Introduction to Euclidean geometry
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Euclid as the Father of Geometry
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Language and Notation of Basic Geometry
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Lines, Line Segments, and Rays
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Measuring segments
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Points, lines, and planes
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Identifying Rays
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Recognizing rays lines and line segments
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Congruent segments
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Segment addition
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Midpoint of a segment
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Language and Notation of the Circle
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The Golden Ratio
Language and Notation of Basic Geometry Understanding basic ideas in geometry and how we represent them with symbols
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- What I wanna do in this video is give an introduction to the language
- Or some of the characters that we use when we talk about geometry
- and I guess the best place to start is even think about
- what geometry means
- As you might recognize the first part of geometry right over here
- You have the root word geo
- The same word that you see in things like Geography and Geology
- And this comes this refers to the earth
- This refers
- My E looks like a C right over there
- This refers to the earth
- And then you see this Metry part
- And you see Metry in things like Trigonometry as well
- And Metry or the metric system and this comes from measurement
- This comes from measurement or measure
- Measurement
- So when someone's talking about Geometry,
- The word itself comes from earth measurement
- And that's kind of not so bad of a name
- Because it is such of a general subject
- Geometry really is the study and trying to understand
- How shapes and space and things that we see relate to each other
- So you know when you start learning about Geometry,
- You learn about lines and triangles and circles
- And you learn about angles
- And we'll define all of these things more and more precisely
- As we go further and further on
- But also encapsulated things like patterns
- And three dimensional shapes
- So it's almost everything that we see
- All the Math- visually Mathematical things that we understand
- Can in someway be categorized in Geometry
- Now with that out of the way, let's just start from the basics
- So basic starting point from Geometry
- And then we can just grow from there
- So if we just start at a dot
- That dot right over there
- It's just a point
- It's just that little point on that screen right over there
- we'd literally call that a point
- And I'd call that a definition
- And the fun thing about Mathematics is
- That you can make definitions
- We could've called this
- We could've called this an armadillo
- But we decide to call this a point which I think make sense
- Because it's just what we would call it in everyday language as well
- That is a point
- Now what's interesting about a point is that
- it is just a position; that you can't move on a point
- If you moved if you were are at this point
- And if you moved in any direction at all
- you would no longer be at that point
- So you cannot move on a point
- Now there are differences between points
- For example that's one point there
- Maybe I have another point over here
- And then I have another point over here
- And then another point over there
- And you want to be able to refer to the different points
- And not everyone has the luxury of a nice colored pen like I do
- Otherwise, they could refer to the green point,
- Or the blue point, or the pink point
- And so in Geometry to refer to points we tend to give them labels
- And the labels tend to have letters
- So for example, this could be point A;
- this could be B; this would be point C;
- and this right over here could be point D
- So if someone says hey circle point C- you know which one to circle
- You know that you would have to circle,
- You would have to circle that point right over there
- Well that so far it's kind of interesting
- You have these things called points
- You can't really move around on a point
- All they do is specify a position
- What if we wanna move around little bit more?
- What if we wanna get from one point to another?
- So what if we took
- We started at one point and we wanted we wanted all of the points
- including that point that connect that point in another point
- So all of these points right over here
- So what would we call what would we call this thing?
- All of the points that connect A and B along a straight
- And I'll use everyday language here
- Along kind of a straight line like this,
- Well we'll call this a line segment
- In everyday language you might call it a line
- But well call it a line segment coz we'll see in ge
- When we talk in Mathematical terms
- A line means something slightly different
- So this is a line segment
- And if we were to connect D and C,
- This would also be another line segment
- A line segment
- And once again, because we always don't have the luxury of colors
- This one is clearly the orange line segment;
- This is clearly the yellow line segment;
- We want to have labels for these line segments
- And the best way to label the line segments are with its end points
- And that's another word here
- so a point is just literally A or B
- But A and B are also the end points of these line segments
- Coz it starts and ends at A and B
- So let me write this A and B
- A and B are end points
- Another definition right over here
- We, once again, we could've called them aardvarks or end armadillos
- But we as mathematicians decide to call them end points
- Because that seems to be a good name for it
- And once again, we need a way to label these line segments
- With that have the end points
- And what's a better way to label a line segment than
- With its actual end points
- So we would refer to this line segment over here-
- We would put its end points there
- And to show that it's a line segment,
- we would draw a line over it just like that
- This line segment down here, we would write it like this
- And we could've just as easily written it like this:
- CD with a line over it
- Would've refer to the same line segment
- BA, BA with a line segment with a line over it
- would refer to that same line segment
- And now you might be saying well I'm not satisfied
- Just travelling in between A and B
- And this is actually another interesting idea
- When you were just on A, when you were just on a point
- And you couldn't travel at all;
- you couldn't travel at all in any direction without,
- while staying on that point,
- That means you have zero options to travel in
- You can't go up or down, left or right, in or out of the page
- And still be on that point
- And so that's why we say a point has zero dimensions
- Zero dimensions
- Now all of a sudden, we have this thing: this line segment here
- And this line segment
- we can at least go to the left and the right
- along this line segment
- We can go towards A or towards B
- So we can go back or forward in one dimension
- So the line segment is a one dimensional
- It is a one dimensional idea almost or one dimensional object
- Although these are more kind of abstract ideas
- There is no such thing as a perfect line segment
- Because everything a line segment you can't move
- You can't move up or down on this line segment while being on it
- While in reality anything that we think is a line segment
- Even you know, a stick of some type-
- a very straight stick or a string that is thought
- that still will have some width
- but the geometrical pure line segment has no width
- It only has a length here so you can only move along the line
- And that's why we say it's one dimensional
- A point you can't move at all;
- a line segment you can only move in that back and forth
- along that same direction
- Now I just said to you that it can actually have a length
- How do you refer to that?
- Well you refer to that by not writing that line on it
- So if I write AB with a line on top of it like that
- That means I'm referring to the actually line segment
- If I say that
- let me do this in a new color
- If I say that AB is equal to five units
- It might be centimeters or meters whatever
- I just the abstract unit is five,
- That means that the distance between A and B is five
- That the length of line segment AB is actually five
- Now let's keep on extending it
- Let's say we wanna just keep going in one direction
- So let's say that I start at A
- Let me do this in a new color
- Let's say I start at A and I wanna go to D
- But I wanna I want the option of keep on
- I wanna keep on going
- So I can't go further in A's direction than A
- But I can go further in D's direction
- So this little this idea that I just showed;
- This essentially a cycle line segment
- But I can keep on going past this end point
- we call this We call this a ray
- and the starting point for a ray is called the vertex
- Not a term that you'll see too often
- You'll see vertex later on in other context but it's good to know
- This is the vertex of the ray
- It's not the vertex of this line segment
- So maybe I shouldn't label it just like that
- And what's interesting about a ray is
- once again its one a dimensional figure
- But you could keep on going in one of the direct-
- You can keep on going toward past one of the end points
- And the way that we would specify a ray is we would say
- We would call it AD and we would put this little arrow over on top of it
- To show that is a ray
- And in this case it matters the order
- That we put the letters in
- If I put DA if I put DA as a ray
- This would mean a different ray
- That would mean we're starting at D and then we're going past A
- So this is not ray DA, this is ray AD
- Now the last idea that I'm sure you're thinking about is
- What if I could keep on going in every, in both directions?
- So let's say I can keep going in
- My diagram is getting messy
- So let's let me introduce some more points
- So let's say I have point E and then I have point F right over here
- And let's say that I have this object that can
- That goes through both E and F
- but this keeps on going in both directions
- This is, when we talk in geometry terms, this is what we call a line
- Now know this a line never ends;
- you can keep going in either direction
- A line segment does end
- It has end points; A line does not
- And actually a line segment can sometimes be called just a segment
- And so you would specify line EF
- you would specify line EF with these arrows just like that
- Now the thing you're gonna see most typically
- When we're studying Geometry are these right over here
- Because we're gonna be concerned
- With sides of shapes, distances between points
- And we're talking about any of those things;
- Things that have finite length;
- things that have an actual length;
- Things that don't go off forever in one or two directions,
- Then you're talking about a segment or a line segment
- Now if we go back to a line segment
- just to kind of keep talking about new words-
- That you might confront in Geometry
- If we go back talking about a line
- That's I was drawing a ray
- So let's say I have point X and point Y
- And so this is line segment XY;
- so I could specify it, denote it just like that
- If I have another point
- Let's say I have another point right over here
- Let's call that point Z
- and I'll introduce another word
- XY And Z are on the same they're all lined on the same line
- If you would imagine that a line
- could keep going on and on forever and ever;
- So we can say that XY and Z are collinear
- So those three points are co- that they are collinear
- They all sit on the same line or
- And they also all sit on the line segment XY
- Now let's say we know, we're told that XZ is equal to ZY
- And they are all collinear
- so that means
- This is telling us that the distance between X and Z
- is the same as the distance between Z and Y
- so sometimes we can we can mark it like that
- This distance is the same as that distance over there
- So that tells us that Z is exactly half way between X and Y
- So in this situation we would call Z the midpoint
- The midpoint of line segment XY coz it's exactly half way between
- Now to finish up,
- we've talked about things that have zero dimensions, points
- We've talked about things that have one dimension-
- a line, a line segment or a ray
- you might say well what has two dimensions?
- Well in order to have two dimensions
- That means I can go backwards and forwards
- in two different directions
- So this page right here or this video
- or the screen that you're looking at is a two dimensional object
- I can go
- I can go right left that is one dimension;
- or I can go up down
- And so this surface of the monitor you're looking at
- Is actually two dimensions
- Two dimensions
- You can go backwards or forwards in two directions
- And things that are two dimensions, we call them planar
- Or we call them planes
- So if you took a if you took a piece of paper
- Then extend it forever; just extended in every direction forever
- That, in a Geometrical sense was a plane
- The piece of paper itself; the thing that's finite
- And you'll never see this talked about in a typical Geometry class
- But I guess if we were to draw the analogy
- You could call a piece of paper, maybe a plane segment
- Because it's a segment of an entire plane
- If you had a third dimension then you're talking about kind of
- A three dimensional space
- In three dimensional space,
- not only could you move left or right along the screen
- Or up and down,
- you could also move in and out of the screen
- You can also have this dimension that I'll try to draw
- You could go into the screen
- or you could go out of the screen like that
- And as we go to higher and higher Mathematics
- Although it becomes very hard to visualize,
- you'll see that we can even start to study things
- That have more than three dimensions
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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