Question
Lesson 10 transformation rotation quick check
Answers
GPT-4o mini
Certainly! Here’s a quick check covering key concepts related to transformations, specifically focusing on rotation.
### Quick Check: Transformation - Rotation
1. **Definition of Rotation**:
- What is a rotation in geometric transformations?
2. **Center of Rotation**:
- Identify the center of rotation in a geometric figure. If a shape is rotated 90° around the origin, what remains unchanged?
3. **Angle of Rotation**:
- If a triangle with vertices A(1, 2), B(3, 2), and C(2, 4) is rotated 90° counterclockwise around the origin, what are the new coordinates of the vertices A', B', and C'?
4. **Counterclockwise vs. Clockwise**:
- Describe the difference between a counterclockwise rotation and a clockwise rotation.
5. **Matrix Representation**:
- Write the rotation matrix for a 180° rotation around the origin.
6. **Effects of Rotation**:
- How does rotation affect the size and shape of a geometric figure?
7. **Real-world Application**:
- Provide an example where rotation is used in real-life situations (e.g., in machinery or design).
8. **Finding the Angle**:
- If a point P(3, 4) is rotated to point P'(-4, 3) around the origin, what is the angle of rotation?
### Answers (for self-check):
1. A rotation is a transformation that turns a figure around a fixed point, called the center of rotation, by a specific angle in a specified direction.
2. The center of rotation is the point around which the shape is rotated. A shape rotated around the origin retains its location relative to the origin.
3. After applying a 90° counterclockwise rotation to the triangle's vertices:
- A'( -2, 1),
- B' ( -2, 3),
- C' ( -4, 2).
4. A counterclockwise rotation turns the figure in the opposite direction of a clock's hands, while a clockwise rotation turns it in the same direction as a clock's hands.
5. The rotation matrix for a 180° rotation is:
\[
R(180°) = \begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
\]
6. Rotation does not affect the size or shape of a geometric figure; it only changes its orientation.
7. An example of rotation in real-life is the motion of the blades of a helicopter, which rotate around a fixed point.
8. To find the angle of rotation, observe the coordinates - a transformation from P(3, 4) to P'(-4, 3) involves a 90° counterclockwise rotation.
Feel free to modify or expand on any of these points as needed for your lesson!
### Quick Check: Transformation - Rotation
1. **Definition of Rotation**:
- What is a rotation in geometric transformations?
2. **Center of Rotation**:
- Identify the center of rotation in a geometric figure. If a shape is rotated 90° around the origin, what remains unchanged?
3. **Angle of Rotation**:
- If a triangle with vertices A(1, 2), B(3, 2), and C(2, 4) is rotated 90° counterclockwise around the origin, what are the new coordinates of the vertices A', B', and C'?
4. **Counterclockwise vs. Clockwise**:
- Describe the difference between a counterclockwise rotation and a clockwise rotation.
5. **Matrix Representation**:
- Write the rotation matrix for a 180° rotation around the origin.
6. **Effects of Rotation**:
- How does rotation affect the size and shape of a geometric figure?
7. **Real-world Application**:
- Provide an example where rotation is used in real-life situations (e.g., in machinery or design).
8. **Finding the Angle**:
- If a point P(3, 4) is rotated to point P'(-4, 3) around the origin, what is the angle of rotation?
### Answers (for self-check):
1. A rotation is a transformation that turns a figure around a fixed point, called the center of rotation, by a specific angle in a specified direction.
2. The center of rotation is the point around which the shape is rotated. A shape rotated around the origin retains its location relative to the origin.
3. After applying a 90° counterclockwise rotation to the triangle's vertices:
- A'( -2, 1),
- B' ( -2, 3),
- C' ( -4, 2).
4. A counterclockwise rotation turns the figure in the opposite direction of a clock's hands, while a clockwise rotation turns it in the same direction as a clock's hands.
5. The rotation matrix for a 180° rotation is:
\[
R(180°) = \begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix}
\]
6. Rotation does not affect the size or shape of a geometric figure; it only changes its orientation.
7. An example of rotation in real-life is the motion of the blades of a helicopter, which rotate around a fixed point.
8. To find the angle of rotation, observe the coordinates - a transformation from P(3, 4) to P'(-4, 3) involves a 90° counterclockwise rotation.
Feel free to modify or expand on any of these points as needed for your lesson!