Problem Analysis:
The task involves performing rotations of points on a coordinate plane and determining the new coordinates after each rotation. The rotations are specified as clockwise or counterclockwise by specific angles (e.g., 90°, 180°, 270°). We will solve each part step by step using the rules for rotating points around the origin.
#### Rotation Rules:
1.
90° Clockwise Rotation:
\[
(x, y) \rightarrow (y, -x)
\]
2.
180° Rotation:
\[
(x, y) \rightarrow (-x, -y)
\]
3.
270° Clockwise Rotation (or 90° Counterclockwise):
\[
(x, y) \rightarrow (-y, x)
\]
4.
360° Rotation:
\[
(x, y) \rightarrow (x, y) \quad \text{(no change)}
\]
Given Points:
- \( A(2, 7) \)
- \( B(7, 7) \)
- \( C(7, 2) \)
- \( D(2, 1) \)
Solution to Each Part:
#### 1. Find \( A' \) after \( A \) is rotated 90° clockwise.
- Original point: \( A(2, 7) \)
- Rotation rule for 90° clockwise: \( (x, y) \rightarrow (y, -x) \)
- Applying the rule:
\[
A'(7, -2)
\]
-
Answer: \( (7, -2) \)
#### 2. Find \( B' \) after \( B \) is rotated 270° clockwise.
- Original point: \( B(7, 7) \)
- Rotation rule for 270° clockwise: \( (x, y) \rightarrow (-y, x) \)
- Applying the rule:
\[
B'(-7, 7)
\]
-
Answer: \( (-7, 7) \)
#### 3. Find \( C' \) after \( C \) is rotated 180° clockwise.
- Original point: \( C(7, 2) \)
- Rotation rule for 180°: \( (x, y) \rightarrow (-x, -y) \)
- Applying the rule:
\[
C'(-7, -2)
\]
-
Answer: \( (-7, -2) \)
#### 4. Find \( D' \) after \( D \) is rotated 90° clockwise.
- Original point: \( D(2, 1) \)
- Rotation rule for 90° clockwise: \( (x, y) \rightarrow (y, -x) \)
- Applying the rule:
\[
D'(1, -2)
\]
-
Answer: \( (1, -2) \)
#### 5. Find \( A' \) after \( A \) is rotated 360° counterclockwise.
- Original point: \( A(2, 7) \)
- Rotation rule for 360°: \( (x, y) \rightarrow (x, y) \) (no change)
- Applying the rule:
\[
A'(2, 7)
\]
-
Answer: \( (2, 7) \)
#### 6. Find \( B' \) after \( B \) is rotated 270° counterclockwise.
- Original point: \( B(7, 7) \)
- Note: 270° counterclockwise is equivalent to 90° clockwise.
- Rotation rule for 90° clockwise: \( (x, y) \rightarrow (y, -x) \)
- Applying the rule:
\[
B'(7, -7)
\]
-
Answer: \( (7, -7) \)
Final Answers:
\[
\boxed{
\begin{array}{ll}
1. & (7, -2) \\
2. & (-7, 7) \\
3. & (-7, -2) \\
4. & (1, -2) \\
5. & (2, 7) \\
6. & (7, -7) \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of rotations in the coordinate plane worksheet.
