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Lesson 8Rotation Patterns

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.

8.1 Building a Quadrilateral

Here is a right isosceles triangle:

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.
  1. Rotate triangle ABC 90 degrees clockwise around B . 
  2. Rotate triangle ABC 180 degrees clockwise around 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 .

open applet in presentation mode

  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 ?

Are you ready for more?

Two parallel line segments are shown on a grid.
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.

open applet in presentation mode
  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.

Lesson 8 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.

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.

Lesson 8 Practice Problems

  1. For the figure shown here,

    Line CMD is shown next to point E.
    1. Rotate segment CD 180^\circ around point D .
    2. Rotate segment CD 180^\circ around point E .
    3. Rotate segment CD 180^\circ around point M .

  2. Here is an isosceles right triangle:

    Draw these three rotations of triangle ABC together.

    1. Rotate triangle ABC 90 degrees clockwise around A .
    2. Rotate triangle ABC 180 degrees around A .
    3. Rotate triangle ABC 270 degrees clockwise around A .
    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.

  3. Each graph shows two polygons ABCD and A’B’C’D’ . In each case, describe a sequence of transformations that takes ABCD to A’B’C’D’ .

    1. Two congruent figures are shown on a graph.
    2. Two congruent figures are shown on a graph.

  4. Lin says that she can map Polygon A to Polygon B using only reflections. Do you agree with Lin? Explain your reasoning.

    Two identical quadrilateral labeled A and B on a square grid are in different orientations and positions. The square grid has 8 horizontal units and 8 vertical units. Starting from the bottom left vertex, polygon A is located 1 unit right and 4 units down from the edges of the square grid. The second vertex is 2 units right and 3 units up from the first vertex. The third vertex is 3 units right and 1 unit up from the first vertex. the fourth vertex is 2 units right from the first vertex. Starting from the bottom vertex polygon B is located 5 units right and 7 units down from the edges of the square grid. The second vertex is 1 unit left and 2 units up from the first vertex. the third vertex is 3 units up from the first vertex. The fourth vertex is 2 units right and 3 units up from the first vertex.