Learn about CA Geometry: Pythagorean Theorem, Compass Constructions Video

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Learn about CA Geometry: Pythagorean Theorem, Compass Constructions

We’re on problem 51. And they say a diagram of A proof of the Pythagorean theorem is pictured below. And they say, which statement would not be used in the proof of the Pythagorean Theorem. Since they’ve drawn this diagram out, I think we must as well just 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. So, 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. So, the way they’ve drawn in – the big insight here is – well, let’s figure out the area of this large square is, right? Well, there’s two way to think about it. You can just say, “Okay. This is 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 squares 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, 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, but the area of this larger square is also the area of these four triangles plus the area of this smaller square. So, this – there, 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. That’s one half base. Let’s just pick this one. One half base times height. So it’s one half times A times B. So, it’s one half AB. So one half AB is one of these and I multiply it by 4 to get all 4 of these triangles. And then we want to add the area of this inside square. And that’s just going to be C squared, right. This side of C, that side of C, so plus C squared. And let’s see if we can simplify this. So, you get A squared plus 2AB plus B squared s equal to 4 times one half is 2AB plus C squared. Well, we could subtract 2AB from both sides of this equation, right? I’ve worked it to the top and bottom of this equation though we have written it. But if we do that, you get subtract 2AB from there. Subtract 2AB from there and you are 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. Which statement would not be used in the proof of the Pythagorean Theorem? The area of a triangle equals one half AB. No, we used that. We had 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. Now, that’s the cracks of the proof. So, we definitely use that. So, C is our answer. That’s the statement that would not be used in the proof. I’m learning the copy and paste ahead of time. And 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? Okay. So, this is 5, 2 and they want to 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. That’s 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 Reserve. The plan showing the old pipeline and the new route is shown below. Okay. How many extra miles will the oil flow once the new route is established? 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? That was the old route. 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, these are a bit of a pain to deal with. Maybe if I can factor out something here, I could 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 just see if I factored out, both of those are divisible by 4, right? Both of those are divisible by 4. And so, 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. 60 squared is 3600. And 32 squared, let see, 32 times 32. 2 times 32 is 64. 3 times 2 is 6. 3 times 3 is 9. So, it’s 4 -12 -1024 plus 1024 is equal to x squared. So let me just wish both sides. X squared is equal to 3600 plus 1024 is 4624. And let me see if I can get an approximate – let’s see 20 times – so, x is going to be the square root of this thing right here. So let’s see if I can get a hand on the magnitude on where this would be. So, 24 times 24, you know, that’s 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 time 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, if 4 , 12, 6, 46, 24, right? So, x is equal to 68. Oh, you know what? I used 68. I should have because they don’t want to know how long was the old pipeline, that’s 68. And that 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. And good thing they have that number there so I could try it out. Now, that is the square root of 4624. So, how much longer is the new one? Well, 92 minus 68 is what? That’s 24 miles, right? 92 minus 68, yeah 24. So, it’s choice A, not choice B. B is how long the old pipeline was. We want to know how much longer the new route is. You know, that was tricky. Well, not tricky, but I kind of fell for it, right? Forgetting wahat the question was about. Anyway, next question. Marsha is using a straight edge in compass to do the construction below. Interesting. Which best described the construction Marsha is doing. So, I assume when they say construction, there, she’s drawing something. And let’s see what is looks like. It looks like she’s taking her compass. She’s finally 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, you know, the line – the reason why she picked this point here is that 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 straight edge. The straight edge is to draw these lines. The compass is to draw these curves. So, she would have drawn another line between these two points. If she would’ve drawn another line between those two points, if you know, it looks something like that, then she would have parallel lines. And the reason why she should 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 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? Okay. And so, this is, you know – 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, you know, it’s like a pivot point. It looks something like this. It looks like you know, it has a pivot point and then on the other side, you can stick your pencil. And the bottom line is that you can adjust it up here. And at the bottom, you pivot around this and you can draw circles of arbitrary radiuses, right? It seems that like with that, that’s what they did here. So, if I wanted to draw the angle bisector of A, just thinking about it, it seems I could put the pivot point here. And then, I could point the put the pencil on and I can draw this circle. I can draw this circle. And it really – as long as I just, you know find the two points that it intersect 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. And they just picked the 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 draw a circle around that, you would have gotten this one, right here, right? And if you were to put your pivot point around here and draw a circle, you would be able to draw that. And then, what they intersect, that would give you the indication of where the angle bisector is, right? And you could then, draw that line to where they intersect. So, let’s see. They say, what is the first in constructing the angle bisector of angle A? So, they say, draw ray AD. Well now, that seemed like that would be the step. Then, you’re done, right? Draw AD. That is the angle bisector. Draw a line segment connecting points B and C. A line segment – No, that’s useful. I mean, useless. You don’t need a line segment. I mean, this – even what they have drawn, that’s an arc. It’s not a line. From point B and C, draw equal arcs that intersect at D. That was the second step. I mean, you have to have points B and C before you can draw those equal arcs. From point A, draw an arc that intersects a side of the angle at points B and C – 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 this arc. And you say, “Okay. This point and this point.” So that would be the first step, D. And Oh, I’m all out of problems and I’m out of time. See you in the next video.

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We’re on problem 51. And they say a diagram of A proof of the Pythagorean theorem is pictured below. And they say, which statement would not be used in the proof of the Pythagorean Theorem. Since they’ve drawn this diagram out, I think we must as well just 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. So, 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.

So, the way they’ve drawn in – the big insight here is – well, let’s figure out the area of this large square is, right? Well, there’s two way to think about it. You can just say, “Okay. This is 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 squares 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, 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, but the area of this larger square is also the area of these four triangles plus the area of this smaller square. So, this – there, 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. That’s one half base. Let’s just pick this one. One half base times height. So it’s one half times A times B. So, it’s one half AB. So one half AB is one of these and I multiply it by 4 to get all 4 of these triangles. And then we want to add the area of this inside square. And that’s just going to be C squared, right. This side of C, that side of C, so plus C squared. And let’s see if we can simplify this. So, you get A squared plus 2AB plus B squared s equal to 4 times one half is 2AB plus C squared. Well, we could subtract 2AB from both sides of this equation, right? I’ve worked it to the top and bottom of this equation though we have written it. But if we do that, you get subtract 2AB from there. Subtract 2AB from there and you are 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. Which statement would not be used in the proof of the Pythagorean Theorem? The area of a triangle equals one half AB. No, we used that. We had 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. Now, that’s the cracks of the proof. So, we definitely use that. So, C is our answer. That’s the statement that would not be used in the proof.

I’m learning the copy and paste ahead of time. And 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? Okay. So, this is 5, 2 and they want to 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. That’s 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 Reserve. The plan showing the old pipeline and the new route is shown below. Okay. How many extra miles will the oil flow once the new route is established? 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? That was the old route.

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, these are a bit of a pain to deal with. Maybe if I can factor out something here, I could 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 just see if I factored out, both of those are divisible by 4, right? Both of those are divisible by 4. And so, 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. 60 squared is 3600. And 32 squared, let see, 32 times 32. 2 times 32 is 64. 3 times 2 is 6. 3 times 3 is 9. So, it’s 4 -12 -1024 plus 1024 is equal to x squared. So let me just wish both sides. X squared is equal to 3600 plus 1024 is 4624. And let me see if I can get an approximate – let’s see 20 times – so, x is going to be the square root of this thing right here. So let’s see if I can get a hand on the magnitude on where this would be. So, 24 times 24, you know, that’s 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 time 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, if 4 , 12, 6, 46, 24, right? So, x is equal to 68.

Oh, you know what? I used 68. I should have because they don’t want to know how long was the old pipeline, that’s 68. And that 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. And good thing they have that number there so I could try it out. Now, that is the square root of 4624. So, how much longer is the new one? Well, 92 minus 68 is what? That’s 24 miles, right? 92 minus 68, yeah 24. So, it’s choice A, not choice B. B is how long the old pipeline was. We want to know how much longer the new route is. You know, that was tricky. Well, not tricky, but I kind of fell for it, right? Forgetting wahat the question was about.

Anyway, next question. Marsha is using a straight edge in compass to do the construction below. Interesting. Which best described the construction Marsha is doing. So, I assume when they say construction, there, she’s drawing something. And let’s see what is looks like. It looks like she’s taking her compass. She’s finally 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, you know, the line – the reason why she picked this point here is that 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 straight edge. The straight edge is to draw these lines. The compass is to draw these curves. So, she would have drawn another line between these two points. If she would’ve drawn another line between those two points, if you know, it looks something like that, then she would have parallel lines. And the reason why she should 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 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? Okay. And so, this is, you know – 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, you know, it’s like a pivot point. It looks something like this. It looks like you know, it has a pivot point and then on the other side, you can stick your pencil. And the bottom line is that you can adjust it up here. And at the bottom, you pivot around this and you can draw circles of arbitrary radiuses, right? It seems that like with that, that’s what they did here. So, if I wanted to draw the angle bisector of A, just thinking about it, it seems I could put the pivot point here. And then, I could point the put the pencil on and I can draw this circle. I can draw this circle. And it really – as long as I just, you know find the two points that it intersect 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. And they just picked the 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 draw a circle around that, you would have gotten this one, right here, right? And if you were to put your pivot point around here and draw a circle, you would be able to draw that. And then, what they intersect, that would give you the indication of where the angle bisector is, right? And you could then, draw that line to where they intersect. So, let’s see.

They say, what is the first in constructing the angle bisector of angle A? So, they say, draw ray AD. Well now, that seemed like that would be the step. Then, you’re done, right? Draw AD. That is the angle bisector. Draw a line segment connecting points B and C. A line segment – No, that’s useful. I mean, useless. You don’t need a line segment. I mean, this – even what they have drawn, that’s an arc. It’s not a line. From point B and C, draw equal arcs that intersect at D. That was the second step. I mean, you have to have points B and C before you can draw those equal arcs. From point A, draw an arc that intersects a side of the angle at points B and C – 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 this arc. And you say, “Okay. This point and this point.” So that would be the first step, D. And Oh, I’m all out of problems and I’m out of time. See you in the next video.

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