Worksheet on describing rotations of geometric shapes on a coordinate plane.

Problem Analysis and Solution

The provided images contain a worksheet titled "Describing Rotations." The task involves identifying and describing the rotations that map one shape onto another. Below, I will solve each problem step by step.

---

#### Page 1: Describing Rotations

##### Problem 1
For each of the pairs of shapes below, describe the single transformation that maps the first shape onto the second shape.

1. A to B
- Observation: Shape A is rotated clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

2. B to C
- Observation: Shape B is rotated counterclockwise around its center.
- Solution: The rotation is 90° counterclockwise about the center of the shape.

3. C to D
- Observation: Shape C is rotated 180° around its center.
- Solution: The rotation is 180° about the center of the shape.

4. D to E
- Observation: Shape D is rotated counterclockwise around its center.
- Solution: The rotation is 90° counterclockwise about the center of the shape.

5. E to F
- Observation: Shape E is rotated clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

6. F to A
- Observation: Shape F is rotated counterclockwise around its center.
- Solution: The rotation is 90° counterclockwise about the center of the shape.

---

#### Page 2: Describing Rotations

##### Problem 2
Describe the single transformation that maps:
1. Shape 1 onto shape 2
- Observation: Shape 1 is rotated 90° clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

2. Shape 2 onto shape 3
- Observation: Shape 2 is rotated 90° clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

3. Shape 3 onto shape 4
- Observation: Shape 3 is rotated 90° clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

4. Shape 4 onto shape 5
- Observation: Shape 4 is rotated 90° clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

5. Shape 5 onto shape 6
- Observation: Shape 5 is rotated 90° clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

6. Shape 6 onto shape 7
- Observation: Shape 6 is rotated 90° clockwise around its center.
- Solution: The rotation is 90° clockwise about the center of the shape.

---

##### Problem 3
The triangle has vertices at (1, 1), (2, 1), and (1, 2). The triangle is rotated 90° clockwise such that the base of the triangle remains horizontal. Find the center of rotation.

1. Identify the Original Triangle:
- Vertices: \( A(1, 1) \), \( B(2, 1) \), \( C(1, 2) \).

2. Rotation Requirement:
- The triangle is rotated 90° clockwise.
- After rotation, the base of the triangle must remain horizontal.

3. Determine the Center of Rotation:
- For a 90° clockwise rotation, the center of rotation is typically the midpoint of the line segment connecting a vertex to its image after rotation.
- Let the center of rotation be \( O(h, k) \).

4. Rotation Formula:
- If a point \( (x, y) \) is rotated 90° clockwise around \( (h, k) \), its new coordinates are:
\[
(x', y') = (k + (y - k), h - (x - h)) = (k + y - k, h - x + h) = (y, 2h - x)
\]

5. Apply the Rotation to One Vertex:
- Consider vertex \( A(1, 1) \).
- After rotation, the new coordinates of \( A \) should be \( (1, 2) \) (since the base remains horizontal).
- Using the rotation formula:
\[
(1, 2) = (1, 2h - 1)
\]
- From \( y' = 2 \):
\[
2 = 2h - 1 \implies 2h = 3 \implies h = \frac{3}{2}
\]

6. Verify with Another Vertex:
- Consider vertex \( B(2, 1) \).
- After rotation, the new coordinates of \( B \) should be \( (2, 1) \).
- Using the rotation formula:
\[
(2, 1) = (1, 2h - 2)
\]
- From \( y' = 1 \):
\[
1 = 2h - 2 \implies 2h = 3 \implies h = \frac{3}{2}
\]

7. Conclusion:
- The center of rotation is \( \left( \frac{3}{2}, \frac{3}{2} \right) \).

Final Answer for Problem 3:
\[
\boxed{\left( \frac{3}{2}, \frac{3}{2} \right)}
\]

---

##### Problem 4
The triangle is rotated 180° to rotate the triangle such that the base of the triangle remains horizontal. Find the center of rotation.

1. Identify the Original Triangle:
- Vertices: \( A(1, 1) \), \( B(2, 1) \), \( C(1, 2) \).

2. Rotation Requirement:
- The triangle is rotated 180°.
- After rotation, the base of the triangle must remain horizontal.

3. Determine the Center of Rotation:
- For a 180° rotation, the center of rotation is the midpoint of the line segment connecting a vertex to its image after rotation.
- Let the center of rotation be \( O(h, k) \).

4. Rotation Formula:
- If a point \( (x, y) \) is rotated 180° around \( (h, k) \), its new coordinates are:
\[
(x', y') = (2h - x, 2k - y)
\]

5. Apply the Rotation to One Vertex:
- Consider vertex \( A(1, 1) \).
- After rotation, the new coordinates of \( A \) should be \( (2, 2) \) (since the base remains horizontal).
- Using the rotation formula:
\[
(2, 2) = (2h - 1, 2k - 1)
\]
- From \( x' = 2 \):
\[
2 = 2h - 1 \implies 2h = 3 \implies h = \frac{3}{2}
\]
- From \( y' = 2 \):
\[
2 = 2k - 1 \implies 2k = 3 \implies k = \frac{3}{2}
\]

6. Verify with Another Vertex:
- Consider vertex \( B(2, 1) \).
- After rotation, the new coordinates of \( B \) should be \( (1, 2) \).
- Using the rotation formula:
\[
(1, 2) = (2h - 2, 2k - 1)
\]
- From \( x' = 1 \):
\[
1 = 2h - 2 \implies 2h = 3 \implies h = \frac{3}{2}
\]
- From \( y' = 2 \):
\[
2 = 2k - 1 \implies 2k = 3 \implies k = \frac{3}{2}
\]

7. Conclusion:
- The center of rotation is \( \left( \frac{3}{2}, \frac{3}{2} \right) \).

Final Answer for Problem 4:
\[
\boxed{\left( \frac{3}{2}, \frac{3}{2} \right)}
\]

---

Final Summary of Answers

1. Problem 1:
- A to B: 90° clockwise
- B to C: 90° counterclockwise
- C to D: 180°
- D to E: 90° counterclockwise
- E to F: 90° clockwise
- F to A: 90° counterclockwise

2. Problem 2:
- All rotations: 90° clockwise

3. Problem 3:
- Center of rotation: \(\boxed{\left( \frac{3}{2}, \frac{3}{2} \right)}\)

4. Problem 4:
- Center of rotation: \(\boxed{\left( \frac{3}{2}, \frac{3}{2} \right)}\)