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Цаг: 0:00Нийт үргэлжлэх хугацаа:8:46

a lot of what geometry is about is proving things about the world in order to really prove things about the world we have to be very careful very precise very exact with our language so that we know what we're proving and we know what we're assuming and and what type of deductions we are making as we prove things and so to get some practice being precise and exact with our language I'm going to go through some exercises from the geometric the geometric definitions exercise on Khan Academy so this first one says three students attempt to define what it means four lines L and M to be perpendicular can you match the teachers comments to the definitions all right so it looks like three different students attempt definitions of what it means to be perpendicular and then there's these teachers comments that we can move around so we're gonna I guess pretend that we're the teacher so Ruby's definition for being perpendicular L and M lines L and M are perpendicular if they never meet well that's not true in fact perpendicular lines for sure will intersect so the fact they intersect at right angles so that is not going to be that's not going to be correct and so actually this looks right were you thinking of parallel lines because that looks like what she was trying to trying to define if things are on the same plane and they never intersect then you are talking about parallel lines I also get shriya's definition L and M are perpendicular if they meet at one point and one of the angles at their point of intersection is a right angle well that seems spot-on so let's see I would say ooh nice work I couldn't have said it better myself now let's just make sure this with this comment matches for this definition Abhishek says L and M are perpendicular if they meet at a single point such that the two lines make a tea well that's in a hand wavy way kind of right when you when you imagine a perpendicular lines you could imagine them kind of forming a cross or I guess part of you know a tea would be part of it but I think this comment is spot-on the teachers comment your definition is kind of correct but it lacks mathematical precision you know what are they what does he mean by a T what is a that what does it mean to make a T frias definition is much more precise so perpendicular if they meet at one point and one of their angles at their point of intersection is a right angle is a 90 degree angle let's check our answer so let's do a few more of these this is actually a lot of fun so once again we're gonna have three students attempting to define but now they're going to define an object called an angle can you match the teachers comments to the definitions so Ruby of these three same three students Ruby says the amount of turn between two straight lines that have a common vertex well this is kind of getting there the definition of an angle we typically talk about two rays with a common vertex she's talking about two lines with a common vertex and she's talking about the amount of turns so she's really talking about more of kind of the measure of an angle so let's see what what comment here so you seem to be getting at the idea of a measure of an angle and not the definition of an angle itself so this is actually right I would put this one right here we just got lucky this this was already aligned so shriya's definition two lines that come together so once again this is kind of the definition of an angle is two raised with a common vertex so two lines that come together this is just intersecting lines I want that happens you might be forming some angles but I would just say were you thinking of intersecting lines and let's see what Abby Shaikh says a figure composed of two rays sharing a common endpoint the common endpoint is known as the vertex yep that's that's a good definition of an angle so Abby she ate God at this time let's do another one so three students are now attempting to define what it means for two lines to be parallel so now let's match the teachers comments so Daniella says two lines are parallel if they are distinct and one can be translated on top of the other all right so that actually seems pretty interesting that's actually not the first way that I would have defined parallel line notice today if they're on the same plane and they don't intersect then they are parallel but this is this seems pretty good because if you're translating something you're not you're not you you aren't going to rotate it you're not going to change its direction I guess one way to think about it and so if you're translating one if you can Tran if there if there are two different lines but you can you can shift them without changing their direction which is what translation is all about on top of each other that actually feels pretty good so I'll put that right over there so Ori says two lines are parallel if they are close together but don't intersect so if you're trying to define parallel lines parallel lines it doesn't matter if they're close together or not it's they just have to be in the same plane and not intersect they could be very far apart and they could still be parallel so this isn't an incorrect statement you could have two lines that are close together and don't intersect on the same plane and they are going to be parallel but this isn't a good definition because you can also have parallel lines that are far apart and so actually I'd go at the statement right here part of your definition is correct but the other part is not parallel lines don't have to be close together so this isn't a good definition of parallel lines and let's see Kaori two lines are parallel as long as they aren't perpendicular well that's just not true because you can intersect you can have two lines that intersect at non right angles and they're not parallel and they're also not perpendicular so this is you know sorry your definition is incorrect this is actually a lot of fun pretending to be the teacher let's do another one all right so three students attempt to define what a line segment is and we have a depiction of a line segment right over here it's we have point P point Q and the line segment is all the points in between P and Q so so let's let's match the teachers comments to the definitions Ivy's definition all of the points in line with P and Q extending infinitely in both directions well that would be the definition of a line that would be the line PQ that would be if you're extending in both directions so I would say are you thinking of a line instead of a line segment Ethan's definition the exact distance from P to Q well that's just a that's just the the that's the length of a line segment that's not exactly what a line segment is let's see if buca's definition the points P and Q which are called endpoints and all of the points in a straight line between points P and Q yep that looks like a good definition for a line segment and so we can just check R we can just check our answer so looking good let's do one more of this I'm just really enjoying pretending to be a teacher all right three students attempted to find what a circle is define what a circle is can you match the teachers comments to the definitions Daru the set of all points in a plane that are the same distance away from some given point which we call the center that actually seems like a pretty good definition of a circle so stupendous well done Oliver's Devon Oliver's definition the set of all points in 3d space that are the same distance from a Centerpoint well for talked about 3d space and the set of all points that are equidistant from that point in 3d space now we're talking about a sphere not a circle and so you seem to be confusing a circle with this year and then finally a perfectly round shape well that's kind of true but if you don't want three dimensions you could be talking about a sphere if you talk about you know if you go beyond three dimensions hyper sphere or whatever else in two dimensions yeah a perfectly round shape most people would call it a circle but that doesn't have a lot of precision to it it doesn't have it doesn't give us a lot that we can work with from a mathematical point of view so I would say actually what the teacher saying your definition needs to be much more precise Durer's definition is is is much much more precise the set of all points that are equidistant from that in a plane that are equidistant away from a given point which we call the center so yeah Carlos could use a little bit more precision and we're all done