Geometry Midpoints and Segment Congruence

Welcome back to Educator.com.0000

This lesson is on midpoints and segment congruence.0002

We are going to talk about more segments.0007

The definition of midpoint: this is very important, the definition of a midpoint.0010

Midpoint M of PQ is the point right between P and Q, such that PM = MQ; that means PM is equal to MQ.0017

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...0033

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.0049

I can write little marks like that to show that this segment right here, PM, and QM are the same.0058

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 10.0069

So then, if it is the midpoint, then this part will have half the measure of the whole thing.0084

OK, some formulas: the first one: this is on a number line--that is very important.0093

Depending on where we are trying to find the midpoint, you are going to use different formulas.0101

On the number line, the coordinates of the midpoint of a segment whose endpoints have coordinates a and b is (a + b) divided by 2.0106

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;0116

if I want to find the midpoint between 1 and 5--so then, this will be a, and this will be b--0132

then I just add up the two numbers, and I divide it by 2.0140

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.0145

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.0151

Now, if you are trying to find the midpoint on a coordinate plane, then it is different, because you have points;0171

you don't have just numbers a and b--you have points.0181

So, to find the midpoint with the endpoints (x₁,y₁) and (x₂,y₂),0184

you are going to use this formula right here: remember from the last lesson: we also used (x₁,y₁)0193

and (x₂,y₂) for the distance formula--remember that, for this one,0200

it is not (x²,y²), because that is very different.0205

This right here, these numbers, 1...it is just saying that this is the first point.0208

And then, (x₂,y₂): it is the second point, because all points are (x,y).0214

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).0221

You are given two points, and you have to find the midpoint.0231

Then, you just take the average, so it is the same as this formula on the number line--0235

you are just adding up the x's, dividing it by 2, and that becomes your x-coordinate;0242

and then you add up the y's, and you divide by 2.0248

You are just taking the average of the x's and the average of the y's.0251

Remember: to find the average, you have to add them up and then divide by however many of them there are.0254

So, in this case, we have two x's, so you add them up and divide by 2.0260

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.0266

So, to find the average, you are finding what is exactly in between them.0272

Let's do a few examples: Use a number line to find the midpoint of AB.0279

Here is A at -2, and B is at 8; and this is supposed to have a 7 above it.0287

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 2.0298

Again, just think of midpoint as average; so add them up and divide by 2.0317

It is going to be 6; -2 + 8 is 6, over 2; and then 3; so right here--that is the midpoint.0324

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.0337

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.0345

The same thing if you get 5 or 6 or even 7--you know that that is the wrong answer.0353

So, it should be right in the middle of those two points.0358

OK, to find the midpoint of AB here--well, we know that we are not going to use the first one, (a + b)/2,0363

because this is in the coordinate plane, and we have to use the second one, where we have (x₁,y₁) and (x₂,y₂).0377

So, in this case, since they don't give us the points, we have to find the points ourselves.0390

So, A is at (1,2); this is 1, 2, 3; and then, B is at (-1,-3).0394

For the midpoint, it is the same concept as finding it on the coordinate plane.0412

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.0418

You just have two steps: all I do is take...0424

Now, it doesn't matter which one you label (x₁,y₁) and which is one is (x₂,y₂).0429

So then, we could just make this (x₁,y₁), and this could be (x₂,y₂); it does not matter.0436

You have to have this thing x and this thing y...both of these thing x's and both of these thing y's.0446

Let's see: x₂ + x₁, or x₁ + x₂, is 2, plus -1, divided by 2;0456

that is the average of the x's; comma; the average of the y's is going to be 3 + -3, over 2.0467

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.0479

After you find the midpoint, kind of look at it and see if it looks like the midpoint.0499

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.0511

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.0529

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.0543

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.0555

They are giving us the midpoint, and they want us to find B--they want us to find one of the endpoints.0562

So, we know the midpoint; so how do we solve this?0570

The midpoint is (2,-1): well, the formula I know is...how did you get (2,-1)?--how did you get the midpoint?0573

You do x₁ + x₂; you add up the x's and divide it by 2;0584

and then to find the y-coordinate, you add up the y's, and you divide it by 2.0590

And that is 2, and this is -1; so this is the formula to find the midpoint; this is the midpoint.0599

All of this equals this, and all of this equals that.0610

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?0617

It is as if our coordinates for B are going to be the x₂ (this is what we are solving for)...0633

we are solving for x₂, and we are solving for y₂.0640

Those are the two points that we need.0644

Going back to this--well, we know what x₁ was, and we know that this whole thing equals this whole thing.0649

So, I am going to make this a 0, plus x₂/2; all of that equals 2.0657

When you found the average of the x's, you got 2; but you just don't know what this value is.0670

Then, if you solve for x₂, you get 4; so x₂ = 4.0681

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 -1.0689

You just write that out; 0 + y₂, divided by 2, equals -1.0703

And this is what I am solving for, again; so what is y₂? y₂ became -2, because you multiply the 2 over: -2.0714

That means that our coordinates for B are (4,-2).0728

So again, you can use the formula to find one of the endpoints, too.0741

If they give you the midpoint, then just use it; you have to use this formula to come up with these values, too.0748

So then, we know that the sum of the x's, divided by 2, is 2; so you do 0 + x₂ = 2.0759

And then, the sum of the y's, divided by 2, is -1; so you do 0 + y₂, divided by 2, equals -1.0768

OK, we will do another example later.0778

The second one: E is the midpoint of DF; let me draw DF; and E is the midpoint.0782

DE, this, is 4x - 1; and EF is 2x...this must be plus 9...find the value of x and the measure of DF.0799

They want us to find x and the value of the whole thing.0816

Since we know that E is the midpoint, we know that DE and EF are exactly the same.0821

So, I can just take these two and make them equal to each other: 4x - 1 = 2x + 9.0830

And then, you solve for x; subtract it over: 2x equals...you add 1...10...x equals 5.0838

Then that is one of the things they wanted us to find: x = 5.0847

And then, find the measure of DF; how do you find the measure of DF?0853

Well, I have x, so I am able to find DE, or I can find EF.0858

Let's plug it into DE: 4 times 5...you substitute 5 in for x...minus 1; there is 20 - 1, which is 19.0866

If this is 19, then what does this have to be? 19.0879

Just to double-check: 2 times 5 is 10, plus 9 is 19.0883

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.0888

Segment bisector: Any segment, line, or plane that intersects a segment at its midpoint.0915

A segment bisector is anything that cuts the segment in half.0922

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.0931

Here, these little marks mean that this segment and this segment are the same.0939

CD is the segment bisector of AB, because CD is the one that cut AB in half.0948

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.0959

This bisected the segment AD; this is a line segment, CD; this segment bisector is a segment.0968

It doesn't have to be just a segment; it could be a segment; it could be a line; it could be a plane.0978

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.0985

The bisector can be anything: it can be in the form of a line, too.0999

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,1003

then a plane could be a segment bisector, because it intersects the segment at its midpoint, point D.1024

Again, a segment bisector is anything that cuts the segment in half, that bisects it, that intersects it at its midpoint.1035

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.1051

And we have to actually prove theorems.1059

Remember: we talked about postulates--how postulates are statements that we can just assume to be true,1062

meaning that once they give us a postulate, then we can just go ahead and use it from there.1069

Theorems, however, have to be proved or justified using definitions, postulates, and previously-proven theorems.1074

So, whenever there is a theorem--some kind of statement--then it has to be backed up by something.1082

It has to show why that is true; and then you prove it.1090

And then, once it is proven, you can start using that theorem from there on, whenever you need to.1094

Now, a proof is a logical argument in which each statement you make is backed up by a statement that is accepted as true.1102

You can't just give a statement; you have to back it up with a reason: why is that statement true?1109

And that is what a proof is.1115

Now, there are different types of proofs; the one that is used the most is called a two-column proof.1116

But we are going to go over that, actually, in the next chapter.1125

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.1128

So, you are just explaining in words why that is true.1140

And a conjecture is an if/then statement: if something is true, then something else.1145

So, actually, you are going to go over that more in the next chapter, too.1153

But that is all you are doing; for a paragraph proof, you are just explaining it in words.1156

So, instead of a two-column proof, where you are listing out each statement, and you are giving the reasons for that,1162

to prove something, in a paragraph proof, you are just writing it as a paragraph to prove that a theorem is true.1168

The first proof that we are going to do is going to be a paragraph proof.1180

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.1186

The midpoint theorem says that, if M is the midpoint of AB (here is AB), then AM is congruent to MB.1196

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.1208

And that is just talking about the measures of them.1223

But "congruent"--to show that AM is congruent to MB--that is the theorem, and we have to prove that first.1226

Given that M is the midpoint of AB, write a paragraph proof to show that AM is congruent to MB.1241

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.1250

And then, from there, we can just use it.1260

We are going to always start with the given: given that M is the midpoint of AB--that is the starting point.1265

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.1278

Now, that means that AM and MB have the same measure--that AM has the same measure as MB.1306

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.1333

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.1352

They are congruent segments--something like that.1379

You don't have to write exactly the same thing, but you are just kind of showing that we know1385

that we went over the definition of midpoint, and that is AM = MB.1389

And then, from there, you use the definition of congruence to show that AM is congruent to MB.1393

They are congruent segments--that is what the definition of congruence says.1401

Now that we have proven that the midpoint theorem is correct, or is true, then now, from now on,1410

for the remainder of the course, you can just use it whenever you need to--the midpoint theorem.1419

Extra Example 1: Use a number line to find the midpoint of each: BD.1427

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.1434

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 7/2.1442

You can leave it like that, or you can just write 3 and 1/2.1456

Between -2 and 9, it is going to be right there: three and a half.1465

CB: you know, I shouldn't write it like this, because it looks like BD...this looks like distance.1475

You can just write the midpoint of BD: so just be careful--don't write BD; don't write it like that.1496

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.1505

CB is right here; now, you can go from B to C, or we can go from C to B.1514

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 1.1520

That means the midpoint from here, C, to B, is 1, right there.1535

You can also check; you can count; this is three units, and then this is three units, so it has to be the same.1545

And then, number 3: AD: A is -5; D is 9; divided by 2...this is 4, divided by 2, which is 2.1553

Here is -5, and here is 9; the midpoint is right here, between those two...from here to here, and from here to here.1568

OK, draw a diagram to show each: AC bisects BD.1586

That means that this is the one that is doing the bisecting; this is the segment bisector.1591

This is the segment that is getting bisected: so BD...here is BD.1597

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.1606

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)...1619

OK, the next one: HI is congruent to IJ.1638

HI: see how this is congruent...this is part of the midpoint theorem...HI and IJ.1644

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.1657

The next one: RT equals half of PT--let me just draw another segment; RT is equal to half of PT.1671

Now, look at what they have in common; here is T, and here is T.1683

Now, if RT is equal to half of PT, that means that you have to divide PT by 2 to get RT.1689

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.1701

And that is going to be RT; that means R is right here--the midpoint.1712

Again, here is PT; so for example, if PT is 12 (say this whole thing, PT, is 12),1720

then you divide it by 2, or you multiply it by 1/2; then you get 6; that means RT has to equal 6.1728

And you don't have to write the numbers; just draw the diagram.1742

You could have just left it at P-R-T, or just showed it, maybe, like this.1747

The next example: Q is the midpoint of RS; if two points are given, find the coordinates of the third point.1754

RS's midpoint is Q; I'll show that; if two points are given, find the coordinates of the third point.1768

R is here; R is at (4,3); S is at (2,1); you have to find the midpoint.1785

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.1799

So, x₁ + x₂, divided by 2, equals your x-coordinate for the midpoint.1808

4 + 2, divided by 2...and then you are going to take the y's, 3 + 1, divided by 2;1817

that is going to be 6 divided by 2, which is 3; so Q is 3, comma...4 divided by 2 is 2;1831

it is going to be at (3,2); there is the midpoint for that one.1844

The next one: let me just redraw R-Q-S...Q...on this one, they give you the midpoint, and they give you this.1851

This right here is going to be...you could make this (x₁,y₁), or (x₂,y₂).1872

And then, when you write it out, it is going to be -5 + x₂ over 2, and then 2 + y₂, divided by 2.1882

Now, we know that all of this equals...the midpoint is (-2,-1).1904

That means that these equal each other and these equal each other.1912

This is going to give you -2; and this is what we are solving for.1915

I can just make this thing equal to this thing: -5 + x₂, over 2.1927

Now, this is not x²; be careful of that.1933

That equals -2; this is going to be -5 + x₂ equals...you multiply the 2 over to the other side...-4.1936

You add the 5 over; so x₂ = 1.1948

I am going to do the y over here; then you make this equal to -1; it is 2 + y₂, over 2, equals -1.1956

You multiply the 2 over; 2 + y₂ = -2; y₂ = -4.1967

All right, S is going to be (1,-4); that is to find this right here.1977

With this one, we had to find Q, the midpoint; and with this one, we had to find S.1995

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.2002

That means that AD is the one that got cut in half at C.2018

That means that this whole thing, AC, and CD are congruent.2022

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.2030

That means that AC is bisected; B is the midpoint.2050

For each, find the value of x and the measure of the segment.2056

That means AC and CD are the same, and then AB and BC are the same.2060

AB equals 3x + 6; BC equals 2x + 14; they want you to find AC.2074

Again, AB and BC have the same measure; so I can just make them equal to each other...it equals 2x + 14.2087

Subtract the 2x over here; subtract the 6 over there; you get 8.2101

And then, find the value of x and the measure of the segment.2109

The segment right here, AC, is AB + BC; or it is just AB times 2, because it is doubled.2113

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.2126

AB is 3 times 8, plus 6; that is 24; that is 30.2142

24 plus 6 is 30; and then, AC is double that, so AC is 60.2152

AB is 30; that means that AC is 60.2161

AD, the whole thing, is 6x - 4; AC is 4x - 3; and you have to find CD.2172

Now, remember how EC bisected AD; that means that AD is cut in half; C is the midpoint.2184

AD, the whole thing, is 6x - 4; and then, AC is 4x - 3.2192

I can do this two ways: AC is 4x - 3--that means DC, or CD, is 4x - 3.2198

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.2208

I am just going to do 2 times (4x - 3), because AC is 4x - 3, and AD is double that.2224

Now, even though this is 60, so you might assume that this whole thing is 120; this is a different problem.2238

So then, it is not going to have the same measure.2244

2 times AC equals AD, which is 6x - 4.2251

If we continue it here, it is going to be 8x (and this is the distributive property), minus 6, equals 6x minus 4.2261

This becomes 2x; if you add the 6 over there, it equals 2; x = 1.2275

So then, here is the x-value; and then, they want you to find CD, this right here.2283

Since CD is the same thing as AC, I can just find AC.2291

AC is 4 times 1, minus 3; so AC is 4 minus 3, is 1; so CD is 1; AC is 1; CD is 1.2298

OK, the last one: AD, the whole thing, is 5x + 2; and BC is 7 - 2x; find CD.2325

Now, this one is a little bit harder, because they give you AD, the whole thing, and they give you BC.2341

Now, remember: if C is the midpoint of AD, that means that this whole thing and this whole thing are the same.2353

Let's look at this number right here; if this is 60, then this will also be 60.2367

That means that the whole thing together is going to be 120.2374

Then this right here is 30; this right here is 30.2380

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.2389

Then, you take that paper, and you fold it in half again; so then, that is when you get point B.2399

So, now your paper is folded into how many parts? 1, 2, 3, 4.2405

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.2418

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, 4.2429

BC times 4: 7 - 2x...multiply it by 4, and you get the whole thing, AD.2446

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 2.2456

And then, find CD: CD was 2 times BC, because BC with another BC is going to equal CD.2478

I am just going to find BC: 7 - 2(2)...just plug in x...so 7 - 4 is 3.2492

If BC is 3...now, these numbers right here; that was actually for number 1;2504

those are the values for number 1, and then I just used it as an example for number 3.2511

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.2516

BC was 3; then what is CD? CD is 2 times BC--it is double BC, so CD is 6.2526

OK, well, that is it for this lesson; thank you for watching Educator.com.2542