High school geometry
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 unit long.
Translate line segment by .
What is the length of the pre-image—the segment before the translation?
What is the length of the image—the segment after the translation?
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 by .
Has the measure of the angle changed after the translation?
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 and by .
Are the two image lines parallel?
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.
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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- will this help me understand lam 1(10 votes)
- it helps best if you put thought in it and work toward it(4 votes)
- Will this help me in the future(3 votes)
- As a gardener -- no; as a computer scientist -- yes.(12 votes)
- how do you do this i kind of get it and kind of dont so teach me the way(0 votes)
- Hi Miguel,
The key to understanding translations is that we are SLIDING a point or vertices of a figure LEFT or RIGHT along the x-axis and UP or DOWN along the y-axis.
On our x-axis, if the translation number is positive, move that point right by the given number of units, and if the translation number is negative, move that point to its left.
Use the same logic for y-axis; if the translation number is positive, move it up, and if the translation number is negative, move the point down.
Let us have a look at an example. We are given a point A, and its position on the coordinate is (2, 5). The translation number is (-1, 3).
So, we will move the point LEFT by 1 unit on the x-axis, as translation number is (-1).
We will then move the point 3 units UP on the y-axis, as the translation number is (+3).
The image will be A prime at (1, 8).
We did this with a point, but the same logic is applicable when you have a line or any kind of figure.
The other two points to remember in a translation are-
1. Lengths don't change
2. Angles don't change
I hope this helped.
- this is differnt then what there teaching us in my class(7 votes)
- these comments are so old(5 votes)
- The site defaults to showing the top rated questions & answers. Since the rating take time to accumulate, they are apt to be older. You should read them, they likely have good things to learn.
If you want to see the more recent questions, just change the sort sequence to "recent".(5 votes)
- From which point on line we start translation(1 vote)
- It doesn't matter where you choose to start. Pick any point on the line and move it to the correct coordinates, and the rest of the line will also be translated correctly.(8 votes)
- hello how are y'all doing today(6 votes)
- I'm doing quite alright sean jeffries. How bout you my boy?(1 vote)
- Can you describe translation as a matrix-vector product?(3 votes)
- It´s very interesting your question, cause a translation means just move around a shape without changing it lengths, but when we do matrix multiplications (linear transformations), we are shrinking or "adjusting" the vectors to fit in a specific point, so we are breaking basically the definition of translation. If you deal with the line segments as vectors. I don´t know if I'm right, but it took me awhile to think about it, but there is maybe a relationship between transformations and linear algebra, but in a different way. Hope that was helpful.(2 votes)
- do the lines always stay parallel??(2 votes)
- Technically, parallel lines meet in infinity, but since you cannot reach infinity, they never meet(0 votes)
- If you translate traingle ABC to A'B'C', and if there were 3 lines drawn - A to A', B to B', and C to C' - how would you know the lines are all parallel to each other?(1 vote)
- A translation will map one point to another. If we draw the line segement between those two points, the length and slope of the segment completely determines the translation. Also vice versa, the translation completely determines the length and slope of that segment.
So the three lines between your triangles all have the same slope, and hence are all parallel.(5 votes)