Κύριο περιεχόμενο

## Εισαγωγή στην Ευκλείδεια γεωμετρία

# Geometric definitions example

## Απομαγνητοφώνηση βίντεο

- [Voiceover] A lot of
what geometry is about is proving things about the world. And in order to really prove
things about the world, we have to be very careful, very precise, very exact with our language so that we know what we're proving and we know what we're assuming and what type of deductions we are making as we prove things. And so, to get some practice being precise and exact with our language, I'm going to go through some exercises from the Geometric Definitions Exercise on Khan Academy. So this first one says,
"Three students attempt "to define what it means for lines l and m "to be perpendicular. "Can you match the teacher's
comments to the definitions?" All right. So looks like
three different students attempt definitions of what
it means to be perpendicular, and then there's these teacher's comments that we can move around. So we're gonna, I guess,
pretend that we're the teacher. So Ruby's definition
for being perpendicular: "l and m," lines l and
m, "are perpendicular "if they never meet." Well, that's not true. In fact, perpendicular lines
for sure will intersect. So, in fact, they
intersect at right angles. So that is not going to be, that's not going to be correct. And so, actually this looks right. "Were you thinking of parallel lines?" Because that's looks like
what she was trying to define. If things are on the same
plane and they never intersect, then you are talking about parallel lines. I also got Shriya's definition. "l and m are perpendicular
if the meet at one point "and one of the angles at
their point of intersection "is a right angle."
Well, that seems spot on. So let's see. I would say, "Woohoo! Nice Work! "I couldn't have said it better myself." Now let's just make sure this comment matches for this definition. Abhishek says, "l and m are perpendicular "if they meet at a single point, "such that the two lines make a 'T'." Well, that's, in a hand-wavy
way, kind of right. When you imagine perpendicular lines, you could imagine them kind of forming a cross or, I guess, part of ... You know, a T would be part of it. But I think this comment is spot-on. The teacher's comment, "Your
definition is kind of correct, "but it lacks mathematical precision." You know, what does he mean by a T? What does it mean to "make a T"? Shriya's definition is much more precise. They're perpendicular "if
they meet at one point "and one of their angles at
their point of intersection "is a right angle," is a 90 degree angle. Let's check our answers. Let's do a few more of these. This is actually a lot of fun. So once again, we're
gonna have three students attempting to define, but
now they're going to define "an object called an angle." "Can you match the teacher's
comments to the definitions?" So Ruby, oh, it's three,
same three students. Ruby says, "The amount of turn
between two straight lines "that have a common vertex." Well, this is kind of getting there. The definition of an angle,
we typically talk about two rays with a common vertex. She's talking about two
lines with a common vertex. And she's talking about
the amount of turn. So she's really talk
about more of, kind of, the measure of an angle. So let's see what comment here. So, "You seem to be getting at the idea "of the measure of an angle, and "not the definition
of the angle itself." So this is actually right. I would put this one right here. We just got lucky. This was already aligned. So Shriya's definition: "Two
lines that come together." So, once again, this is kind of ... The definition of an angle is
two rays with a common vertex. So, "Two lines that come together," this is just intersecting lines. Now, when that happens, you
might be forming some angles, but I would just say, "Were you thinking of intersecting lines?" And let's see what Abhishek says. "A figure composed of two rays
sharing a common endpoint. "The common endpoint is
known as the vertex." Yup. That's a good definition of an angle. So Abhishek got it this time. Let's do another one. So three students are
now attempting to define "what it means for two
lines to be parallel." So now let's match the teacher's comments. So Daniela says, "Two lines are parallel "if they are distinct
and one can be translated "on top of the other." All right. So that actually seems pretty interesting. That's actually not the first way that I would have defined parallel lines. I would have said, "Hey, if
they're on the same plane "and they don't intersect,
then they are parallel." But this is, this seems pretty good because, if you're translating something, you're not, you aren't going to rotate it, you're not going to change its direction, I guess, one way to think about it. And so, if you're translating one ... If you can trans- If they're two different lines, but you can shift them without
changing their direction, which is what translation is all about, on top of each other, that
actually feels pretty good. So, I'll put that right over there. So Ori says, "Two lines are parallel "if they are close together
but don't intersect." So, if you're trying to
define parallel lines, parallel lines, it doesn't matter if they're close together or not. It's, they just have to be in the same plane and not intersect. They can be very far apart and
they could still be parallel. So this isn't an incorrect statement. You could have two lines
that are close together and don't intersect on the same plane, and they are going to be parallel. But this isn't a good definition, because you can also have parallel lines that are far apart. And so, actually I'd go with
this statement right here: "Part of your definition is correct, "but the other part is not. "Parallel lines don't have
to be close together." So this isn't a good
definition of parallel lines. So let's see. Kaori:
"Two lines are parallel "as long as they aren't perpendicular." Well, that's just not true, because you can intersect, you can have two lines that intersect at non-right angles, and
they're not parallel, and they're also not perpendicular. So this is, you know, "Sorry,
your definition is incorrect." This is actually a lot of fun, pretending to be the teacher. Let's do another one. All right. So "Three students attempt to define "what a line segment is." And we have a depiction of a
line segment right over here. We have point P, point
Q, and the line segment is all the points in between P and Q. So, so let's match the teacher's comments to the definitions. Ivy's definition: "All of the points "in line with P and Q,
extending infinitely "in both directions." Well, that would be the
definition of a line. That would be the line P, Q. That would be, if you're
extending infinitely in both directions, so ... I would say, "Are you thinking of a line "instead of a line segment?" Ethan's definition: "The
exact distance from P to Q. Well, that's just a ... that's the length of a line segment. That's not exactly what a line segment is. And see, Ebuka's definition. "The points P and Q, which
are called endpoints, "and all of the points in a straight line "between points P and Q. Yep. That looks like a good definition for a line segment. So we can just check our, we can just check our answer. So, looking good. Let's do one more of this. I'm just really enjoying
pretending to be a teacher. All right. "Three
students attempt to define "what a circle is." Define what a circle is. "Can you match the teacher's
comments to the definitions?" Duru. "The set of all points in a plane "that are the same distance
away from some given point, "which we call the center." That just seems like a pretty
good definition of a circle. So, I would, you know,
"Stupendous! Well done." Oliver's definition. "The set of all points in 3D space "that are the same distance
from a center point." If we're talking about 3D space and the set of all points
that are equidistant from that point in 3D space, now we're talking about
a sphere, not a circle. And so, "You seem to be confusing "a circle with a sphere." And then, finally, "A
perfectly round shape." Well, that's kind of true. But if you're talk about three dimensions, you could be talking about a sphere. If you're talking about, if
you go beyond three dimensions, hypersphere, whatever else. In two dimensions, yeah,
perfectly round shape, most people would call it a circle. But that doesn't have a
lot of precision to it. It doesn't have, it doesn't give us a lot that we can work with from a
mathematical point of view. So I would say, actually,
what the teacher's saying: "Your definition needs
to be much more precise." Duru's definition is
much, much more precise. The set of all points that
are equidistant from ... in a plane, that are equidistant away from a given point,
which we call the center. So yep. Carlos could use a
little bit more precision. We're all done.