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  • - [Instructor] So, what I have here are a few definitions

  • that will be useful for a proof we're going to do

  • that connects the worlds of congruence of line segments

  • to the idea of them having the same length.

  • So, first of all, there's this idea

  • of rigid transformations, which we've talked about

  • in other videos, but just as a refresher,

  • these are transformations

  • that preserve distance between points.

  • So, for example, if I have points A and B,

  • a rigid transformation could be something like a translation

  • because after I've translated them, notice the distance

  • between the points is still the same.

  • It could be like that.

  • It includes rotation.

  • Let's say I rotated about point A as the center of rotation.

  • That still would not change, that still would not change

  • my distance between points A and B.

  • It could even be things like taking the mirror image.

  • Once again, that's not going to change the distance

  • between A and B.

  • What's not a rigid transformation?

  • Well, one thing you might imagine is dilating,

  • scaling it up or down.

  • That is going to change the distance,

  • so rigid transformation are any transformations

  • that preserve the distance between points.

  • Now, another idea is congruence,

  • and in the context of this video, we're going to be viewing

  • the definition of congruence as two figures are congruent

  • if and only if there exists a series

  • of rigid transformations which will map one figure

  • onto the other.

  • You might see other definitions of congruence in your life,

  • but this is the rigid transformation definition

  • of congruence that we will use, and we're going to use

  • these two definitions to prove the following,

  • to prove that saying two segments are congruent

  • is equivalent to saying that they have the same length.

  • So, let me get some space here to do that in.

  • So first, let me prove that if segment AB

  • is congruent to segment CD,

  • then the length of segment AB,

  • which we'll just denote as AB without the line over it,

  • is equal to the length of segment CD.

  • How do we do that?

  • Well, the first thing to realize is if AB,

  • if AB is

  • congruent to CD,

  • then AB can

  • be mapped onto CD

  • with rigid transformations, rigid transformations.

  • That comes out of the definition of congruence.

  • And then we could say, "Since

  • "the transformations are rigid,

  • "distance is preserved,

  • "preserved," and so, that would imply

  • that the distance between the points

  • are going to be the same.

  • AB, the distance between points AB, or the length

  • of segment AB, is equal to the length of segment CD.

  • That might almost seem too intuitive for you,

  • but that's all we're talking about.

  • So, now, let's see if we can prove the other way.

  • Let's see if we can prove that if the length of segment AB

  • is equal to the length of segment CD, then segment AB

  • is congruent to segment CD, and let me draw them

  • right over here, just to, so, let's say I have segment AB

  • right over there, and I'll draw another segment

  • that has the same length, so maybe it looks something

  • like this, and this is obviously hand-drawn.

  • So, then, let's call this CD.

  • So, in order to prove this, I have to show,

  • "Hey, if I have two segments with the same length,

  • "that there's always a set of rigid transformations

  • "that will map one segment onto the other,

  • "which means, by definition, they are congruent."

  • So, let me just construct those transformations.

  • So, my first rigid transformation that I could do

  • is to translate, translate, and I'll underline the name

  • of the transformation, segment AB, so that point A is

  • on top of point C, or A is mapped onto C,

  • and you could see that there's always a translation

  • to do that.

  • It would be doing that, and of course we would translate.

  • B would end up like that, and so, after this translation,

  • it's going to be A right over there.

  • A is going to be there, and then B is going

  • to be right over there.

  • Now, the second step I would do is then rotate AB

  • about A, so A is the center of rotation,

  • so I'm gonna rotate it so that

  • point B lies on ray CD.

  • Well, what does this transformation do?

  • Well, since point A is the center of rotation,

  • A is going to stay mapped on top of C

  • from our first translation, but now B is rotated,

  • so it sits on top of the ray that starts at C

  • and goes through D and keeps going,

  • and where will B be on that ray?

  • Well, since the distance between B and A is the same

  • as the distance between D and C,

  • and A and C are the same point,

  • and now B sits on that ray, B will now sit right on top

  • of D because AB is equal to CD.

  • B will be mapped onto,

  • onto D, and just like that,

  • we've shown that if the segment lengths are equal,

  • there is always a set of rigid transformations

  • that will map one segment onto the other.

  • Therefore, since A and B have been mapped onto C and D,

  • we know that A, that segment AB is congruent to segment CD,

  • and we are done.

  • We have proven what we set out to prove both ways.

- [Instructor] So, what I have here are a few definitions

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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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