# Lesson 7: No Bending or Stretching

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

## 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.
2. Find the length of this segment to the nearest 0.1 of a unit.
3. Estimate the length of this segment to the nearest $\frac18$ of a unit.
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.

GeoGebra Applet VVZzNwgg

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.

GeoGebra Applet FnsBW2Rq

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.

GeoGebra Applet thAERnHY

## 7.3: Which One?

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

GeoGebra Applet ZgZrzwkN

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.

## 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.

measurements in triangle $ABC$ corresponding measurements in image $EFD$
row 1 $AB = 2.24$ $EF = 2.24$
row 2 $BC = 2.83$ $FD = 2.83$
row 3 $CA = 3.00$ $DE = 3.00$
row 4 $m\angle ABC = 71.6^\circ$ $m\angle EFD= 71.6^\circ$
row 5 $m\angle BCA = 45.0^\circ$ $m\angle FDE= 45.0^\circ$
row 6 $m\angle CAB = 63.4^\circ$ $m\angle DEF= 63.4^\circ$

## Glossary

corresponding

#### corresponding

If a part of the original figure matches up with a part of the copy, we call them corresponding parts. The part could be an angle, point, or side, and you can have corresponding angles, corresponding points, or corresponding sides.

If you have a distance between two points in the original figure, then the distance between the corresponding points in the copy is called the corresponding distance.

rigid transformation

#### rigid transformation

A rigid transformation is a sequence of translations, rotations, and reflections. If a rigid transformation is applied to a geometric figure, the resulting figure is called the image of the original figure under the transformation.

The diagram shows a rigid transformation consisting of a translation (from A to B) followed by a rotation (from B to C) followed by a reflection (from C to D). The last triangle is the image of the first triangle under this rigid transformation.