Demonstrating the properties of rotations, if a line segment with endpoints (0,-3) and (0,-7) is rotated 90 degrees clockwise, what is an endpoint of this rotated segment?
To rotate a point 90 degrees clockwise, we can use the following formula:
(x', y') = (xcosθ + ysinθ, -xsinθ + ycosθ)
Where (x, y) represents the original point, (x', y') represents the rotated point, and θ represents the angle of rotation in radians.
Let's apply this formula to the given line segment:
Endpoint 1: (0, -3)
Endpoint 2: (0, -7)
Since both endpoints have the same x-coordinate, we only need to rotate the y-coordinate.
For Endpoint 1:
x = 0, y = -3
θ = 90 degrees = π/2 radians
Using the formula:
x' = x*cosθ + y*sinθ
y' = -x*sinθ + y*cosθ
Calculating the rotated coordinates:
x' = 0*cos(π/2) + (-3)*sin(π/2)
= 0 + (-3)
= -3
y' = -0*sin(π/2) + (-3)*cos(π/2)
= 0 + 0
= 0
Thus, Endpoint 1 after a 90-degree clockwise rotation is (-3, 0).
Using the same calculation method for Endpoint 2:
For Endpoint 2:
x = 0, y = -7
θ = 90 degrees = π/2 radians
Calculating the rotated coordinates:
x' = 0*cos(π/2) + (-7)*sin(π/2)
= 0 + (-7)
= -7
y' = -0*sin(π/2) + (-7)*cos(π/2)
= 0 + 0
= 0
Thus, Endpoint 2 after a 90-degree clockwise rotation is (-7, 0).
Therefore, an endpoint of the rotated line segment is (-7, 0).