Rotations worksheet - Free Printable

Problem Overview:

The task involves rotating a triangle labeled \( \triangle TRY \) around the origin using specific rules of rotation. The coordinates of the vertices \( T, R, \) and \( Y \) are given in the image, and we need to apply the rotation rules to find the new coordinates after each rotation.

#### Step 1: Identify the Original Coordinates
From the image:
- \( T = (-4, 2) \)
- \( R = (-6, 7) \)
- \( Y = (-2, 5) \)

#### Step 2: Apply the Rotation Rules
The rotation rules provided are:
1. 90° CW or 270° CCW: \( (x, y) \rightarrow (y, -x) \)
2. 180° CW or 180° CCW: \( (x, y) \rightarrow (-x, -y) \)
3. 90° CCW or 270° CW: \( (x, y) \rightarrow (-y, x) \)

We will use these rules to solve each part of the problem.

---

Part 1: Rotate \( \triangle TRY \) 90° CW from the origin

#### Rotation Rule: \( (x, y) \rightarrow (y, -x) \)

- Vertex \( T(-4, 2) \):
\[
T' = (2, -(-4)) = (2, 4)
\]

- Vertex \( R(-6, 7) \):
\[
R' = (7, -(-6)) = (7, 6)
\]

- Vertex \( Y(-2, 5) \):
\[
Y' = (5, -(-2)) = (5, 2)
\]

#### Final Coordinates:
- \( T' = (2, 4) \)
- \( R' = (7, 6) \)
- \( Y' = (5, 2) \)

---

Part 2: Rotate \( \triangle TRY \) 90° CCW from the origin

#### Rotation Rule: \( (x, y) \rightarrow (-y, x) \)

- Vertex \( T(-4, 2) \):
\[
T' = (-2, -4)
\]

- Vertex \( R(-6, 7) \):
\[
R' = (-7, -6)
\]

- Vertex \( Y(-2, 5) \):
\[
Y' = (-5, -2)
\]

#### Final Coordinates:
- \( T' = (-2, -4) \)
- \( R' = (-7, -6) \)
- \( Y' = (-5, -2) \)

---

Part 3: Rotate \( \triangle TRY \) 180° CW from the origin

#### Rotation Rule: \( (x, y) \rightarrow (-x, -y) \)

- Vertex \( T(-4, 2) \):
\[
T' = (-(-4), -(2)) = (4, -2)
\]

- Vertex \( R(-6, 7) \):
\[
R' = (-(-6), -(7)) = (6, -7)
\]

- Vertex \( Y(-2, 5) \):
\[
Y' = (-(-2), -(5)) = (2, -5)
\]

#### Final Coordinates:
- \( T' = (4, -2) \)
- \( R' = (6, -7) \)
- \( Y' = (2, -5) \)

---

Summary of Results

1. 90° CW Rotation:
- \( T' = (2, 4) \)
- \( R' = (7, 6) \)
- \( Y' = (5, 2) \)

2. 90° CCW Rotation:
- \( T' = (-2, -4) \)
- \( R' = (-7, -6) \)
- \( Y' = (-5, -2) \)

3. 180° CW Rotation:
- \( T' = (4, -2) \)
- \( R' = (6, -7) \)
- \( Y' = (2, -5) \)

---

Final Answer

\[
\boxed{
\begin{array}{ll}
\text{1. 90° CW:} & T'(2, 4), R'(7, 6), Y'(5, 2) \\
\text{2. 90° CCW:} & T'(-2, -4), R'(-7, -6), Y'(-5, -2) \\
\text{3. 180° CW:} & T'(4, -2), R'(6, -7), Y'(2, -5)
\end{array}
}
\]