Bam!
Mr. Tarrou.
In this video and the next video to follow, we are going to do four basic constructions
and learn how to use a couple of basic tools in geometry; tools that in very, very early
geometry way back when it was developed by the Greeks and the Babylonians and such, these were very very
important tools, now we have compasses and rulers and protractors and it makes measurements
easier.
These are core important tools in geometry.
And there’s actually studies in college on plane geometry and Euclidean plane geometry
where the whole course is basically just using these two tools to do things like bisect angles
and other what nots.
So were just going to do some basic constructions, so a compass and a straight edge.
Now, this is a yard-stick, okay.
I’m sure you can see in the camera that has measurements on it.
A straight edge does not have measurements.
That's what the compass does- is take measurements.
A straight edge really doesn’t have inches or centimeters marked on it, to take measurements.
So we’re going to talk about constructing a congruent segment, and a congruent angle
in this video.
And in the next video we are going to bisect those.
So I have a segment already given, and we are going to walk through these three steps
to use a compass and straight edge to draw a congruent segment, I already have part of
this set up.
Now, nowadays and if you have a ruler, you know we can just go well let’s see here,
okay that appears to be twelve inches long, let’s put this down and go chhh and mark
off twelve inches and were done, but again that’s using a ruler.
And let’s not forget, congruent segments are simply two segments that have the same
length.
It’s a very simple geometric structure, so length is all we can measure, and that’s
all we need.
Two congruent segments simply have the same measure or same length.
Okay.
So it doesn’t matter what direction it goes in.
Draw a ray.
Okay, a ray, we need an endpoint, and it just extends forever.
We can measure length of segments I just did, it was approximately 12 inches long, we can
do that because it has two endpoints, and thus, it does not extend forever.
The length of this ray right now with the arrow indicating the direction of movement
has a length of infinity, it goes on forever.
It doesn’t matter that I can put a ruler on here and say it’s you know x number of
inches or centimeters long, its extending forever with the arrow.
But we have set our one endpoint.
Measure the given segment with your compass.
Now if we have a compass were going to put, this is a rubber tip, so this would be like
the sharp side of your compass, and the chalk for the pencil lead.
We’re going to open this up, and put the point on one end, and the chalk at the end,
well that was a lucky guess, at the end of the segment that were given.
Now if you don’t have a compass, or if your saying like doing construction, we can just
use like string or I’m going to use my lanyard here, and I can put my thumb here and measure
off that length with my lanyard, and set that.
And then this can kind of be my compass on my other segment, if you want to sort of improvise
with some just like random string maybe and let that along with your finger act as a compass,
but we have a compass, so we are going to measure that.
So viola, were going to put the point, this would be our rubber side, put the point of
your compass, the sharp needle-ish point of your compass, on the end point of your ray,
and mark.
we’re going to do that by rotating your compass and make an arc, which is going to
be our mark.
The intersection between the arc that you draw and the ray, is going to be the new end
point of your congruent segment, which you’re attempting to draw.
This is very, very quick, all we have to do is point on the one end point that we have
so far, make an arc through that ray, and if you have a good compass with the screw
bolt that goes between, it’s impossible to change the adjustment of that compass,
if you have just a compass that works on friction, you’re going to have to work a little bit
hard on holding it carefully, I like to hold on the pivot point to make sure, attempt to
try to make sure, it doesn’t open or close after you take that measurement.
Because its endpoint, chalk at the other endpoint, or your pencil, measure that, and then you’re
trying to transfer that equal measure onto the ray.
Of course, if your compass opens and closes, you’re going to get the wrong measurement,
so try to hold your compass carefully, or maybe even have a really good one, that’s
almost impossible to miss a line, or change its setting without intention.
What you could also do with a pencil, if your just using string, is, thumb here, measure
it, and then bring that thumb down here, and again make your arc, it’s going to be a
bit more difficult to make that exact measurement, but it’s a you know, if you need to you
can do that.
Well now, this intersection is going to be my new endpoint, and once I have my initial
endpoint from the ray, and this new endpoint of the intersection of the arc or the mark
that you made, and the ray will just come with our pencil, erase that, and now we have
a congruent segment, an exact, pretty much as best we can with a piece of pencil lead,
an exact match of the original segment given to us.
Now, let’s talk about drawing a congruent ray.
Certainly a few more steps, but hopefully will be easy to understand.
Okay we have five steps for drawing a congruent angle using a compass, and straight edge,
We’re going to draw a ray, with the point of our compass, on the vertex of the given
angle, we’re going to make an arc through that angle, and label the intersection points.
So let’s get that part done right now.
So were going to take our straight edge, were going to just draw a ray somewhere, I’m
going to try not to have that going the exact same direction, because I want you to realize
that’s not really necessary, so let’s just say the ray is going this way.
Fairly straight.
So now we have our ray.
Okay.
Step 1: With the point of our compass, on the vertex of the given angle, make an arc
through the angle.
Alright now I’m not taking any measurements, I just need an arc going through this given
angle that I can use as reference.
So I can’t have an extended course beyond the angles, so I’m going to shrink this
up a little bit, & that’s loosening up my compass as I go so let’s tighten it.
Okay.
My rubber side, the point of your compass on the vertex, and draw an arc through that
given angle.
Now, were going to label those intersection points, so let’s call that A & B. Okay.
So this is going to be point A, and this point B. Okay now what?
Make an arc through your ray with the same compass settings.
So, thankful I didn’t move this around too much, I’m going to double check it anyways.
Make an arc through the ray, that would be here, with the same compass setting.
Let’s make sure that was still correct.
Okay.
Yep.
Still hitting.
So now I’m going to put the point of my compass on the endpoint of my ray, and just
make an arc, and also, want to copy this ray, so you need your arc length to be about the
same, so point on the endpoint, and let’s just swing this around a little bit, oh I’m
also leaning the compass sort of in the direction, and I’m trying to pull it around, you don’t
want to push it, that’s going to make the pencil bounce, so I’m kind of like leaning
it this way, and I’m pulling it across the board, and that should be fine.
Okay?
[mumbling] Measure the distance between the marked point on you given angles, so in that
case it would be A and B, so let’s do that, so point where the arc intersects the other
ray of your angle, and measure that, get it to be as equal as you can, that’s pretty
good.
Measure the distance between the mark points and the given angle.
With that setting, so don’t move your compass, put the point on the intersection of that
arc, and ray, and make another arc through the first arc.
What?
What am I trying to say?
I want you to take this setting that we just measured from point A to point B, go to the
ray that we were trying to draw the new one, put the point of your compass on that intersection
point.
Probably would have been easier if I labeled it.
Let’s call that point C. And I want to make an arc going through the
first arc that we just drew a second ago.
So that’s going to go.. okay.
About right there.
The intersection of those two arcs needs to be marked again with another variable or with
another letter, so let’s call this point D, and now that I’m trying to label everything,
let’s see here, this is point O, we’ve got D and C, how about we call this first
endpoint E. Okay.
So, measure the distance… we did that.
Setting… did that.
Make an arc… and now, oh, were done!
We’re going to draw a ray from the endpoint of the ray, or were trying to copy the angle,
through the intersecting arcs, which we now have labeled as D. Okay?
And uh that’s going to be it.
So we are going to take our straight edge.
We are going to again put it on the endpoint of our ray, (the right colored chalk), also
having to go to through the point defined by the two intersecting arcs that we just
made and viola.
We have copied a congruent angle, without using a protractor to measure off the number
of degrees that are in the angle, and just simply copy it again using these more basic
tools that were used to help develop geometry of a compass and a straight edge.
If I did have a gigantic protractor, the number of degrees inside this angle BOA or Angle
AOB would be the same if I used my two tools correctly as angle CED or of course you could
have the letters in reverse order, angle DEC.
And if their measures are equal that means those angles are congruent.
So, I’m Mr. Tarrou, we’re going to move onto our next video, where were going to bisect
an angle, and bisect, make a perpendicular bisector of a segment.
If you don’t need that, well I’m Mr. Tarrou.
Bam.
Go do your homework!