Demonstrating the properties of rotations, if a line segment with endpoints (0,-3) and (0,-7) is rotated 90 degrees clockwise, what is the endpoint of this rotated segment?
To understand the properties of rotations, we need to visualize the process.
We have a line segment with endpoints (0,-3) and (0,-7). This line segment is vertical and aligned with the y-axis. To rotate this line segment 90 degrees clockwise, we need to imagine rotating it around the origin (0,0) in a clockwise direction.
First, let's find the midpoint of the line segment. The x-coordinate of the midpoint will be the same as the x-coordinate of the endpoints (0), and the y-coordinate will be the average of the y-coordinates of the endpoints:
Midpoint y-coordinate = (y1 + y2) / 2 = (-3 + -7) / 2 = -10 / 2 = -5
So, the midpoint of the line segment is (0, -5).
Next, we rotate the line segment 90 degrees clockwise around the origin (0,0). This particular rotation swaps the x and y coordinates and changes the sign of the new x-coordinate.
The original line segment is vertical, so after the rotation, it will become horizontal. The midpoint will still be the same (0, -5). Now, we need to find the new endpoint.
The new endpoint will have the same y-coordinate as the midpoint (-5) and the x-coordinate will have the opposite sign of the original y-coordinate of the endpoint (-7). Therefore, the new endpoint after the 90-degree clockwise rotation is:
Endpoint: ( - (-7), -5) = (7, -5)
So, the endpoint of the rotated line segment is (7, -5).