Describe the process of rotating quadrilateral ABCD counterclockwise by 90 degrees centered at point (1,1).
To rotate a quadrilateral counterclockwise by 90 degrees centered at a given point, you can follow these steps:
1. Find the coordinates of the center of rotation: In this case, the center of rotation is given as (1, 1).
2. Translate the coordinates of the vertices of the quadrilateral to have the center of rotation as the origin. This involves subtracting the x-coordinate of the center of rotation from all x-coordinates of the vertices, and subtracting the y-coordinate of the center of rotation from all y-coordinates of the vertices.
3. Apply the rotation matrix for counterclockwise rotation by 90 degrees: A rotation matrix can be used to rotate points in a coordinate plane. For a counterclockwise rotation by 90 degrees, the rotation matrix is:
R = | 0 -1 |
| 1 0 |
Multiply the rotation matrix by the translated coordinates of the vertices. This will give you the new coordinates after the rotation.
4. Translate the coordinates back: Add the x-coordinate of the center of rotation to the x-coordinates of the rotated vertices, and add the y-coordinate of the center of rotation to the y-coordinates of the rotated vertices.
Let's calculate the final coordinates of the rotated quadrilateral ABCD centered at (1, 1):
Given the original coordinates of ABCD:
A(2, 2), B(2, 4), C(5, 4), D(5, 2)
1. Find the center of rotation: The given center of rotation is (1, 1).
2. Translate the coordinates of ABCD to have (1, 1) as the origin:
Translated A: (2 - 1, 2 - 1) = (1, 1)
Translated B: (2 - 1, 4 - 1) = (1, 3)
Translated C: (5 - 1, 4 - 1) = (4, 3)
Translated D: (5 - 1, 2 - 1) = (4, 1)
3. Apply the rotation matrix:
Rotated A: (0 * 1 - (-1) * 1, 1 * 1 + 0 * 1) = (1, 1)
Rotated B: (0 * 1 - (-1) * 3, 1 * 1 + 0 * 3) = (3, 1)
Rotated C: (0 * 4 - (-1) * 3, 1 * 4 + 0 * 3) = (3, 4)
Rotated D: (0 * 4 - (-1) * 1, 1 * 4 + 0 * 1) = (1, 4)
4. Translate the coordinates back:
Final A: (1 + 1, 1 + 1) = (2, 2)
Final B: (3 + 1, 1 + 1) = (4, 2)
Final C: (3 + 1, 4 + 1) = (4, 5)
Final D: (1 + 1, 4 + 1) = (2, 5)
Therefore, the rotated quadrilateral ABCD counterclockwise by 90 degrees centered at (1, 1) has vertices:
A'(2, 2), B'(4, 2), C'(4, 5), D'(2, 5).
To rotate a quadrilateral counterclockwise by 90 degrees, centered at a point (1,1), follow these steps:
1. Plot the coordinates of the vertices of quadrilateral ABCD. Let's assume that the coordinates of the vertices are:
- A(2,3)
- B(4,3)
- C(4,1)
- D(2,1)
2. Calculate the translation factor by subtracting the center point coordinates from each vertex of the quadrilateral. In this case, subtract (1,1) from each vertex:
- A(2,3) - (1,1) = (1,2)
- B(4,3) - (1,1) = (3,2)
- C(4,1) - (1,1) = (3,0)
- D(2,1) - (1,1) = (1,0)
3. Perform a counter-clockwise rotation of 90 degrees on each translated point. Use the following formula for a 90-degree rotation counterclockwise about the origin:
- x' = -y
- y' = x
Apply this formula to each translated point:
- A'(1,2): x' = -2, y' = 1
- B'(3,2): x' = -2, y' = 3
- C'(3,0): x' = 0, y' = 3
- D'(1,0): x' = 0, y' = 1
4. Finally, apply the reverse translation by adding the center coordinates (1,1) to each point.
- A''(1,2) + (1,1) = (2,3)
- B''(3,2) + (1,1) = (4,3)
- C''(3,0) + (1,1) = (4,1)
- D''(1,0) + (1,1) = (2,1)
So, the rotated quadrilateral ABCD counterclockwise by 90 degrees, centered at (1,1), has the following vertices:
A''(2,3), B''(4,3), C''(4,1) and D''(2,1).