# 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 $AB$.

GeoGebra Applet YF2EDCTt

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 $180^\circ$?

## 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.

GeoGebra Applet Ccv3FucS

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.