise about the origin 270 degrees, we need to use the following rotation formula:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
In this case, since we're rotating counterclockwise 270 degrees, we have θ = 270 degrees.
Converting θ to radians: 270 degrees * (π/180) radians/degree = 3π/2 radians
Substituting the values into the formulas:
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 about the origin 270 degrees is (-y, -x).