1.2.1: No Bending or Stretching

Lesson

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

Exercise \(\PageIndex{1}\): 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 \(\frac{1}{8}\) of a unit.

Figure \(\PageIndex{1}\): A line segment measured by a ruler above it. The ruler has units represented by large tick marks with whole numbers from 1 to 6. Each unit is evenly divided into 8 parts. The segment measures 4 and the fraction 5 over 8 units.

2. Find the length of this segment to the nearest \(0.1\) of a unit.

Figure \(\PageIndex{2}\): A line segment measured by a ruler above it. The ruler has units represented by large tick marks with whole numbers from 1 to 5. Each unit is evenly divided into 10 parts. The segment measures 4 and the fraction 7 over 10 units.

3. Estimate the length of this segment to the nearest \(\frac{1}{8}\) of a unit.

Figure \(\PageIndex{3}\): A line segment measured by a ruler above it. The ruler has units represented by large tick marks with whole numbers from 1 to 5. The segment measures near 3 and the fraction 3 over 4 units.

4. Estimate the length of the segment in the prior question to the nearest \(0.1\) of a unit.

Exercise \(\PageIndex{2}\): Sides and Angles

1. Translate Polygon \(A\) so point \(P\) goes to point \(P'\). In the image, write 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 \(l\).
   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.

Exercise \(\PageIndex{3}\): 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.

Are you ready for more?

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?

Figure \(\PageIndex{4}\)

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.

Figure \(\PageIndex{5}\)

Table \(\PageIndex{1}\)

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

Definition: 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 \(DF\).

Figure \(\PageIndex{6}\)

Definition: 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.

Practice

Exercise \(\PageIndex{4}\)

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

Figure \(\PageIndex{7}\)

Exercise \(\PageIndex{5}\)

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

Figure \(\PageIndex{8}\): Triangle A and its image triangle B on a coordinate plane, origin \(O\). Horizontal and vertical axis scale negative 5 to 5 by 1’s. Triangle A has coordinates (0 comma 0), (3 comma 2) and (3 comma 0). Triangle B has coordinates (3 comma 2), (6 comma 4) and (6 comma 2).

Exercise \(\PageIndex{6}\)

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

Figure \(\PageIndex{9}\):

Exercise \(\PageIndex{7}\)

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 \(A'\).

Figure \(\PageIndex{10}\)

2. Rotate the shape 180 degrees counterclockwise around \(B\).

Figure \(\PageIndex{11}\)

3. Reflect the shape over the line shown.

Figure \(\PageIndex{12}\)

(From Unit 1.1.4)