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.(-3, 0)
2. (4, -6)
3. The reflected figure has a clockwise orientation.
4. (8, 1), (12, 1), (11, 3)
5. translation and rotation.
those are the right answers!!
for rotations quick check for connexus students.<3
Vex is right, just question 3 isnt. 4/5 isnt bad though!
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?(
To find the endpoint of a line segment after rotation, we can follow these steps:
1. Determine the midpoint of the line segment.
- The midpoint is found by averaging the x-coordinates and the y-coordinates of the endpoints.
- In this case, the x-coordinates are both 0, and the y-coordinates are -3 and -7. So the midpoint is (0, (-3 + -7)/2) = (0, -5).
2. Construct a vector from the midpoint to one of the endpoints.
- In this case, let's choose the vector from the midpoint (0, -5) to the endpoint (0, -7).
- The vector is obtained by subtracting the coordinates of the midpoint from the coordinates of the endpoint: (0 - 0, -7 - (-5)) = (0, -2).
3. Apply a clockwise rotation of 90° to the vector.
- To rotate a vector clockwise by 90°, we swap the x and y coordinates and change the sign of the new x-coordinate.
- Applying this rotation to the vector (0, -2), we obtain: (2, 0).
4. Add the rotated vector to the midpoint to get the endpoint after rotation.
- Add the rotated vector (2, 0) to the midpoint (0, -5): (0+2, -5+0) = (2, -5).
Therefore, the endpoint of the rotated line segment is (2, -5).
The endpoint of the rotated segment would be (-7, 0).
To rotate a line segment 90° clockwise, we can use the following steps:
Step 1: Determine the center of rotation.
Step 2: Calculate the distance of each endpoint from the center of rotation.
Step 3: Apply the rotation formula to each endpoint.
Step 4: Find the new coordinates of the rotated endpoint.
Let's solve this step-by-step:
Step 1: Determine the center of rotation.
Since we are not given any specific center of rotation, we can assume the origin (0,0) as the center of rotation.
Step 2: Calculate the distance of each endpoint from the center of rotation.
The first endpoint (0, -3) is 3 units away from the center of rotation.
The second endpoint (0, -7) is 7 units away from the center of rotation.
Step 3: Apply the rotation formula to each endpoint.
The rotation formula for a 90° clockwise rotation is:
x' = x * cosθ - y * sinθ
y' = x * sinθ + y * cosθ
In this case, since we are rotating 90° clockwise, θ (theta) will be -90°.
For the first endpoint:
x' = (0) * cos(-90°) - (-3) * sin(-90°)
= 0 - (-3) * 1
= 0 + 3
= 3
y' = (0) * sin(-90°) + (-3) * cos(-90°)
= 0 + (-3) * 0
= 0
For the second endpoint:
x' = (0) * cos(-90°) - (-7) * sin(-90°)
= 0 - (-7) * 1
= 0 + 7
= 7
y' = (0) * sin(-90°) + (-7) * cos(-90°)
= 0 + (-7) * 0
= 0
Step 4: Find the new coordinates of the rotated endpoint.
The new coordinates of the rotated endpoints are:
For the first endpoint: (3, 0)
For the second endpoint: (7, 0)
Therefore, the endpoint of the rotated segment is (7, 0).