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  • - [Instructor] In this video,

  • we're gonna talk a little bit about segment congruence.

  • And what we have here, let's call this statement one,

  • this is the definition of line segment congruence,

  • or at least the one that we will use.

  • Two segments are congruent, that means that we can map

  • one segment onto the other using rigid transformations.

  • And examples of rigid transformations are reflections,

  • rotations, translations, and combinations of them.

  • Now what we're going to see in this video

  • is that statement one

  • is actually equivalent to statement two.

  • Or another way of saying it is if statement one is true,

  • then statement two is true,

  • and if statement two is true, then statement one is true.

  • Or we can write it like this,

  • we can map one segment onto another

  • using rigid transformations if and only, only if,

  • the two segments have the same length.

  • So how do we go about proving it?

  • Well the first thing that we'd want to prove is

  • that if statement one is true, then,

  • then statement two is true.

  • So how would we go about doing this?

  • And like always, I encourage you

  • to pause the video and have a go at it.

  • All right, now let's work through it together.

  • Some proofs like this might be difficult

  • because they feel so intuitive.

  • But one way to prove this is to first say that by definition

  • rigid transformations preserve length.

  • So by definition, by definition, definition,

  • rigid transformations, that's what makes them rigid,

  • rigid transformations preserve length.

  • So if one segment, if one segment,

  • can be mapped onto,

  • onto a second segment

  • with rigid transformations, with rigid transformations,

  • they must have had the original, the same original length,

  • they must have had the same original length,

  • they must have had same original length.

  • Or another way to say it is, then two is true.

  • Then we can try to do it the other way around.

  • So let's see if we can prove that if two,

  • if statement two is true, then statement one is true,

  • then statement one, and why don't you pause this video

  • and have a go at that as well.

  • So let's assume, assume, I have segment AB

  • and then I have another segment, let's call it CD,

  • have the same length, have same length.

  • So they meet the number two statement right over there.

  • To map, to map AB onto CD,

  • all I have to do, I can do this in two rotations every time,

  • I first will, I will translate so that A is on top of C,

  • so I will translate, translate AB,

  • so that point A is on top of point C.

  • And then the next thing I would do is rotate,

  • rotate AB so that point B,

  • point B, is on top of point D.

  • And there you have it.

  • For any two segments with the same length,

  • I can always translate it

  • so that I have one set of points overlap,

  • and then to get the other points to overlap

  • I just have to rotate it.

  • I know that's going to work

  • because they have the same length.

  • So I've just shown you, if we can map one segment

  • onto another using rigid transformations,

  • then we know they have the same length,

  • and if two segments have the same length,

  • then we know that we can map one segment

  • onto the other using rigid transformations.

- [Instructor] In this video,

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長さが同じであることと同等のセグメントの一致を示す (Showing segment congruence equivalent to having same length)

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    林宜悉 に公開 2021 年 01 月 14 日
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