# Lesson 8

Rotation Patterns

Let’s rotate figures in a plane.

### 8.1: Building a Quadrilateral

Here is a right isosceles triangle:

1. Rotate triangle $$ABC$$ 90 degrees clockwise around $$B$$.
2. Rotate triangle $$ABC$$ 180 degrees clockwise round $$B$$.
3. Rotate triangle $$ABC$$ 270 degrees clockwise around $$B$$.
4. What would it look like when you rotate the four triangles 90 degrees clockwise around $$B$$? 180 degrees? 270 degrees clockwise?

### 8.2: Rotating a Segment

Create a segment $$AB$$ and a point $$C$$ that is not on segment $$AB$$.

1. Rotate segment $$AB$$ $$180^\circ$$ around point $$B$$

2. Rotate segment $$AB$$ $$180^\circ$$ around point $$C$$

Construct the midpoint of segment $$AB$$ with the Midpoint tool.

1. Rotate segment $$AB$$ $$180^\circ$$ around its midpoint. What is the image of A?

2. What happens when you rotate a segment $$180^\circ$$?

Here are two line segments. Is it possible to rotate one line segment to the other? If so, find the center of such a rotation. If not, explain why not.

### 8.3: A Pattern of Four Triangles

Here is a diagram built with three different rigid transformations of triangle $$ABC$$.

Use the applet to answer the questions. It may be helpful to reset the image after each question.

1. Describe a rigid transformation that takes triangle $$ABC$$ to triangle $$CDE$$.
2. Describe a rigid transformation that takes triangle $$ABC$$ to triangle $$EFG$$.
3. Describe a rigid transformation that takes triangle $$ABC$$ to triangle $$GHA$$.
4. Do segments $$AC$$, $$CE$$, $$EG$$, and $$GA$$ all have the same length? Explain your reasoning.

### Summary

When we apply a 180-degree rotation to a line segment, there are several possible outcomes:

• The segment maps to itself (if the center of rotation is the midpoint of the segment).
• The image of the segment overlaps with the segment and lies on the same line (if the center of rotation is a point on the segment).
• The image of the segment does not overlap with the segment (if the center of rotation is not on the segment).

We can also build patterns by rotating a shape. For example, triangle $$ABC$$ shown here has $$m(\angle A) = 60$$. If we rotate triangle $$ABC$$ 60 degrees, 120 degrees, 180 degrees, 240 degrees, and 300 degrees clockwise, we can build a hexagon.

For example, point $$B$$ in the first triangle corresponds to point $$E$$ in the second triangle. Segment $$AC$$ corresponds to segment $$DF$$.