Math worksheet focused on rotations, including labeling rotation directions, identifying rotation images, and rotating shapes on a coordinate plane.

Problem Analysis and Solution

The worksheet focuses on understanding and applying rotations in geometry. Let's solve each part step by step.

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#### 1. Label the following as either clockwise (CW) or counter-clockwise (CCW)

We are given four curved arrows and need to determine whether they represent clockwise or counter-clockwise rotation.

- First arrow: The arrow curves to the right, indicating a clockwise rotation.
- Second arrow: The arrow curves to the left, indicating a counter-clockwise rotation.
- Third arrow: The arrow curves to the right, indicating a clockwise rotation.
- Fourth arrow: The arrow curves to the right, indicating a clockwise rotation.

Answer:
1. CW
2. CCW
3. CW
4. CW

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#### 2. Which pictures show a rotation?

We need to identify which of the given figures demonstrate a rotation. Rotation means that the shape is turned around a point, but its size and orientation relative to the grid remain consistent.

- a) The figure shows two parallelograms that are mirror images of each other. This is not a rotation; it is a reflection.
- b) The figure shows two L-shaped figures that are not aligned in a way that suggests rotation. This is not a rotation.
- c) The figure shows two identical shapes that appear to be rotated around a central point. This is a rotation.
- d) The figure shows two triangles that are not aligned in a way that suggests rotation. This is not a rotation.

Answer:
- c) shows a rotation.

Explanation: In option c, the shapes are identical and appear to have been rotated around a central point, maintaining their orientation relative to the grid.

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#### 3. Perform the specified rotations

We are given six geometric figures and asked to perform specific rotations about certain points. Let’s solve each one step by step.

##### a) Rotate ¼ turn cw about Point R

- Shape: A quadrilateral with vertices P, Q, R, S.
- Rotation: ¼ turn clockwise (90° clockwise) about point R.
- Steps:
1. Identify point R as the center of rotation.
2. For each vertex (P, Q, S), find its new position after rotating 90° clockwise around R.
- To rotate a point (x, y) 90° clockwise around another point (h, k), use the formula:
\[
(x', y') = (k + (y - k), h - (x - h))
\]
- Apply this formula to each vertex relative to R(7, 6):
- P(2, 8):
\[
x' = 6 + (8 - 6) = 8, \quad y' = 7 - (2 - 7) = 12 \quad \Rightarrow \quad P'(8, 12)
\]
- Q(5, 8):
\[
x' = 6 + (8 - 6) = 8, \quad y' = 7 - (5 - 7) = 9 \quad \Rightarrow \quad Q'(8, 9)
\]
- S(2, 6):
\[
x' = 6 + (6 - 6) = 6, \quad y' = 7 - (2 - 7) = 12 \quad \Rightarrow \quad S'(6, 12)
\]
3. Plot the new vertices P'(8, 12), Q'(8, 9), R(7, 6), S'(6, 12) to complete the rotated shape.

##### b) Rotate ½ turn about Point P

- Shape: A triangle with vertices O, P, N.
- Rotation: ½ turn (180°) about point P.
- Steps:
1. Identify point P as the center of rotation.
2. For each vertex (O, N), find its new position after rotating 180° around P.
- To rotate a point (x, y) 180° around another point (h, k), use the formula:
\[
(x', y') = (2h - x, 2k - y)
\]
- Apply this formula to each vertex relative to P(6, 7):
- O(4, 10):
\[
x' = 2(6) - 4 = 8, \quad y' = 2(7) - 10 = 4 \quad \Rightarrow \quad O'(8, 4)
\]
- N(11, 7):
\[
x' = 2(6) - 11 = 1, \quad y' = 2(7) - 7 = 7 \quad \Rightarrow \quad N'(1, 7)
\]
3. Plot the new vertices O'(8, 4), P(6, 7), N'(1, 7) to complete the rotated shape.

##### c) Rotate ¼ turn ccw about Point C

- Shape: A pentagon with vertices C, D, E, F, G.
- Rotation: ¼ turn counter-clockwise (90° counter-clockwise) about point C.
- Steps:
1. Identify point C as the center of rotation.
2. For each vertex (D, E, F, G), find its new position after rotating 90° counter-clockwise around C.
- To rotate a point (x, y) 90° counter-clockwise around another point (h, k), use the formula:
\[
(x', y') = (k - (y - k), h + (x - h))
\]
- Apply this formula to each vertex relative to C(4, 5):
- D(3, 7):
\[
x' = 5 - (7 - 5) = 3, \quad y' = 4 + (3 - 4) = 3 \quad \Rightarrow \quad D'(3, 3)
\]
- E(6, 8):
\[
x' = 5 - (8 - 5) = 2, \quad y' = 4 + (6 - 4) = 6 \quad \Rightarrow \quad E'(2, 6)
\]
- F(10, 6):
\[
x' = 5 - (6 - 5) = 4, \quad y' = 4 + (10 - 4) = 10 \quad \Rightarrow \quad F'(4, 10)
\]
- G(8, 4):
\[
x' = 5 - (4 - 5) = 6, \quad y' = 4 + (8 - 4) = 8 \quad \Rightarrow \quad G'(6, 8)
\]
3. Plot the new vertices D'(3, 3), E'(2, 6), F'(4, 10), G'(6, 8), C(4, 5) to complete the rotated shape.

##### d) Rotate 90° ccw about Point R

- Shape: A quadrilateral with vertices S, T, U, R.
- Rotation: 90° counter-clockwise about point R.
- Steps:
1. Identify point R as the center of rotation.
2. For each vertex (S, T, U), find its new position after rotating 90° counter-clockwise around R.
- Using the formula for 90° counter-clockwise rotation:
\[
(x', y') = (k - (y - k), h + (x - h))
\]
- Apply this formula to each vertex relative to R(6, 8):
- S(3, 4):
\[
x' = 8 - (4 - 8) = 12, \quad y' = 6 + (3 - 6) = 3 \quad \Rightarrow \quad S'(12, 3)
\]
- T(3, 8):
\[
x' = 8 - (8 - 8) = 8, \quad y' = 6 + (3 - 6) = 3 \quad \Rightarrow \quad T'(8, 3)
\]
- U(5, 10):
\[
x' = 8 - (10 - 8) = 6, \quad y' = 6 + (5 - 6) = 5 \quad \Rightarrow \quad U'(6, 5)
\]
3. Plot the new vertices S'(12, 3), T'(8, 3), U'(6, 5), R(6, 8) to complete the rotated shape.

##### e) Rotate 270° ccw about Point J

- Shape: A triangle with vertices K, L, J.
- Rotation: 270° counter-clockwise about point J.
- Steps:
1. Note that 270° counter-clockwise is equivalent to 90° clockwise.
2. Identify point J as the center of rotation.
3. For each vertex (K, L), find its new position after rotating 90° clockwise around J.
- Using the formula for 90° clockwise rotation:
\[
(x', y') = (k + (y - k), h - (x - h))
\]
- Apply this formula to each vertex relative to J(6, 5):
- K(2, 4):
\[
x' = 5 + (4 - 5) = 4, \quad y' = 6 - (2 - 6) = 10 \quad \Rightarrow \quad K'(4, 10)
\]
- L(5, 8):
\[
x' = 5 + (8 - 5) = 8, \quad y' = 6 - (5 - 6) = 7 \quad \Rightarrow \quad L'(8, 7)
\]
4. Plot the new vertices K'(4, 10), L'(8, 7), J(6, 5) to complete the rotated shape.

##### f) Rotate 180° about Point L

- Shape: A quadrilateral with vertices K, L, M, N.
- Rotation: 180° about point L.
- Steps:
1. Identify point L as the center of rotation.
2. For each vertex (K, M, N), find its new position after rotating 180° around L.
- Using the formula for 180° rotation:
\[
(x', y') = (2h - x, 2k - y)
\]
- Apply this formula to each vertex relative to L(7, 5):
- K(5, 3):
\[
x' = 2(7) - 5 = 9, \quad y' = 2(5) - 3 = 7 \quad \Rightarrow \quad K'(9, 7)
\]
- M(8, 7):
\[
x' = 2(7) - 8 = 6, \quad y' = 2(5) - 7 = 3 \quad \Rightarrow \quad M'(6, 3)
\]
- N(11, 5):
\[
x' = 2(7) - 11 = 3, \quad y' = 2(5) - 5 = 5 \quad \Rightarrow \quad N'(3, 5)
\]
3. Plot the new vertices K'(9, 7), L(7, 5), M'(6, 3), N'(3, 5) to complete the rotated shape.

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Final Answer:

\[
\boxed{
\begin{array}{l}
1. \text{CW, CCW, CW, CW} \\
2. \text{c) shows a rotation.} \\
3. \text{See detailed steps above for each rotation.}
\end{array}
}
\]