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Lesson 7No Bending or Stretching

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

Learning Targets:

  • I can describe the effects of a rigid transformation on the lengths and angles in a polygon.

7.1 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 is measured to the 5th notch after unit 4 on the ruler.
  2. Find the length of this segment to the nearest 0.1 of a unit.
    A line is measured to the 7th notch after unit 4 on the ruler.
  3. Estimate the length of this segment to the nearest \frac18 of a unit.
    5 units of measurement sre shown. The line being measured goes from 0 to 4
  4. Estimate the length of the segment in the prior question to the nearest 0.1 of a unit.

7.2 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.
    open applet in presentation mode
  2. Rotate Triangle B   90^\circ clockwise using R as the center of rotation. In the image, write the measure of each angle in its interior using the draw tool.
    open applet in presentation mode
  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.
    open applet in presentation mode

7.3 Which One?

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

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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.

Lesson 7 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 Terms

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 .

Corresponding triangles ABC and DEF are shown.
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. 

Lesson 7 Practice Problems

  1. Is there a rigid transformation taking Rhombus P to Rhombus Q? Explain how you know.

    Two parallelograms are shown.

  2. Describe a rigid transformation that takes Triangle A to Triangle B.

    Two right triangles are formed off of a linear function on a graph.

  3. Is there a rigid transformation taking Rectangle A to Rectangle B? Explain how you know.

    Rectangle A and B are shown. They have different side lengths.

  4. For each shape, draw its image after performing the transformation. If you get stuck, consider using tracing paper.

    1. Translate the shape so that A goes to A’ .
    A figure is shown on a grid with Point A'.
    1. Rotate the shape 180 degrees counterclockwise around B .
    A figure is shown on a grid.
    1. Reflect the shape over the line shown.
    A figure is shown on a grid with one point on line l