WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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This lesson is on midpoints and segment congruence.
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We are going to talk about more segments.
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The definition of midpoint: this is very important, the definition of a midpoint.
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Midpoint M of PQ is the point right between P and Q, such that PM = MQ; that means PM is equal to MQ.
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So, if I say that M, the midpoint...it is the point right in the middle of P and Q, so it is the "midpoint," the middle point--this is M...
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then I can say that PM, this right here, is equal to MQ, because you are just cutting it in half exactly, so it is two equal parts.
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I can write little marks like that to show that this segment right here, PM, and QM are the same.
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That is the definition of midpoint; that means that, if PQ, let's say, is 20, and M is the midpoint of PQ, then PM is going to be 10; this is 10, and this is 10.
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So then, if it is the midpoint, then this part will have half the measure of the whole thing.
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OK, some formulas: the first one: this is on a number line--that is very important.
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Depending on where we are trying to find the midpoint, you are going to use different formulas.
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On the number line, the coordinates of the midpoint of a segment whose endpoints have coordinates a and b is (a + b) divided by 2.
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So, again, only on a number line: if I have a number line like this, say...this is 0; this is 1; 2, 3, 4, 5;
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if I want to find the midpoint between 1 and 5--so then, this will be a, and this will be b--
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then I just add up the two numbers, and I divide it by 2.
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That is the same...just think of it as average: whenever you try to find the midpoint on a number line, you are finding the average.
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You add them up, and you divide by 2: so it is just 1 + 5, divided by 2, which is 6/2, and that is 3; so the midpoint is right here.
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Now, if you are trying to find the midpoint on a coordinate plane, then it is different, because you have points;
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you don't have just numbers a and b--you have points.
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So, to find the midpoint with the endpoints (x₁,y₁) and (x₂,y₂),
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you are going to use this formula right here: remember from the last lesson: we also used (x₁,y₁)
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and (x₂,y₂) for the distance formula--remember that, for this one,
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it is not (x²,y²), because that is very different.
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This right here, these numbers, 1...it is just saying that this is the first point.
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And then, (x₂,y₂): it is the second point, because all points are (x,y).
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So, they are just saying, "OK, well, then, if this is (x,y) and this is (x,y)..." we are just saying that that is the first (x,y) and these are the second (x,y).
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You are given two points, and you have to find the midpoint.
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Then, you just take the average, so it is the same as this formula on the number line--
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you are just adding up the x's, dividing it by 2, and that becomes your x-coordinate;
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and then you add up the y's, and you divide by 2.
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You are just taking the average of the x's and the average of the y's.
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Remember: to find the average, you have to add them up and then divide by however many of them there are.
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So, in this case, we have two x's, so you add them up and divide by 2.
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For the y's, to find the average, you are going to add them up and divide by 2; and that is going to be your midpoint.
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So, to find the average, you are finding what is exactly in between them.
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Let's do a few examples: Use a number line to find the midpoint of AB.
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Here is A at -2, and B is at 8; and this is supposed to have a 7 above it.
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AB: to find the midpoint, to find the point that is right between A and B, I am going to add them up and divide by 2: so -2 + 8, divided by 2.
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Again, just think of midpoint as average; so add them up and divide by 2.
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It is going to be 6; -2 + 8 is 6, over 2; and then 3; so right here--that is the midpoint.
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You can kind of tell if it is going to be the right answer; that is the midpoint, the point right in the middle of those two.
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If I got 0 as my answer, well, 0 is too close to A and far away from B, so you know that that is not the right answer.
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The same thing if you get 5 or 6 or even 7--you know that that is the wrong answer.
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So, it should be right in the middle of those two points.
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OK, to find the midpoint of AB here--well, we know that we are not going to use the first one, (a + b)/2,
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because this is in the coordinate plane, and we have to use the second one, where we have (x₁,y₁) and (x₂,y₂).
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So, in this case, since they don't give us the points, we have to find the points ourselves.
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So, A is at (1,2); this is 1, 2, 3; and then, B is at (-1,-3).
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For the midpoint, it is the same concept as finding it on the coordinate plane.
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When you are finding the average, you are just finding the average of the x's; and then you find the average of the y's.
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You just have two steps: all I do is take...
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Now, it doesn't matter which one you label (x₁,y₁) and which is one is (x₂,y₂).
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So then, we could just make this (x₁,y₁), and this could be (x₂,y₂); it does not matter.
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You have to have this thing x and this thing y...both of these thing x's and both of these thing y's.
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Let's see: x₂ + x₁, or x₁ + x₂, is 2, plus -1, divided by 2;
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that is the average of the x's; comma; the average of the y's is going to be 3 + -3, over 2.
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Here we have 1/2; and 3 + -3 (that is 3 - 3) is 0; our midpoint is going to be at (1/2,0); so this right here is our midpoint.
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After you find the midpoint, kind of look at it and see if it looks like the midpoint.
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A couple more problems: If C is at (2,-1), that is the midpoint of AB, and point A is on the origin, find the coordinates of B.
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C is the midpoint of AB; so if I have AB there, and C is the midpoint right there (there is C), point A is on the origin; find the coordinates of B.
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Then C is (2,-1), and then A is (0,0): now this is obviously not what it looks like; it is on the coordinate plane, since we are dealing with points.
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But this is just so I get an idea of what I am supposed to be doing, because in this type of problem, they are not asking us to find the midpoint.
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They are giving us the midpoint, and they want us to find B--they want us to find one of the endpoints.
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So, we know the midpoint; so how do we solve this?
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The midpoint is (2,-1): well, the formula I know is...how did you get (2,-1)?--how did you get the midpoint?
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You do x₁ + x₂; you add up the x's and divide it by 2;
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and then to find the y-coordinate, you add up the y's, and you divide it by 2.
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And that is 2, and this is -1; so this is the formula to find the midpoint; this is the midpoint.
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All of this equals this, and all of this equals that.
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I can just say, if we make A (0,0)--if this is our (x₁,y₁)...then what is this going to be? (x₂,y₂), right?
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It is as if our coordinates for B are going to be the x₂ (this is what we are solving for)...
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we are solving for x₂, and we are solving for y₂.
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Those are the two points that we need.
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Going back to this--well, we know what x₁ was, and we know that this whole thing equals this whole thing.
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So, I am going to make this a 0, plus x₂/2; all of that equals 2.
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When you found the average of the x's, you got 2; but you just don't know what this value is.
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Then, if you solve for x₂, you get 4; so x₂ = 4.
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Then, you have to do the same thing for the y's; so you add up the y's; this plus this, divided by 2, equals -1.
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You just write that out; 0 + y₂, divided by 2, equals -1.
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And this is what I am solving for, again; so what is y₂? y₂ became -2, because you multiply the 2 over: -2.
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That means that our coordinates for B are (4,-2).
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So again, you can use the formula to find one of the endpoints, too.
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If they give you the midpoint, then just use it; you have to use this formula to come up with these values, too.
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So then, we know that the sum of the x's, divided by 2, is 2; so you do 0 + x₂ = 2.
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And then, the sum of the y's, divided by 2, is -1; so you do 0 + y₂, divided by 2, equals -1.
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OK, we will do another example later.
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The second one: E is the midpoint of DF; let me draw DF; and E is the midpoint.
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DE, this, is 4x - 1; and EF is 2x...this must be plus 9...find the value of x and the measure of DF.
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They want us to find x and the value of the whole thing.
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Since we know that E is the midpoint, we know that DE and EF are exactly the same.
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So, I can just take these two and make them equal to each other: 4x - 1 = 2x + 9.
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And then, you solve for x; subtract it over: 2x equals...you add 1...10...x equals 5.
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Then that is one of the things they wanted us to find: x = 5.
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And then, find the measure of DF; how do you find the measure of DF?
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Well, I have x, so I am able to find DE, or I can find EF.
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Let's plug it into DE: 4 times 5...you substitute 5 in for x...minus 1; there is 20 - 1, which is 19.
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If this is 19, then what does this have to be? 19.
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Just to double-check: 2 times 5 is 10, plus 9 is 19.
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You can just do 19 + 19, or you can do 19 times 2; you are going to get DF = 38, because it is this times 2...19 + 19...it is the whole thing.
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Segment bisector: Any segment, line, or plane that intersects a segment at its midpoint.
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A segment bisector is anything that cuts the segment in half.
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It bisects the segment, meaning that it cuts it in half; so bisecting just means that it cuts it in half; think of it that way.
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Here, these little marks mean that this segment and this segment are the same.
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CD is the segment bisector of AB, because CD is the one that cut AB in half.
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Now, the one that is doing the cutting--the one that is bisecting, or the one that is cutting in half--is the segment bisector.
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This bisected the segment AD; this is a line segment, CD; this segment bisector is a segment.
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It doesn't have to be just a segment; it could be a segment; it could be a line; it could be a plane.
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In this case, it is a segment; if you draw it out like this, it is still a segment bisector, because it is the segment that was bisected.
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The bisector can be anything: it can be in the form of a line, too.
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It can also be in the form of a plane; so if I have (I am a horrible draw-er, but) something like this, and it bisects it right there,
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then a plane could be a segment bisector, because it intersects the segment at its midpoint, point D.
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Again, a segment bisector is anything that cuts the segment in half, that bisects it, that intersects it at its midpoint.
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So, we are going to just talk a little bit about proofs, because the next thing we are going to go over is a theorem.
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And we have to actually prove theorems.
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Remember: we talked about postulates--how postulates are statements that we can just assume to be true,
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meaning that once they give us a postulate, then we can just go ahead and use it from there.
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Theorems, however, have to be proved or justified using definitions, postulates, and previously-proven theorems.
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So, whenever there is a theorem--some kind of statement--then it has to be backed up by something.
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It has to show why that is true; and then you prove it.
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And then, once it is proven, you can start using that theorem from there on, whenever you need to.
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Now, a proof is a logical argument in which each statement you make is backed up by a statement that is accepted as true.
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You can't just give a statement; you have to back it up with a reason: why is that statement true?
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And that is what a proof is.
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Now, there are different types of proofs; the one that is used the most is called a two-column proof.
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But we are going to go over that, actually, in the next chapter.
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The paragraph proof is one type of proof, in which you write a paragraph to explain why a conjecture for a given situation is true.
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So, you are just explaining in words why that is true.
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And a conjecture is an if/then statement: if something is true, then something else.
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So, actually, you are going to go over that more in the next chapter, too.
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But that is all you are doing; for a paragraph proof, you are just explaining it in words.
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So, instead of a two-column proof, where you are listing out each statement, and you are giving the reasons for that,
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to prove something, in a paragraph proof, you are just writing it as a paragraph to prove that a theorem is true.
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The first proof that we are going to do is going to be a paragraph proof.
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And that is to prove the midpoint theorem; so again, this is a theorem; it is not a postulate; so we can't just assume that this statement is true.
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The midpoint theorem says that, if M is the midpoint of AB (here is AB), then AM is congruent to MB.
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Now, we know, from the definition of midpoint, that if M is the midpoint of AB, then they have equal measures; that is the definition of midpoint.
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And that is just talking about the measures of them.
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But "congruent"--to show that AM is congruent to MB--that is the theorem, and we have to prove that first.
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Given that M is the midpoint of AB, write a paragraph proof to show that AM is congruent to MB.
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And then, the only purpose of this proof that we are going to do right now is to prove that this midpoint theorem is true.
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And then, from there, we can just use it.
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We are going to always start with the given: given that M is the midpoint of AB--that is the starting point.
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Let's just write it out: From the definition of midpoint, we know that, since M is the midpoint of AB, AM is equal to MB.
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Now, that means that AM and MB have the same measure--that AM has the same measure as MB.
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Then, by the definition of congruence...the definition of congruence is just when you switch it from equal to congruent, or from congruent to equal.
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So, by the definition of congruence, if AM is equal to MB, then (and I can just switch it over to congruent), AM is congruent to MB.
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They are congruent segments--something like that.
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You don't have to write exactly the same thing, but you are just kind of showing that we know
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that we went over the definition of midpoint, and that is AM = MB.
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And then, from there, you use the definition of congruence to show that AM is congruent to MB.
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They are congruent segments--that is what the definition of congruence says.
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Now that we have proven that the midpoint theorem is correct, or is true, then now, from now on,
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for the remainder of the course, you can just use it whenever you need to--the midpoint theorem.
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Extra Example 1: Use a number line to find the midpoint of each: BD.
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We are going to find the midpoint of BD; again, to find the midpoint, you want to find the point that is right in the middle.
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So, you are going to add up the two points and divide it by 2; so it is -2 + 9, divided by 2; that is 7/2.
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You can leave it like that, or you can just write 3 and 1/2.
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Between -2 and 9, it is going to be right there: three and a half.
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CB: you know, I shouldn't write it like this, because it looks like BD...this looks like distance.
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You can just write the midpoint of BD: so just be careful--don't write BD; don't write it like that.
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Just write midpoint of BD, or...I am just going to write the number 2, just so that we know that that is the midpoint of CB.
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CB is right here; now, you can go from B to C, or we can go from C to B.
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It doesn't matter; if I go from C to B, 4 + -2, over 2, is going to be 2 divided by 2, which is 1.
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That means the midpoint from here, C, to B, is 1, right there.
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You can also check; you can count; this is three units, and then this is three units, so it has to be the same.
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And then, number 3: AD: A is -5; D is 9; divided by 2...this is 4, divided by 2, which is 2.
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Here is -5, and here is 9; the midpoint is right here, between those two...from here to here, and from here to here.
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OK, draw a diagram to show each: AC bisects BD.
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That means that this is the one that is doing the bisecting; this is the segment bisector.
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This is the segment that is getting bisected: so BD...here is BD.
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Now, it doesn't have to look like mine, just as long as BD is getting bisected, or AC is intersecting BD at BD's midpoint.
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You can just...if I say that this is the midpoint, then AC is the one that is cutting it like this; it is going to be A, and then C (it cuts it)...
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OK, the next one: HI is congruent to IJ.
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HI: see how this is congruent...this is part of the midpoint theorem...HI and IJ.
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Here is HI; I has to be that in the middle; and J; so HI is congruent to IJ, and that is how you would want to show it; OK.
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The next one: RT equals half of PT--let me just draw another segment; RT is equal to half of PT.
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Now, look at what they have in common; here is T, and here is T.
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Now, if RT is equal to half of PT, that means that you have to divide PT by 2 to get RT.
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That means that the whole thing is going to be PT, because, if you have to divide it by 2, you cut it in half, meaning at its midpoint.
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And that is going to be RT; that means R is right here--the midpoint.
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Again, here is PT; so for example, if PT is 12 (say this whole thing, PT, is 12),
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then you divide it by 2, or you multiply it by 1/2; then you get 6; that means RT has to equal 6.
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And you don't have to write the numbers; just draw the diagram.
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You could have just left it at P-R-T, or just showed it, maybe, like this.
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The next example: Q is the midpoint of RS; if two points are given, find the coordinates of the third point.
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RS's midpoint is Q; I'll show that; if two points are given, find the coordinates of the third point.
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R is here; R is at (4,3); S is at (2,1); you have to find the midpoint.
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To find the midpoint, you are going to take the sum of the x's, divided by 2, and that is going to be your x-coordinate.
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So, x₁ + x₂, divided by 2, equals your x-coordinate for the midpoint.
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4 + 2, divided by 2...and then you are going to take the y's, 3 + 1, divided by 2;
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that is going to be 6 divided by 2, which is 3; so Q is 3, comma...4 divided by 2 is 2;
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it is going to be at (3,2); there is the midpoint for that one.
00:30:51.100 --> 00:31:12.500
The next one: let me just redraw R-Q-S...Q...on this one, they give you the midpoint, and they give you this.
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This right here is going to be...you could make this (x₁,y₁), or (x₂,y₂).
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And then, when you write it out, it is going to be -5 + x₂ over 2, and then 2 + y₂, divided by 2.
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Now, we know that all of this equals...the midpoint is (-2,-1).
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That means that these equal each other and these equal each other.
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This is going to give you -2; and this is what we are solving for.
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I can just make this thing equal to this thing: -5 + x₂, over 2.
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Now, this is not x²; be careful of that.
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That equals -2; this is going to be -5 + x₂ equals...you multiply the 2 over to the other side...-4.
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You add the 5 over; so x₂ = 1.
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I am going to do the y over here; then you make this equal to -1; it is 2 + y₂, over 2, equals -1.
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You multiply the 2 over; 2 + y₂ = -2; y₂ = -4.
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All right, S is going to be (1,-4); that is to find this right here.
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With this one, we had to find Q, the midpoint; and with this one, we had to find S.
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The last example: EC bisects AD; that means that EC is the one is doing the bisecting, and AD is the one that got bisected.
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That means that AD is the one that got cut in half at C.
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That means that this whole thing, AC, and CD are congruent.
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EF bisects...this is supposed to be a line...and let's make this F, right here; that means that line EF bisects AC at B.
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That means that AC is bisected; B is the midpoint.
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For each, find the value of x and the measure of the segment.
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That means AC and CD are the same, and then AB and BC are the same.
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AB equals 3x + 6; BC equals 2x + 14; they want you to find AC.
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Again, AB and BC have the same measure; so I can just make them equal to each other...it equals 2x + 14.
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Subtract the 2x over here; subtract the 6 over there; you get 8.
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And then, find the value of x and the measure of the segment.
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The segment right here, AC, is AB + BC; or it is just AB times 2, because it is doubled.
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So, AC...here is my x, and AC equals...we will have to find AB first--or we can find BC; it doesn't matter.
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AB is 3 times 8, plus 6; that is 24; that is 30.
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24 plus 6 is 30; and then, AC is double that, so AC is 60.
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AB is 30; that means that AC is 60.
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AD, the whole thing, is 6x - 4; AC is 4x - 3; and you have to find CD.
00:36:24.300 --> 00:36:32.400
Now, remember how EC bisected AD; that means that AD is cut in half; C is the midpoint.
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AD, the whole thing, is 6x - 4; and then, AC is 4x - 3.
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I can do this two ways: AC is 4x - 3--that means DC, or CD, is 4x - 3.
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So, I can just do 4x - 3 + 4x - 3, or I can do (4x - 3) times 2; or we can do the whole thing, AD, 6x - 4, minus this one.
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I am just going to do 2 times (4x - 3), because AC is 4x - 3, and AD is double that.
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Now, even though this is 60, so you might assume that this whole thing is 120; this is a different problem.
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So then, it is not going to have the same measure.
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2 times AC equals AD, which is 6x - 4.
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If we continue it here, it is going to be 8x (and this is the distributive property), minus 6, equals 6x minus 4.
00:37:55.600 --> 00:38:02.900
This becomes 2x; if you add the 6 over there, it equals 2; x = 1.
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So then, here is the x-value; and then, they want you to find CD, this right here.
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Since CD is the same thing as AC, I can just find AC.
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AC is 4 times 1, minus 3; so AC is 4 minus 3, is 1; so CD is 1; AC is 1; CD is 1.
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OK, the last one: AD, the whole thing, is 5x + 2; and BC is 7 - 2x; find CD.
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Now, this one is a little bit harder, because they give you AD, the whole thing, and they give you BC.
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Now, remember: if C is the midpoint of AD, that means that this whole thing and this whole thing are the same.
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Let's look at this number right here; if this is 60, then this will also be 60.
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That means that the whole thing together is going to be 120.
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Then this right here is 30; this right here is 30.
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It is as if you take a piece of paper and you fold it in half; if you fold it in half, that is like getting your C, your midpoint.
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Then, you take that paper, and you fold it in half again; so then, that is when you get point B.
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So, now your paper is folded into how many parts? 1, 2, 3, 4.
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One of these, AB...if you compare AB or, let's say, BC, to the whole thing, AD, then this is a fourth of the whole thing.
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So, this into four parts becomes the whole thing; so you can take BC and multiply it by 4, because this is one part, 2, 3, 4.
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BC times 4: 7 - 2x...multiply it by 4, and you get the whole thing, AD.
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This is 28 - 8x = 5x + 2; I am going to subtract the 2 over; then I get 26; I am going to add the 8x over, and I get 13x; so x is 2.
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And then, find CD: CD was 2 times BC, because BC with another BC is going to equal CD.
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I am just going to find BC: 7 - 2(2)...just plug in x...so 7 - 4 is 3.
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If BC is 3...now, these numbers right here; that was actually for number 1;
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those are the values for number 1, and then I just used it as an example for number 3.
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But don't think that these are the actual values (30 for BC, and then 120 for the whole thing); this is a different problem.
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BC was 3; then what is CD? CD is 2 times BC--it is double BC, so CD is 6.
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OK, well, that is it for this lesson; thank you for watching Educator.com.