Lesson 7Rotation Patterns

Learning Goal

Let’s rotate figures in a plane.

Learning Targets

  • I can describe how to move one part of a figure to another using a rigid transformation.

Lesson Terms

  • corresponding
  • rigid transformation

Warm Up: Building a Quadrilateral

Problem 1

  1. Using the right isosceles triangle, perform the following rotations:

    • Rotate triangle degrees clockwise around .

    • Rotate triangle degrees clockwise around .

    • Rotate triangle degrees clockwise around .

    Right isosceles triangle A B C has horizonatl side A B with point A to the right of B, and has vertical side B C with point C directly above point B.

  2. What would it look like when you rotate the four triangles 90 degrees clockwise around ? 180 degrees? 270 degrees clockwise?

Activity 1: Rotating a Segment

Problem 1

Create a segment and a point that is not on segment .

open applet in presentation mode

    • Rotate segment around point .

    • Rotate segment around point .

    Construct the midpoint of segment with the Midpoint tool.

    A grid with line segment CD and midpoint G. There is a point E that is diagonal to point D on the same grid square.

  2. Rotate segment around its midpoint. What is the image of A?

  3. What happens when you rotate a segment ?

Print Version

  1. Rotate segment 180 degrees around point . Draw its image and label the image of as

    Rotate segment 180 degrees around point . Draw its image and label the image of as and the image of as .

    A grid with line segment CD and midpoint G. There is a point E that is diagonal to point D on the same grid square.

  2. Rotate segment 180 degrees around its midpoint, What is the image of ?

  3. What happens when you rotate a segment 180 degrees around a point?

Are you ready for more?

Problem 1

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.

A grid with two line segments - both at a diagonal and one lower and to the left.

Activity 2: A Pattern of Four Triangles

Problem 1

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

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

open applet in presentation mode

  1. Describe a rigid transformation that takes triangle to triangle .

  2. Describe a rigid transformation that takes triangle to triangle .

  3. Describe a rigid transformation that takes triangle to triangle .

  4. Do segments , , , and all have the same length? Explain your reasoning.

Print Version

You can use rigid transformations of a figure to make patterns. Here is a diagram built with three different transformations of triangle .

Square BDFH with square ACGE inside. Four right triangles are formed - ABC, CDE, EFG, GHA.

  1. Describe a rigid transformation that takes triangle to triangle .

  2. Describe a rigid transformation that takes triangle to triangle .

  3. Describe a rigid transformation that takes triangle to triangle .

  4. Do segments , , , and all have the same length? Explain your reasoning.

Lesson 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 shown here has . If we rotate triangle 60 degrees, 120 degrees, 180 degrees, 240 degrees, and 300 degrees clockwise, we can build a hexagon.

Six identical equilateral triangles are drawn such that each triangle is aligned to another triangle created a hexagon. One of the triangle is labeled A B C and all 6 triangles meet at the common point of A.