Demonstrating the properties of rotations if a line segment with any points 0, -3 and 0, -7 Is rotating 90 clockwise. What is the end point of this rotated segment?
To rotate a line segment 90 degrees clockwise, we can use the rotation matrix:
```
[x'] [ 0 -1 ] [x]
[y'] = [ 1 0 ]*[y]
```
Given the points (0, -3) and (0, -7), we need to apply the rotation matrix to each point.
For the point (0, -3):
```
[x'] [ 0 -1 ] [0]
[y'] = [ 1 0 ]*[-3]
```
Solving this matrix product, we get:
```
[x'] [ 0 -1 ] [0] [ 0 ]
[y'] = [ 1 0 ]*[-3] = [ 3 ]
```
So, the rotated point is (0, 3).
Similarly, for the point (0, -7), we have:
```
[x'] [ 0 -1 ] [0]
[y'] = [ 1 0 ]*[-7]
```
Solving this matrix product, we get:
```
[x'] [ 0 -1 ] [0] [ 0 ]
[y'] = [ 1 0 ]*[-7] = [ 7 ]
```
So, the rotated point is (0, 7).
Therefore, the end point of the rotated segment is (0, 7).
Rotate a point on a line at. -4, 6 180 counter clockwise, what is the resulting point?
To rotate a point (-4, 6) by 180 degrees counter-clockwise, we can use the rotation matrix:
```
[x'] [ cos(theta) -sin(theta) ] [x]
[y'] = [ sin(theta) cos(theta) ]*[y]
```
Since we want to rotate the point by 180 degrees counter-clockwise, we can substitute theta with pi (π) radians.
```
[x'] [ cos(pi) -sin(pi) ] [-4]
[y'] = [ sin(pi) cos(pi) ]*[6]
```
Simplifying this, we have:
```
[x'] [ -1 0 ] [-4] [ 4 ]
[y'] = [ 0 -1 ]*[6] = [ -6 ]
```
Therefore, the resulting point after rotating (-4, 6) by 180 degrees counter-clockwise is (4, -6).
Jack discovers that the orientation of a triangle is counter. Clockwise, he then reflects the triangle over the y-axis. What is the orientation of the reflected figure?
When a figure is reflected over the y-axis, the orientation of the figure is reversed or flipped horizontally. In other words, the vertices of the figure that were previously on the left side of the y-axis will now be on the right side, and vice versa. The direction of rotation remains the same (clockwise or counterclockwise), but the figure itself becomes a mirror image of its original orientation.
Therefore, if the original triangle had a counterclockwise orientation, reflecting it over the y-axis would result in a triangle with a clockwise orientation.
If a Triangle with vertical's 4, 3 8, 3 and 7, 5 is translated to the right 4 units and down 2 units. What are the verticals of the translated figurer
To translate a figure to the right or left and up or down, we simply add or subtract the same value from each vertex of the figure.
Let's translate the given triangle (with vertices A(4, 3), B(8, 3), and C(7, 5)) to the right 4 units and down 2 units.
For vertex A(4, 3):
- Translate to the right by adding 4: x-coordinate becomes 4 + 4 = 8
- Translate down by subtracting 2: y-coordinate becomes 3 - 2 = 1
So, the translated vertex A' becomes (8, 1).
For vertex B(8, 3):
- Translate to the right by adding 4: x-coordinate becomes 8 + 4 = 12
- Translate down by subtracting 2: y-coordinate becomes 3 - 2 = 1
So, the translated vertex B' becomes (12, 1).
For vertex C(7, 5):
- Translate to the right by adding 4: x-coordinate becomes 7 + 4 = 11
- Translate down by subtracting 2: y-coordinate becomes 5 - 2 = 3
So, the translated vertex C' becomes (11, 3).
Therefore, the vertices of the translated figure are A'(8, 1), B'(12, 1), and C'(11, 3).