GEO.1.1
Summary
TLDRThis lesson explains the basic concepts of geometry, including points, lines, and planes. It covers the definitions, properties, and methods for identifying and naming these geometric elements. Key points include understanding that a point represents a location with no size, a line is formed by connecting at least two points and has no thickness, and a plane is a flat surface extending indefinitely. The lesson also discusses collinear and non-collinear points, coplanar points, and how lines and planes intersect.
Takeaways
- 📍 A point is a location with no size or shape, denoted by a capital letter.
- 📏 A line is composed of an infinite number of points and has no thickness; it's identified by two points or a lowercase script letter.
- 🔄 The error to avoid when identifying a line is assuming only the named points exist on the line; there are infinitely more.
- 🔺 Points that lie on the same line are called collinear, while those not on the same line are non-collinear.
- 🏞 A plane is an infinite, flat surface made up of points, requiring at least three non-collinear points to define it, named by an uppercase script letter.
- 🔄 The term 'coplanar' refers to points that lie on the same plane, contrasting with 'non-coplanar' points.
- 🔼 Two intersecting lines meet at a point, while two intersecting planes meet along a line.
- 🔍 Practice involves identifying collinear points, naming lines containing specific points, and finding intersections of lines and planes.
- 📝 When naming a line, it's crucial to remember that the line extends infinitely in both directions beyond the named points.
- 📐 In diagrams, dashed lines indicate that they are not part of the plane but are shown for visual context, similar to a pencil behind a sheet of paper.
Q & A
What is a point, and how is it represented in geometry?
-A point is a location in geometry with no size or shape. It is represented by a capital letter.
How is a line formed, and what are its characteristics?
-A line is formed by connecting two points and consists of an infinite number of points. It has no thickness or width and is usually named by any two points on the line or a lowercase script letter.
What is the common misconception about points on a line?
-The common misconception is that only the two named points are on the line, but in reality, there are an infinite number of points between them.
What are collinear points?
-Collinear points are points that lie on the same line.
How do you define a plane in geometry?
-A plane is a flat surface that extends indefinitely in all directions. It is made up of at least three non-collinear points.
How can you name a plane?
-A plane can be named using any three non-collinear points on the plane or by an uppercase script letter.
What happens when two lines intersect in geometry?
-When two lines intersect, they intersect at a point.
What is the result when two planes intersect?
-When two planes intersect, they intersect at a line.
How many points are needed to define a plane, and what is the relationship between points, lines, and planes?
-At least three non-collinear points are needed to define a plane. A point represents a location, a line is formed by two points, and a plane is formed by at least three points.
What is the difference between collinear and coplanar points?
-Collinear points lie on the same line, while coplanar points lie on the same plane.
Outlines
📍 Introduction to Points, Lines, and Planes
This paragraph introduces the concepts of points, lines, and planes in geometry. It explains that a point is merely a location with no size or shape, often named with a capital letter. A line is defined by two points and can contain an infinite number of points between them. The paragraph also clarifies common misconceptions, such as the belief that a line consists solely of the named points, and discusses different ways to name a line, including using a lowercase script letter.
📏 Understanding Collinear and Non-Collinear Points
This section delves into the definitions of collinear and non-collinear points. Collinear points lie on the same line, while non-collinear points do not. The paragraph uses examples to explain how to identify these types of points, emphasizing that collinear points must be on the same line, whereas non-collinear points are those that lie off that line.
🛠️ Defining Planes and Intersecting Lines
This paragraph describes the nature of planes as flat surfaces extending infinitely in all directions. It highlights that three non-collinear points are needed to define a plane, and that a plane can be named using any three such points or an uppercase script letter. The paragraph also discusses the intersection of lines and planes, explaining that two lines intersect at a point, while two planes intersect at a line, building on the previous concepts of points and lines.
📚 Practical Application of Points, Lines, and Planes
This final section applies the concepts of points, lines, and planes to specific geometric problems. It guides the reader through exercises involving collinear points, identifying lines that contain specific points, and naming lines and planes using different points. The paragraph concludes by addressing the intersection of lines and planes, emphasizing how they connect through points and lines, and recaps the key concepts covered in the unit.
Mindmap
Keywords
💡Point
💡Line
💡Plane
💡Collinear Points
💡Non-collinear Points
💡Intersection
💡Coplanar Points
💡Non-coplanar Points
💡Script Letters
💡Infinite
Highlights
A point is a location with no size or shape, represented by a capital letter.
A line is made up of points and is formed by at least two points; it has no thickness or width.
A common misconception is that a line consists of only the two points used to name it, but it actually contains an infinite number of points.
Collinear points are points that lie on the same line, whereas non-collinear points do not lie on the same line.
A plane is a flat surface made up of at least three non-collinear points and extends indefinitely in all directions.
A plane can be named using any three non-collinear points or an uppercase script letter.
Coplanar points lie on the same plane, while non-coplanar points do not.
Two intersecting lines meet at a point, while two intersecting planes meet at a line.
The relationship between points, lines, and planes is hierarchical, with a point being the most basic, followed by lines and then planes.
When identifying collinear points, it’s important to focus on whether they lie on the same line, regardless of where the lines intersect.
A line containing a specific point can be named in multiple ways, using any two points on the line or a lowercase script letter.
Understanding the intersection of planes and lines requires visualizing planes extending indefinitely, even if they appear to be bounded in diagrams.
Three points are needed to define a plane, and the intersection of planes or lines follows specific geometric rules.
The concept of intersecting planes can be complex, requiring careful consideration of how they extend and intersect in three-dimensional space.
Identifying planes and their intersections involves understanding that planes can contain multiple lines and points, and their intersections define geometric relationships.
Transcripts
points lines and planes so let's look at
what a point line a plan is to find them
and how see how we can identify them
okay so a point is a location
it's not what you think it is in regards
when you see a point like over here
because it actually has no size and it
has no shape it's just a location that's
like saying I'm going to my friend's
house if it's a location right always
use a capital letter to name a point
example one
now looking at a line the line is made
up of points it's made up of many points
actually but the second that you have
two points you form a line you can have
an infinite number of points between two
points as well but the second you have
two points that's when we have that's
when we can actually call it a line a
line has no thickness and it has no
width
and we want to nail line by any two
points on the line or a lowercase script
letter okay so looking over here now one
thing to point out is just because you
see the point X and you see the point
why I can name it as the line X Y this
is one way of writing the line is this
one right here okay but just because I'm
using X Y doesn't mean that there's not
an infinite number of points between
those two points okay so the number one
error if that's often found when
identifying a line is that some people
will just think x and y are the only two
points no it's just the only two points
I've named but they're not the only two
points
okay so other ways of naming points you
could do the first symbol with XY with a
line on top with the arrows you can say
line n because if you notice I can also
call it by the lowercase script letter
there or I can say line ex-wife
any of those three work : ear points
look at this definition I hear : your
points are points that lie on the same
line so if I have three points or five
points or six points whatever it doesn't
matter but as long as they are on the
same line we can say they are collinear
non collinear our points that do not lie
on the same line so if I was to throw
another point out here and let's call
this guy Z right I can say that x y and
z are non collinear
now just because x and y are on the same
line ii I throw Z in there I say XY and
Z well XY and z are not on the same line
so therefore I can say XY and z are non
collinear
let's move on to plain let's define a
plane
so a plane is a flat surface that's made
up of points
made up of points okay you need at least
three points to make up a plane and you
can kind of see the connection here
between a point line is playing at the
point is one point obviously a plane is
our line is at least two and a plane you
need at least three okay a plane its
extent indefinitely in all directions so
even though right here we have this
plane that has these borders the true
plane doesn't have any borders it's
going to extend indefinitely in all
directions you can name a plane by any
three non collinear points on the plane
or an uppercase script letter so they
can't be on the same line they have to
be non collinear
so over here KJ and L bar non collinear
so I can name this plane okay J bow or I
can use the upper upper case script
letter so if you see here that one does
not have a dot next to it so that's not
a point so if you do not see the dot
next to it that says this is a point
it's not a point that's actually the
name of the plane so I could save plane
and just like collinear SoCo means
together and obviously linear means line
so collinear points that lie on the same
line we have coplanar which is points
that lie on the same plane and the
second we don't have lines that lay on
the same plane they're non coplanar and
of course you must have at least four
points to make that statement now
remember a plane can be made of an
infinite number of points but you need
at least three to make up a plane
so when I have intersecting lines and
planes you can see that two lines are
going to intersect at a point that's
kind of you know something we've seen
before in our previous learning
but here we can see that two planes two
planes are going to intersect at a line
right now when you think about these
three things you can think about them as
I'm building up and going back so you
have a point you have a line the line is
bigger than a point and then you have a
plane that's right so the two lines
intersect at a point and two planes are
going to intersect at a line so they
build off of each other so when you're
asking yourself okay where do two planes
intersect well the thing behind a plane
is a line and therefore two planes
intersect at a line where did two lines
intersect well the thing behind the line
is a point so two lines are going to
intersect at a point
so let's look at number one two three
and four in our practice I'm going to do
number one with you then I'm going to
let you do two three four and then we're
going to come back and look at the
answers so in number 1a we want four
collinear points so I need four points
that are on the exact same line and it
doesn't matter where the lines intersect
all we care about is are they on the
same line
and if you notice I only have one line
that has four points so those are my
only four collinear points
h/n okay and
so those are our four collinear points
now we want a line that contains a point
O so where's em okay there's actually a
few ways that you can name this line you
can name it like that and like them
Oh any of those things but because it
wanted to know a line that contains
point am I'm gonna call this line the
lowercase P got keep that in mind it's
always lower case when we're naming a
line by that script we want a line that
contains points H and K so let's find H
let's find K again you can name that
multiple ways but I'm gonna name it line
are they're on the same exact line it
doesn't matter that K is also on cue
that's doesn't matter it's also online
are so H and K are both together
now we want another name for line Q so
let's look at one Q here's my Q these
are the points that make up like you you
can name this three different ways okay
actually yeah got a little more than
three but
we're gonna state it kind of alphabetic
order if you will
so I'm gonna do all the three different
ways without switching the letters you
know which could give you more
combinations but just not switching
letters I'm gonna give you all three but
you could just write one down you don't
have to write all three if you don't
want to write so the first would be
let's do JL the first and the last I'm
going to put my line at the top because
it's one then we have JK
and then finally I can do Kay
you can also do okay remember you can
switch that could be kt j LJ okay in
this particular case because a line goes
both ways but without having to write
them all down let's face it lee what you
have
and then finally we want the
intersections of intersection of lines P
and R so here's P here's our that's
where they intersect they intersect at
the point
so now try two three and four on your
own and then we will come back and look
at the answers together pause it right
here
let's look at the answers to number two
so using the diagram one furthest to a B
C D and E so in a we want to know a line
containing point F you should have line
J another name for line K could be b eb
c or easy you only had to write one a
plane containing point a plane m an
example of three non collinear points
could be a B or D you could have had a
FC a DC a B D which we did have a b e or
a bf any of those would be three non
collinear points now the intersection of
plane em in line K can often be
confusing okay when you see this these
little dash marks that is basically
meaning that the line is actually not on
the plane but behind the plane you kind
of want to see it like
that's like a pencil that has been
shoved through a piece of paper like you
would see in this image here I've got a
little bigger for you and so the pencil
isn't actually on the piece of paper
back here it's actually behind it and
that's what's actually going here so the
intersection of plain mm1 k would be
looking at number three we want to name
three coupe planar points
I chose vxy being apart
of this plane
this whole plan right here you can could
have picked VW c 4 t WX any of those
that you thought made up a plane a plane
containing point X you have three
options but because it was on this
lovely plane
I picked plane are the intersection of
plane R and plane Z V Y so here is the V
Y so it's this plane right here
and so at plane ours is the guy at the
bottom so this is its intersection the
line view I remember a plane is
connected by two lines when two lines
are connected by a point our plane is
connected by a line tube lines are
connected by a point going down down
that's how I like to look at it makes it
easier
and then how many planes peer in the
figure there's actually five planes you
have the plane that's over here you have
a plane over here that's two planes then
we have we have the plane that stick
better over here that work so there's
three planes
we have the plane that here that's for
planes
and now we have this big old plane all
the way on the bottom that is five
planes now you might be thinking that
there was a plane at the bottom over
here and then this one but now you kind
of think about it as this a witch's hat
in a way kind of an odd looking for just
hop but you get what I'm saying except
this isn't a whole it's just pyramid on
top of this plane making up this whole
bottom to be one plane it's only five
planes and then how many planes contain
point W well W is on this plane it is a
part of this plane and it's a part of
this place so three planes
talk about my messy highlighter work
there we go
finally let's look at number four and
the first one the intersection of lines
L and M well here is Ellen M and they're
intersecting their back EEP
another name for plain cute I chose I G
you just need three points that are non
collinear to name a plane our points D
and E collinear or coplanar so well
here's D and here's e are they on the
same line
yes they are so therefore they are
collinear and we could tell just by
looking at it well coplanar or its plane
points to be coplanar you need mr. right
and then how many times do P and Q
intersect
well here's P
and if you can't tell just by looking at
them coloring them in is actually quite
helpful
and here's cute
and do those two planes intersect at the
colors intercept no they do not
there forth the planes do not intersect
at all
and we are done with the unit 1 1 points
lines and planes notes
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