Geometric Transformation: Rotation worksheet - Free Printable

Problem Overview:

The task involves determining the coordinates of the vertices of geometric shapes after they are rotated about the origin. The rotations are specified as:
1. 90° clockwise rotation
2. 180° rotation
3. 90° counterclockwise rotation
4. 180° rotation

We will solve each part step by step.

---

Rotation Rules:

1. 90° Clockwise Rotation:
- The rule is: $(x, y) \rightarrow (y, -x)$

2. 180° Rotation:
- The rule is: $(x, y) \rightarrow (-x, -y)$

3. 90° Counterclockwise Rotation:
- The rule is: $(x, y) \rightarrow (-y, x)$

4. 180° Rotation (same as above):
- The rule is: $(x, y) \rightarrow (-x, -y)$

---

Solution for Each Part:

#### 1. 90° Clockwise Rotation
The original coordinates of the vertices are:
- $ A(-3, 2) $
- $ B(3, 4) $
- $ C(1, -2) $

Using the rule $(x, y) \rightarrow (y, -x)$:

- $ A' = (2, 3) $
- $ B' = (4, -3) $
- $ C' = (-2, -1) $

Thus, the coordinates are:
$$
A'(2, 3), \quad B'(4, -3), \quad C'(-2, -1)
$$

#### 2. 180° Rotation
The original coordinates of the vertices are:
- $ U(-2, 4) $
- $ V(4, 3) $
- $ W(-3, 1) $

Using the rule $(x, y) \rightarrow (-x, -y)$:

- $ U' = (2, -4) $
- $ V' = (-4, -3) $
- $ W' = (3, -1) $

Thus, the coordinates are:
$$
U'(2, -4), \quad V'(-4, -3), \quad W'(3, -1)
$$

#### 3. 90° Counterclockwise Rotation
The original coordinates of the vertices are:
- $ E(1, 1) $
- $ F(4, 2) $
- $ G(5, 4) $
- $ H(3, 5) $

Using the rule $(x, y) \rightarrow (-y, x)$:

- $ E' = (-1, 1) $
- $ F' = (-2, 4) $
- $ G' = (-4, 5) $
- $ H' = (-5, 3) $

Thus, the coordinates are:
$$
E'(-1, 1), \quad F'(-2, 4), \quad G'(-4, 5), \quad H'(-5, 3)
$$

#### 4. 180° Rotation
The original coordinates of the vertices are:
- $ P(-2, 1) $
- $ Q(-4, -3) $
- $ R(4, -5) $

Using the rule $(x, y) \rightarrow (-x, -y)$:

- $ P' = (2, -1) $
- $ Q' = (4, 3) $
- $ R' = (-4, 5) $

Thus, the coordinates are:
$$
P'(2, -1), \quad Q'(4, 3), \quad R'(-4, 5)
$$

---

Final Answer:

$$
\boxed{
\begin{aligned}
1.) & \quad A'(2, 3), \, B'(4, -3), \, C'(-2, -1) \\
2.) & \quad U'(2, -4), \, V'(-4, -3), \, W'(3, -1) \\
3.) & \quad E'(-1, 1), \, F'(-2, 4), \, G'(-4, 5), \, H'(-5, 3) \\
4.) & \quad P'(2, -1), \, Q'(4, 3), \, R'(-4, 5)
\end{aligned}
}
$$