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Properties of translations

Experimentally verify the effect of geometric translations on segment length, angle measure, and parallel lines.
When you translate something in geometry, you're simply moving it around. You don't distort it in any way. If you translate a segment, it remains a segment, and its length doesn't change. Similarly, if you translate an angle, the measure of the angle doesn't change.
These properties may seem obvious, but they're important to keep in mind later on when we do proofs. To make sure we understand these properties, let's walk through a few examples.

Property 1: Line segments are taken to line segments of the same length.

Each square in the grid is 1 unit long.
Translate line segment start overline, S, T, end overline by open angle, 2, comma, minus, 7, close angle.
What is the length of the pre-image—the segment before the translation?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units
What is the length of the image—the segment after the translation?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units

As you can see for yourself, the pre-image and the image are both line segments with the same length. This is true for any line segment that goes under any translation.

Property 2: Angles are taken to angles of the same measure.

Translate angle, M, N, P by open angle, minus, 5, comma, minus, 6, close angle.
Has the measure of the angle changed after the translation?
Choose 1 answer:
Choose 1 answer:

As you can see for yourself, the pre-image angle and the image angle have the same measure. This is true for any angle that goes under any translation.

Property 3: Lines are taken to lines, and parallel lines are taken to parallel lines.

Translate the parallel lines F, G, with, \overleftrightarrow, on top and H, J, with, \overleftrightarrow, on top by open angle, minus, 4, comma, 3, close angle.
Are the two image lines parallel?
Choose 1 answer:
Choose 1 answer:

As you can see for yourself, each line is taken to another line, and the image lines remain parallel to each other. This is true for any line—or lines—that go under any translation.

Conclusion

We found that translations have the following three properties:
  • line segments are taken to line segments of the same length;
  • angles are taken to angles of the same measure; and
  • lines are taken to lines and parallel lines are taken to parallel lines.
This makes sense because a translation is simply like taking something and moving it up and down or left and right. You don't change the nature of it, you just change its location.
It's like taking the elevator or going on a moving walkway: you start in one place and end in another, but you are the same as you were before, right?

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