CA Geometry: Pythagorean Theorem, Compass Constructions 51-55, Pythagorean Theorem, compass constructions
CA Geometry: Pythagorean Theorem, Compass Constructions
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- We're on problem 51.
- And they say, a diagram from the proof or from a proof of the Pythagorean theorem is pictured below.
- And they said, 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, hopefully, they do it the same way.
- So this, this is a pretty neat proof
- of the Pythagorean, Pythagorean theorem. I don't think I've done it yet. So I might as well do it now.
- So, the way they've drawn it the big inside here is,
- 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...
- ...right? So
- we could say (a) plus (b), the whole square's area,
- is (a) plus (b) squared.
- ..(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,
- that's bit of an optical illusion, 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 the...
- 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 it's 1/2 base, lets just pick this one.
- 1/2 base times height. So it's 1/2
- times (a) times (b).
- So it's 1/2 ab. So 1/2 ab is one of these and I multiplied it by 4 to get all four of these
- And then we want to add the area of this inside, this inside square.
- That's just going to be (c) squared, right? this side is (c), that side is (c),
- So plus (c) squared.
- Now 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, right?
- I, I worked at the top and the bottom of this equation the way I've written it.
- But if we do that, you get 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.
- 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.
- No, we used that;
- we have to use 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.
- No, we didn't use that. I think this is the one that
- would not be used in the proof. Let's see
- 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's, that was the crux of the, 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.
- and I'm,
- I'm learning to copy and paste ahead of time.
- So I don't waste your time.
- Alright, a right triangle's hypotenuse has length 5.
- If one leg has length 2, what is the length of the other leg?
- ok, so this is, so its 5, 2, and they wanna know 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, right?
- So we could say, 60, 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, uh,
- 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 let me just, see if i can .
- if I factored out
- Both of those are divisible by 4.
- right? 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.
- lets see, twenty times,
- 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.
- So x is 68. So x is equal to 68.
- Oh, you know what? 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, right?
- 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, right? Yeah, 24. 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.
- 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,
- if, you know, 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.
- Given angle A.
- So given this angle.
- What is the first step in constructing the angle
- bisector of angle A?
- what is the first step constructing angle bisector
- OK, this is, 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. right?
- 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 intersect, that would that would give you
- 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.
- And I'm all out of problems and I'm out of time.
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