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)
1.(-3,0)
2.(3,0)
3.(0,3)
4.(0,7)
To rotate a point 90° clockwise around the origin, we can use the following rotation formula:
(x', y') = (xcosθ - ysinθ, xsinθ + ycosθ)
where (x, y) are the original coordinates, (x', y') are the coordinates after rotation, and θ is the angle of rotation (in radians).
In this case, we want to rotate the line segment with endpoints (0,-3) and (0,-7) 90° clockwise. Since the x-coordinate remains the same, we only need to calculate the new y-coordinate.
For the first endpoint:
(x', y') = (0*cos(90°) - (-3)*sin(90°), 0*sin(90°) + (-3)*cos(90°))
= (0 + 3, 0 - 0)
= (3, 0)
Therefore, the endpoint of the rotated segment is (3, 0).
So the correct answer is: 2. (3, 0)
To determine the endpoint of the line segment after rotating it 90° clockwise, we can follow these steps:
1. Identify the midpoint of the line segment. Given the endpoints (0, -3) and (0, -7), the midpoint can be found by taking the average of the x-coordinates and the y-coordinates. So, the midpoint is (0, (-3 + -7)/2) = (0, -5).
2. Draw a vector from the midpoint to one of the endpoints. In this case, let's choose the point (0, -3). The vector from the midpoint to this endpoint is (-3, -3) - (0, -5) = (-3, 2).
3. Rotate this vector 90° clockwise. To rotate a vector 90° clockwise, we can swap the x and y coordinates and negate the new y-coordinate. So, (-3, 2) becomes (2, 3).
4. Add the rotated vector to the midpoint. The endpoint of the rotated line segment is found by adding the rotated vector (2, 3) to the midpoint (0, -5). (0, -5) + (2, 3) = (0+2, -5+3) = (2, -2).
Therefore, the endpoint of the line segment after rotating it 90° clockwise is (2, -2).
Answer: (2, -2)