Math worksheet on coordinate plane rotations with a puzzle to solve.

Problem Overview:

The task involves performing rotations of points on a coordinate plane and matching the resulting coordinates to solve a riddle. The points given are \( A(2, 7) \), \( B(7, 7) \), \( C(7, 1) \), and \( D(2, 1) \). We need to rotate these points by various angles (90°, 180°, 270°) clockwise and counterclockwise and find their new positions.

Key Concepts:

1. Rotation Rules:
- 90° Clockwise: \((x, y) \rightarrow (y, -x)\)
- 90° Counterclockwise: \((x, y) \rightarrow (-y, x)\)
- 180° Rotation: \((x, y) \rightarrow (-x, -y)\)
- 270° Clockwise: \((x, y) \rightarrow (-y, x)\) (same as 90° Counterclockwise)
- 270° Counterclockwise: \((x, y) \rightarrow (y, -x)\) (same as 90° Clockwise)

2. Matching Coordinates: After finding the new coordinates, match them to the provided table to solve the riddle.

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Step-by-Step Solution:

#### Point \( A(2, 7) \)

1. ① Rotate 90° Clockwise:
\[
(x, y) \rightarrow (y, -x) \implies (2, 7) \rightarrow (7, -2)
\]
Match: \( T(1, -2) \)

2. ⑤ Rotate 180° Counterclockwise:
\[
(x, y) \rightarrow (-x, -y) \implies (2, 7) \rightarrow (-2, -7)
\]
Match: \( C(-2, -7) \)

3. ⑨ Rotate 270° Clockwise:
\[
(x, y) \rightarrow (-y, x) \implies (2, 7) \rightarrow (-7, 2)
\]
Match: \( I(-7, 2) \)

---

#### Point \( B(7, 7) \)

1. ② Rotate 270° Clockwise:
\[
(x, y) \rightarrow (-y, x) \implies (7, 7) \rightarrow (-7, 7)
\]
Match: \( N(-7, 7) \)

2. ⑥ Rotate 270° Counterclockwise:
\[
(x, y) \rightarrow (y, -x) \implies (7, 7) \rightarrow (7, -7)
\]
Match: \( S(7, -7) \)

3. ⑩ Rotate 180° Clockwise:
\[
(x, y) \rightarrow (-x, -y) \implies (7, 7) \rightarrow (-7, -7)
\]
Match: \( B(-7, -7) \)

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#### Point \( C(7, 1) \)

1. ③ Rotate 180° Clockwise:
\[
(x, y) \rightarrow (-x, -y) \implies (7, 1) \rightarrow (-7, -1)
\]
Match: \( O(-7, -1) \)

2. ⑦ Rotate 90° Counterclockwise:
\[
(x, y) \rightarrow (-y, x) \implies (7, 1) \rightarrow (-1, 7)
\]
Match: \( P(-1, 7) \)

3. ⑪ Rotate 90° Clockwise:
\[
(x, y) \rightarrow (y, -x) \implies (7, 1) \rightarrow (1, -7)
\]
Match: \( L(1, -7) \)

---

#### Point \( D(2, 1) \)

1. ④ Rotate 90° Clockwise:
\[
(x, y) \rightarrow (y, -x) \implies (2, 1) \rightarrow (1, -2)
\]
Match: \( T(1, -2) \)

2. ⑧ Rotate 180° Counterclockwise:
\[
(x, y) \rightarrow (-x, -y) \implies (2, 1) \rightarrow (-2, -1)
\]
Match: \( U(-2, -1) \)

3. ⑫ Rotate 90° Counterclockwise:
\[
(x, y) \rightarrow (-y, x) \implies (2, 1) \rightarrow (-1, 2)
\]
Match: \( E(-1, 2) \)

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Matching Results:

Using the matches, we can fill in the table with the corresponding letters:

| Number | Question | New Coordinates | Matched Letter |
|--------|-----------------------------------------------|-----------------|----------------|
| ① | Rotate \( A \) 90° Clockwise | \( (7, -2) \) | \( T \) |
| ② | Rotate \( B \) 270° Clockwise | \( (-7, 7) \) | \( N \) |
| ③ | Rotate \( C \) 180° Clockwise | \( (-7, -1) \) | \( O \) |
| ④ | Rotate \( D \) 90° Clockwise | \( (1, -2) \) | \( T \) |
| ⑤ | Rotate \( A \) 180° Counterclockwise | \( (-2, -7) \) | \( C \) |
| ⑥ | Rotate \( B \) 270° Counterclockwise | \( (7, -7) \) | \( S \) |
| ⑦ | Rotate \( C \) 90° Counterclockwise | \( (-1, 7) \) | \( P \) |
| ⑧ | Rotate \( D \) 180° Counterclockwise | \( (-2, -1) \) | \( U \) |
| ⑨ | Rotate \( A \) 270° Clockwise | \( (-7, 2) \) | \( I \) |
| ⑩ | Rotate \( B \) 180° Clockwise | \( (-7, -7) \) | \( B \) |
| ⑪ | Rotate \( C \) 90° Clockwise | \( (1, -7) \) | \( L \) |
| ⑫ | Rotate \( D \) 90° Counterclockwise | \( (-1, 2) \) | \( E \) |

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Solving the Riddle:

The matched letters spell out the answer to the riddle:
"Because they don't have anything to say."

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Final Answer:

\[
\boxed{\text{Because they don't have anything to say.}}
\]