Demonstrating the properties of rotations, if a line segment with endpoints (0, -3) and (0, -7) is rotated 90° clockwise, what is an endpoint of this rotated segment? (1 point)
O (3, 0)
O(0, 3)
O(-3, 0)
O (0, 7)
To find the endpoint of the rotated segment, we need to apply the properties of rotations.
First, let's find the midpoint of the line segment by taking the average of the x-coordinates and the average of the y-coordinates:
Midpoint = ((0+0)/2, (-3+(-7))/2)
Midpoint = (0, -5)
Next, let's find the distance from the midpoint to one of the endpoints:
Distance = (-7 - (-5))
Distance = -2
Since we are rotating 90° clockwise, the new y-coordinate will be the negative of the distance, and the new x-coordinate will be the same as the midpoint's y-coordinate.
Endpoint = (0, -5) rotated 90° clockwise
Endpoint = (5, 0)
Therefore, the endpoint of the rotated segment is O(5, 0).
are you sure?
that isnt one of the answers
Apologies for the incorrect response. Let's reassess the problem correctly.
To rotate a point in the coordinate plane 90 degrees clockwise, we can use the following transformation:
For a point (x, y), the rotated point (x', y') can be found using the following equations:
x' = y
y' = -x
Let's apply this transformation to the given endpoints (0, -3) and (0, -7):
For the point (0, -3):
x' = -3
y' = 0
For the point (0, -7):
x' = -7
y' = 0
Therefore, the endpoint of the rotated segment is O(-3, 0).
Apologies for the confusion in my previous response. Let's correctly determine the endpoint of the rotated segment.
To rotate a point clockwise about the origin (0,0) by 90 degrees, we can use the following rotation matrix:
[x', y'] = [x*cos(θ) - y*sin(θ), x*sin(θ) + y*cos(θ)]
For a 90 degree clockwise rotation, θ = -90 degrees or -π/2 radians.
Plugging in the coordinates (0, -7) into the rotation matrix, we get:
[x', y'] = [0*cos(-π/2) - (-7)*sin(-π/2), 0*sin(-π/2) + (-7)*cos(-π/2)]
= [0 - (-7*1), 0 + 0]
= [7, 0]
Therefore, the endpoint of the line segment after a 90 degree clockwise rotation would be O(7, 0).