Videos > CA Geometry: Pythagorean theorem, compass constructions
CA Geometry: Pythagorean theorem, compass constructions Video Lecture
FAQs on CA Geometry: Pythagorean theorem, compass constructions Video Lecture
| 1. What is the Pythagorean theorem? |  |
| 2. How can I apply the Pythagorean theorem to solve problems? | |
Ans. To apply the Pythagorean theorem, you need to identify a right triangle and know the lengths of two of its sides. Once you have this information, you can use the theorem to find the length of the third side. For example, if you know the lengths of the legs of a right triangle, you can square each length, sum them together, and then take the square root of the sum to find the length of the hypotenuse.
| 3. What are compass constructions in geometry? | |
Ans. Compass constructions in geometry involve using a compass and straightedge (ruler) to construct geometric figures or perform specific constructions. These constructions are based on specific rules and procedures and are used to create precise shapes or angles. Examples of compass constructions include bisecting angles, constructing perpendicular lines, and drawing circles with specific radii.
| 4. How do I perform a compass construction? | |
Ans. To perform a compass construction, you typically start with a given geometric figure or situation and follow a set of steps using a compass and a straightedge. These steps may involve drawing circles, bisecting angles, or constructing specific lines. It is important to carefully follow the instructions provided for each construction and use accurate measurements to achieve the desired result.
| 5. Can compass constructions be used to solve problems involving the Pythagorean theorem? | |
Ans. Yes, compass constructions can be utilized to solve certain problems related to the Pythagorean theorem. For instance, if you are given the lengths of two sides of a right triangle, you can use compass constructions to accurately draw the triangle and then apply the Pythagorean theorem to find the length of the third side. The compass and straightedge allow you to construct the triangle precisely, ensuring accurate measurements for the theorem's application.
Text Transcript from Video
We're on problem 51.
And they say, a diagram from
a proof of the Pythagorean
theorem is pictured below.
And they they say, which
statement would not be used in
the proof of the Pythagorean
theorem?
So since they have drawn this
diagram out, I think we might
as well just kind of do the
proof and then we can look at
their choices and see which
ones kind of match
up to what we did.
Hopefully, they do
it the same way.
And this is a pretty
neat proof of
the Pythagorean theorem.
I don't think I've
done it yet.
So I might as well do it now.
Well, let's figure out
what the area of this
large square is right.
Well there's two ways to think
about it you could just say,
OK, this is a square.
That's a, that's b.
Well this is going
to be b as well.
This is going to be a as well.
So the area of the square is
going to be the length of one
of its sides squared.
So we could say the
whole square's
area is a plus b squared.
And that's equal to a squared
plus 2ab plus b squared.
Fair enough.
Now we can also say that the
area of this larger square,
and it's a bit of an optical
illusion, it looks like it's
tilted to the left because
of the way it's drawn.
But anyway, that the area of
this larger square is also the
area of these four triangles
plus the area of
this smaller square.
So this, the area of the
larger square, which we
figured out just by taking one
side of it and squaring it,
that should be equal
to the area of the
four smaller triangles.
So there's four of them.
And what's the area
of each of them.
Let's see, let's just
pick this one.
1/2 base times height.
So it's 1/2 times a times b.
So 1/2 ab is one of these and
I multiply by 4 to get all
four of these triangles.
And then we want to add the area
of this inside square.
And that's just going
to be c squared.
So plus c squared.
Let's see if we can
simplify this.
So you get a squared plus 2ab
plus b squared is equal to 4
times 1/2 is 2ab
plus c squared.
Well, we could subtract
2ab from both
sides of this equation.
The top and the bottom of
this equation the way
I've written it.
But if we do that, subtract 2ab
from there, subtract 2ab
from there, and you're left with
a squared plus b squared
is equal to c squared, which
is the Pythagorean theorem.
And we've proved it.
So let's see which of their
choices matches what we did.
OK, which statement would not
be used in the proof of the
Pythagorean theorem.
The area of a triangle
equals 1/2 ab.
We used that.
The four right triangles
are congruent.
No, we used that.
The area of the inner square is
equal to half of the area
of the larger square.
We didn't use that.
I think this is the
one that would not
be used in the proof.
Choice D, the area of the larger
square is equal to the
sum of the squares of the
smaller square and the four
congruent triangles.
No, that that was the
crux of the proof.
So we definitely used that.
So C is our answer.
That's the statement
that would not
be used in the proof.
I'm learning to copy and
paste ahead of time.
So I don't waste your time.
All right, a right triangle's
hypotenuse has length 5.
If one leg has length 2,
what is the length
of the other leg?
Pythagorean theorem, x squared
plus 2 squared is equal to 5
squared, because 5 is
the hypotenuse.
x squared plus 4
is equal to 25.
Subtract 4 from both sides.
x squared is equal to 21.
So x is equal to the
square root of 21.
So choice B.
Next question.
A new pipeline is being
constructed to reroute oil
flow around the exterior of a
national wildlife preserve.
I guess that's the national
wildlife preserve.
The plan showing the old
pipeline and the new route is
shown below.
OK, how many extra miles will
the oil flow once the new
route is establised.
So the new route is going to
be 60 miles plus 32 miles.
So the new route is 92 miles.
So what was the old route?
Well the old route was the
hypotenuse of this triangle.
So we could say, let's
call that x.
60 squared plus 32 squared
is equal to x squared.
Because that's the hypotenuse.
And these numbers, that's a bit
of a pain to deal with.
Maybe if I can factor out
something here I can make it
more interesting.
So I don't have to multiply out
60 squared and 32 squared
and all of the rest.
Well, let me see.
Both of those are
divisible by 4.
So then I would have 15 and 8.
Yeah, that still doesn't
make it that useful.
So I'll just multiply
them out.
So this is 3600.
At 32 squared, let's
see, 32 times 32.
2 times 32 is 64.
3 times 2 is 6.
3 times 3 is 9.
So it's 1024.
Plus 1024 is equal
to x squared.
So let me just switch
both sides.
x squared is equal to 3600
plus 1024 is 4624.
Let me see if I can get
an approximate.
So x is going to be the square
root of this thing right here.
So let's see if I can get
a handle at least on the
magnitude of where
this would be.
So 20 times 20 is 400.
So this is way too small.
60 times 60 is 3600.
So 68 times 68, this
looks right.
Especially because 8 times
8 should end in a 4.
Let me try that out.
68 times 68.
8 times 8 is 64.
8 times 6 is 48 plus 6 is 54.
6 times 8, 48.
6 times 6, 36 plus 4 is 40.
4624.
So x is 68.
Oh, I used 68, I shouldn't have.
Because they don't want
to know how long was
the old pipeline.
That's 68.
It just happened to be
one of the choices.
That's just to make
sure that you read
the question properly.
But they want to know how
much longer is going
to be the new pipeline.
So the new one was 92.
And the old one is 68.
Good thing they had that
number there so I
could try it out.
That was the square
root of 4624.
So how much longer
is the new one?
Well 92 minus 68 that's
24 miles.
So choice A.
Not choice B.
B is how long the old
pipeline was.
We want to know how much longer
the new route is.
That was tricky.
Well not tricky, but I kind of
fell for it by forgetting what
the question was about.
Anyway, next question.
Marcia is using a straightedge
and compass to do the
construction below.
Interesting.
Which best describes the
construction Marcia is doing.
So, I assume when they
say construction
she's drawing something.
Let's see what it looks like.
It looks she's taking her
compass, she's probably
putting one of the points
here, she put one of the
points there and then she
kind of drew this arc.
And then it looks like she put
the point there and then she
drew that arc.
And then she put the point
here and drew that arc.
And then put the point there
and drew that arc.
And the end result, it seems
like the reason why she picked
this point here is it goes
through this line L.
So she's probably trying to find
another point here, so
that she can draw
another line.
Because they say she
has a straightedge.
A straightedge is to
draw these lines.
A compass is to draw
these curves.
So if she were to draw another
line between these two points,
it looks something like
that, then she would
have parallel lines.
The reason why she would have
parallel lines is because
these would be corresponding
angles and
they would be congruent.
And so if you have a transversal
the corresponding
angles are congruent, you're
dealing with parallel lines.
So my read of this question is
that she's probably trying to
draw a line that is
parallel to L.
A line through P parallel
to line L.
Yeah, that's what I think
she's trying to do.
All right, choice A.
55.
Given angle A.
So given this angle.
What is the first step in
constructing the angle
bisector of angle A?
OK, well actually I've
never done this.
But I can assume that
if I have a compass.
You know what a compass is,
it has those two points.
One of them is like
a pivot point.
It looks something like this.
It looks like it has a little
pivot point, and then on the
other side you can stick
your pencil.
And you can adjust it up here.
And the bottom line, you pivot
around this and then you can
draw circles of arbitrary
radiuses.
It seems like that's
what they did here.
So if I want to draw the angle
bisector of a, just thinking
about it, it seems I could put
the pivot point here, and then
I can put the pencil and
I can draw this circle.
And really, as long as I just
find the two points that it
intersects those two lines
or those two rays,
then I'll be fine.
And I could have done
it anywhere.
I could have done it here.
I could have done it out here.
They just picked
points B and C.
And then from each of those
points, you can
put your pivot here.
If you put your pivot here, and
then you were to draw a
circle around that, you
would have gotten
this one right here.
And then if you were to put your
pivot point right here,
draw a circle, you would
be able to draw that.
And then where they interect,
that would that would give you
an indication of where the
angle bisector is.
And you could then draw that
line to where they intersect.
So let's see, they say what is
the first step in constructing
the angle bisector of angle a.
So they say draw ray AD.
Well that seems like that
would be the last step.
Then you're done.
Draw AD, that is the
angle biector.
Draw a line segment connecting
points B and C.
No, that's useless.
You don't need a line segment.
I mean even what they have
drawn, that's an arc.
It's not a line.
From points B and C, draw
equal arcs that
intersect at D.
That was the second step.
You have to have points B and
C before you can draw those
equal arcs.
From point A, draw an arc that
intersects the side of the
angle at points B and C.
Yeah, that's what we said.
That was the first step.
Put your pivot here, and use
your pencil to draw the arc.
You say OK, this point
and this point.
So that would be
the first step.
D.
And I'm all out of problems
and I'm out of time.
See you in the next video.
And they say, a diagram from
a proof of the Pythagorean
theorem is pictured below.
And they they say, which
statement would not be used in
the proof of the Pythagorean
theorem?
So since they have drawn this
diagram out, I think we might
as well just kind of do the
proof and then we can look at
their choices and see which
ones kind of match
up to what we did.
Hopefully, they do
it the same way.
And this is a pretty
neat proof of
the Pythagorean theorem.
I don't think I've
done it yet.
So I might as well do it now.
Well, let's figure out
what the area of this
large square is right.
Well there's two ways to think
about it you could just say,
OK, this is a square.
That's a, that's b.
Well this is going
to be b as well.
This is going to be a as well.
So the area of the square is
going to be the length of one
of its sides squared.
So we could say the
whole square's
area is a plus b squared.
And that's equal to a squared
plus 2ab plus b squared.
Fair enough.
Now we can also say that the
area of this larger square,
and it's a bit of an optical
illusion, it looks like it's
tilted to the left because
of the way it's drawn.
But anyway, that the area of
this larger square is also the
area of these four triangles
plus the area of
this smaller square.
So this, the area of the
larger square, which we
figured out just by taking one
side of it and squaring it,
that should be equal
to the area of the
four smaller triangles.
So there's four of them.
And what's the area
of each of them.
Let's see, let's just
pick this one.
1/2 base times height.
So it's 1/2 times a times b.
So 1/2 ab is one of these and
I multiply by 4 to get all
four of these triangles.
And then we want to add the area
of this inside square.
And that's just going
to be c squared.
So plus c squared.
Let's see if we can
simplify this.
So you get a squared plus 2ab
plus b squared is equal to 4
times 1/2 is 2ab
plus c squared.
Well, we could subtract
2ab from both
sides of this equation.
The top and the bottom of
this equation the way
I've written it.
But if we do that, subtract 2ab
from there, subtract 2ab
from there, and you're left with
a squared plus b squared
is equal to c squared, which
is the Pythagorean theorem.
And we've proved it.
So let's see which of their
choices matches what we did.
OK, which statement would not
be used in the proof of the
Pythagorean theorem.
The area of a triangle
equals 1/2 ab.
We used that.
The four right triangles
are congruent.
No, we used that.
The area of the inner square is
equal to half of the area
of the larger square.
We didn't use that.
I think this is the
one that would not
be used in the proof.
Choice D, the area of the larger
square is equal to the
sum of the squares of the
smaller square and the four
congruent triangles.
No, that that was the
crux of the proof.
So we definitely used that.
So C is our answer.
That's the statement
that would not
be used in the proof.
I'm learning to copy and
paste ahead of time.
So I don't waste your time.
All right, a right triangle's
hypotenuse has length 5.
If one leg has length 2,
what is the length
of the other leg?
Pythagorean theorem, x squared
plus 2 squared is equal to 5
squared, because 5 is
the hypotenuse.
x squared plus 4
is equal to 25.
Subtract 4 from both sides.
x squared is equal to 21.
So x is equal to the
square root of 21.
So choice B.
Next question.
A new pipeline is being
constructed to reroute oil
flow around the exterior of a
national wildlife preserve.
I guess that's the national
wildlife preserve.
The plan showing the old
pipeline and the new route is
shown below.
OK, how many extra miles will
the oil flow once the new
route is establised.
So the new route is going to
be 60 miles plus 32 miles.
So the new route is 92 miles.
So what was the old route?
Well the old route was the
hypotenuse of this triangle.
So we could say, let's
call that x.
60 squared plus 32 squared
is equal to x squared.
Because that's the hypotenuse.
And these numbers, that's a bit
of a pain to deal with.
Maybe if I can factor out
something here I can make it
more interesting.
So I don't have to multiply out
60 squared and 32 squared
and all of the rest.
Well, let me see.
Both of those are
divisible by 4.
So then I would have 15 and 8.
Yeah, that still doesn't
make it that useful.
So I'll just multiply
them out.
So this is 3600.
At 32 squared, let's
see, 32 times 32.
2 times 32 is 64.
3 times 2 is 6.
3 times 3 is 9.
So it's 1024.
Plus 1024 is equal
to x squared.
So let me just switch
both sides.
x squared is equal to 3600
plus 1024 is 4624.
Let me see if I can get
an approximate.
So x is going to be the square
root of this thing right here.
So let's see if I can get
a handle at least on the
magnitude of where
this would be.
So 20 times 20 is 400.
So this is way too small.
60 times 60 is 3600.
So 68 times 68, this
looks right.
Especially because 8 times
8 should end in a 4.
Let me try that out.
68 times 68.
8 times 8 is 64.
8 times 6 is 48 plus 6 is 54.
6 times 8, 48.
6 times 6, 36 plus 4 is 40.
4624.
So x is 68.
Oh, I used 68, I shouldn't have.
Because they don't want
to know how long was
the old pipeline.
That's 68.
It just happened to be
one of the choices.
That's just to make
sure that you read
the question properly.
But they want to know how
much longer is going
to be the new pipeline.
So the new one was 92.
And the old one is 68.
Good thing they had that
number there so I
could try it out.
That was the square
root of 4624.
So how much longer
is the new one?
Well 92 minus 68 that's
24 miles.
So choice A.
Not choice B.
B is how long the old
pipeline was.
We want to know how much longer
the new route is.
That was tricky.
Well not tricky, but I kind of
fell for it by forgetting what
the question was about.
Anyway, next question.
Marcia is using a straightedge
and compass to do the
construction below.
Interesting.
Which best describes the
construction Marcia is doing.
So, I assume when they
say construction
she's drawing something.
Let's see what it looks like.
It looks she's taking her
compass, she's probably
putting one of the points
here, she put one of the
points there and then she
kind of drew this arc.
And then it looks like she put
the point there and then she
drew that arc.
And then she put the point
here and drew that arc.
And then put the point there
and drew that arc.
And the end result, it seems
like the reason why she picked
this point here is it goes
through this line L.
So she's probably trying to find
another point here, so
that she can draw
another line.
Because they say she
has a straightedge.
A straightedge is to
draw these lines.
A compass is to draw
these curves.
So if she were to draw another
line between these two points,
it looks something like
that, then she would
have parallel lines.
The reason why she would have
parallel lines is because
these would be corresponding
angles and
they would be congruent.
And so if you have a transversal
the corresponding
angles are congruent, you're
dealing with parallel lines.
So my read of this question is
that she's probably trying to
draw a line that is
parallel to L.
A line through P parallel
to line L.
Yeah, that's what I think
she's trying to do.
All right, choice A.
55.
Given angle A.
So given this angle.
What is the first step in
constructing the angle
bisector of angle A?
OK, well actually I've
never done this.
But I can assume that
if I have a compass.
You know what a compass is,
it has those two points.
One of them is like
a pivot point.
It looks something like this.
It looks like it has a little
pivot point, and then on the
other side you can stick
your pencil.
And you can adjust it up here.
And the bottom line, you pivot
around this and then you can
draw circles of arbitrary
radiuses.
It seems like that's
what they did here.
So if I want to draw the angle
bisector of a, just thinking
about it, it seems I could put
the pivot point here, and then
I can put the pencil and
I can draw this circle.
And really, as long as I just
find the two points that it
intersects those two lines
or those two rays,
then I'll be fine.
And I could have done
it anywhere.
I could have done it here.
I could have done it out here.
They just picked
points B and C.
And then from each of those
points, you can
put your pivot here.
If you put your pivot here, and
then you were to draw a
circle around that, you
would have gotten
this one right here.
And then if you were to put your
pivot point right here,
draw a circle, you would
be able to draw that.
And then where they interect,
that would that would give you
an indication of where the
angle bisector is.
And you could then draw that
line to where they intersect.
So let's see, they say what is
the first step in constructing
the angle bisector of angle a.
So they say draw ray AD.
Well that seems like that
would be the last step.
Then you're done.
Draw AD, that is the
angle biector.
Draw a line segment connecting
points B and C.
No, that's useless.
You don't need a line segment.
I mean even what they have
drawn, that's an arc.
It's not a line.
From points B and C, draw
equal arcs that
intersect at D.
That was the second step.
You have to have points B and
C before you can draw those
equal arcs.
From point A, draw an arc that
intersects the side of the
angle at points B and C.
Yeah, that's what we said.
That was the first step.
Put your pivot here, and use
your pencil to draw the arc.
You say OK, this point
and this point.
So that would be
the first step.
D.
And I'm all out of problems
and I'm out of time.
See you in the next video.
About this Video
Mar 06, 2025
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