Lesson 6

No Bending or Stretching

Let’s compare measurements before and after translations, rotations, and reflections.

6.1: Measuring Segments

For each question, the unit is represented by the large tick marks with whole numbers.

  1. Find the length of this segment to the nearest \(\frac18\) of a unit.
    A line segment measured by a ruler.
  2. Find the length of this segment to the nearest 0.1 of a unit.
    A line segment measured by a ruler.
  3. Estimate the length of this segment to the nearest \(\frac18\) of a unit.
    A line segment measured by a ruler.
  4. Estimate the length of the segment in the prior question to the nearest 0.1 of a unit.

6.2: Sides and Angles

  1. Translate Polygon \(A\) so point \(P\) goes to point \(P'\). In the image, write in the length of each side, in grid units, next to the side using the draw tool.

     
  2. Rotate Triangle \(B\) 90 degrees clockwise using \(R\) as the center of rotation. In the image, write the measure of each angle in its interior using the draw tool.

     
  3. Reflect Pentagon \(C\) across line \(\ell\).
    1. In the image, write the length of each side, in grid units, next to the side.
    2. In the image, write the measure of each angle in the interior.

     

6.3: Which One?

Here is a grid showing triangle \(ABC\) and two other triangles.

You can use a rigid transformation to take triangle \(ABC\) to one of the other triangles.

  1. Which one? Explain how you know.

     

  2. Describe a rigid transformation that takes \(ABC\) to the triangle you selected.



A square is made up of an L-shaped region and three transformations of the region. If the perimeter of the square is 40 units, what is the perimeter of each L-shaped region?

A square formed from a L-shaped figure and three transformations of the figure.

 

Summary

The transformations we’ve learned about so far, translations, rotations, reflections, and sequences of these motions, are all examples of rigid transformations. A rigid transformation is a move that doesn’t change measurements on any figure.

Earlier, we learned that a figure and its image have corresponding points. With a rigid transformation, figures like polygons also have corresponding sides and corresponding angles. These corresponding parts have the same measurements.

For example, triangle \(EFD\) was made by reflecting triangle \(ABC\) across a horizontal line, then translating. Corresponding sides have the same lengths, and corresponding angles have the same measures.

Triangle A, B, C and its image after reflection and translation.
measurements in triangle \(ABC\) corresponding measurements in image \(EFD\)
\(AB = 2.24\) \(EF = 2.24\)
\(BC = 2.83\) \(FD = 2.83\)
\(CA = 3.00\) \(DE = 3.00\)
\(m\angle ABC = 71.6^\circ\) \(m\angle EFD= 71.6^\circ\)
\(m\angle BCA = 45.0^\circ\) \(m\angle FDE= 45.0^\circ\)
\(m\angle CAB = 63.4^\circ\) \(m\angle DEF= 63.4^\circ\)

Glossary Entries

  • corresponding

    When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.

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

    2 triangles with corresponding parts
  • rigid transformation

    A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations, as is any sequence of these.