Geometry Arcs and Chords

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For the next lesson, we are going to go over arcs and chords of circles.0002

An arc of the chord: first of all, we know that chords are segments within a circle whose endpoints are on the circle.0011

So then, here, this is a chord, AB; remember: it has to be a line segment.0024

There have to be two endpoints, and those endpoints have to be on the circle; that is a chord.0030

Now, the arc of the chord would be the intercepted arc, and it would be the minor arc.0037

It wouldn't be this major arc over here; it would be the minor arc, AB.0047

This is a chord, and this would be the arc of the chord; it shares the same endpoints.0054

In a circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.0063

This is the first theorem; we are actually going to go over a few different theorems today.0073

The first theorem is saying that it could be within one circle or congruent circles.0076

Here, I do congruent circles; and if this chord is congruent to this chord right here (again, it has to be congruent), then their arcs will be congruent.0084

So, by saying "if and only if," it is saying it could work both ways, including vice versa.0105

If the arcs are congruent, then their corresponding chords will be congruent.0111

Since these chords are congruent, I know that these arcs are congruent.0120

An inscribed polygon is when you have a polygon within a circle with all of the vertices lying on the circle.0132

It could be this right here; this is a quadrilateral; but it could be a triangle; it could be any type of polygon0143

that is inside the circle, with all of the vertices touching.0152

So, if you have something like this, that would not be an inscribed polygon, because the vertices are not on the circle.0156

This is an inscribed polygon; and you are going to see this word a lot, maybe on the SAT's or any test; they will use the word "inscribed."0170

A polygon is inscribed in a circle; just remember that, when you see that word "inscribed," all of the vertices are lying on the circle.0184

Arcs and chords: this next theorem is saying that, if you have a diameter (you know that AC is a diameter,0199

because it is passing through the center) that is perpendicular to a chord0209

(here is chord BD; they are perpendicular), then the diameter will bisect that chord.0214

Remember: "bisect" means to cut it in half, so the diameter is cutting this chord in half, into two equal segments.0221

BE then will be congruent to DE; this little piece and this little piece will be congruent.0232

And then, these intercepted arcs will also be congruent.0245

Then, if this BE is congruent to DE, then this arc BC is going to be congruent to DC.0255

So again, if the diameter is perpendicular to a chord, then the chord, along with the intercepted arcs, are bisected.0266

But it has to be perpendicular, and it has to be a diameter; a diameter has to be perpendicular with a chord.0276

If I just have any other chord (even the diameter is also known as a chord--it has to be the diameter)0285

that is bisecting a chord, then that is not part of the theorem.0294

It has to be a diameter perpendicular to a chord; then it is bisected.0299

The next theorem: In a circle, or in congruent circles, two chords (here is one and here is the other)0307

are congruent, if and only if they are equidistant from the center.0314

Here, we talked about this in the beginning of the course.0322

If you want to find the distance between a line and a point--you want to see how far away this line is from this point--0328

well, let's say that this is you, standing in a room, and this is a wall.0339

If you want to find the distance, how far away you are standing from the wall--you want to measure the distance--0349

you would have to measure the distance from where you are standing to the wall so that it is perpendicular.0357

If you wanted to find your distance from the wall, you would not find it like that; you wouldn't measure this right here; it has to be perpendicular.0366

Any time you want to find the distance between a line and a point, it has to be perpendicular.0376

Here, to find the distance between this chord and this point, I am going to draw the distance so that it is perpendicular.0386

I know that it doesn't look like it, because it is slightly angled; but that would be the perpendicular distance.0399

The same thing works for this one: to find the distance between this chord and this point, I would have to find it so that it would be a perpendicular distance.0408

This represents the distance of this chord from the point, and this represents the distance from the point to the chord.0421

If they have the same distance--if they are equally distant from the center--this chord and this chord0430

(that means if this is congruent to this), then the chords are congruent;0437

this chord will be congruent to this chord, but only if they are equidistant.0445

Now, if they just showed you...if you get a problem like this, ever (my circle is...OK): let's say we have a chord here,0454

and we have a chord here; there is the center; now, they draw lines like this to show the distance;0469

and they are asking if this chord is congruent to this chord: is AB congruent to CD--are they congruent?0479

Well, if this is all they give you, then you would have to say either "no" or "not enough information,"0490

because even though this is representing the distance, you don't know if they are congruent.0500

You don't know that they are the same distance; it just shows lines, but how far is this, and how long is that line?0507

You don't know; so then here, you would probably have to say either "no" or "not enough information."0514

Now, how about this? Let's say you get a problem like that.0520

Are these chords, AB and CD, congruent?0527

Well, again, you would have to say either "no" or "not enough information."0531

Why is that? They show you the two chords, and they are showing you that it is the perpendicular distance.0536

But they don't give you that they are equidistant from each other.0543

So again, it has to be that this distance is congruent to this distance; they have to show you that it is equally distant from the center.0549

Only if they show you that, then you can say that AB is congruent to CD.0562

If it doesn't show you that it is perpendicular, and they just do that; then this is not enough information.0568

You can't assume that this distance is equal to this distance; so then, you don't know.0577

This could be 6 centimeters, and this could be 7 centimeters; you don't know, so in that case, you can't say yes.0583

Also, if they give you this, showing you the distance, and this: well, it looks like it is equidistant,0591

because they are saying that this segment and this segment are congruent.0604

But again, you can't say "yes" to this, because you don't know if it is perpendicular.0609

They could have just found the distance from here to here, like that, instead of finding it perpendicularly.0614

Here, again, you can't say "yes"; you can't say that AB is congruent to CD.0620

It has to be congruent, but it has to be showing you that it is the distance away from them; this one is "yes."0625

And then, now let's go over examples: Name two pairs of congruent arcs.0636

If we look at the circle, we have a whole bunch of stuff; I see a whole bunch of chords here.0645

OK, two congruent arcs: now, it doesn't show me that anything is congruent, so how am I supposed to know which arcs are congruent?0652

Let's see, I have a diameter here; here is my center, diameter, center, diameter;0667

and then, I have a chord that is perpendicular to this diameter, and then I have this chord that is perpendicular to this diameter.0675

So, remember the theorem that said that, if a diameter is perpendicular to a chord, then the diameter will bisect the chord and its arcs.0684

I can say that AH (because BD was the diameter, and BC is the chord, so AC is being bisected) is congruent to HC,0696

and then that means that the intercepted arc (this) is congruent to this.0712

I can also say, since I have another diameter perpendicular to another chord, that this is going to be congruent to this,0720

which means that this arc is congruent to this arc.0730

Name two pairs of congruent arcs: the first pair would be arc AB, which is congruent to arc CB.0739

And the next pair would be arc AF, which is congruent to arc EF; those are my two pairs.0756

The next one: If AJ equals 6, then find EJ.0771

Well, we already wrote that it is congruent; so if AJ is 6, then EJ has to also be 6.0780

And then, we know that AE is going to be 12.0794

The name of the segment congruent to BD: BD is a diameter, so then we know that all other diameters in the circle will be congruent.0799

What is congruent to BD? CF is the segment congruent to BD.0812

And if you were to explain why, then you would say, "Well, all of the diameters within the same circle are congruent."0821

The next example: Find the measure of the given arc.0830

They are asking for the measure of arc AC, right here; you know that they are not talking about AC here,0835

the major arc, because it is just using two variables; if they wanted you to find this major arc,0843

then it would have to be measure of arc ABC, using three variables.0851

Here, the measure of this arc is 130; we know that, since BC is congruent to AC, that theorem says that,0858

if two chords are within the same circle (or congruent circles), the intercepted arcs of the chords are also congruent.0871

So then, this arc will be congruent to this arc; this is 130; this is congruent to this.0883

The whole circle has a measure of 360; so let's say we are going to label this x; then it would be 130, plus x plus x,0895

so it is 2x, is going to equal 360; so then, 2x is going to equal 230, so x is going to be 115.0912

This is 115 degrees, and this is 115 degrees; and that is the measure of arc AC, 115.0933

The measure of arc BC: they don't give us any measures for this one, but since they give us0960

that all of these four chords are congruent, you can say that their arcs are also congruent.0970

That means that it is just 360, divided by 4; and if, let's say, BC is x, then it is 4x, so 4x = 360, so you can divide the 4, and this is going to be 90 degrees.0983

So, the measure of arc BC is 90.1013

The third example: Determine if AB, this chord, and CD are congruent.1032

Here, are they congruent? Well, no, we can't say that they are congruent,1040

because (number 1) we don't know if these chords are equidistant from the center.1045

Here is the center; what is the distance of this, and what is the distance of that?1057

We don't know, so this one would be "no," or you don't know--"not enough information."1063

What if they did that?--there is still not enough information; it has to show you that the chord is perpendicular,1069

that this has a perpendicular distance, because distance from this line to this point has to be perpendicular distance; and it is the same.1081

If it is like that, then it would be "yes"; so in this case, it would be "AB is congruent to CD."1094

And this one here: they give you that arc AB is congruent to arc BC.1108

Well, if these arcs are congruent, then their corresponding chords have to be congruent; so if this is 3, then this will be 3.1116

And then, BC is congruent to CD, which means that this intercepted arc is also congruent to this intercepted arc.1129

It is like saying, "Well, AB is congruent to BC, and BC is congruent to CD, so like the transitive property, AB is congruent to CD."1144

And then, with the arcs and the chords that are corresponding, if the arcs are congruent, then their chords are congruent;1158

if the chords are congruent, then their intercepted arcs are congruent.1166

In this case, it is "yes"; AB is congruent to CD.1171

And then, the last one: Find the value of x.1186

Here, you can't say that these chords are congruent; you would have to say that they are equidistant, like that.1191

Then, I can say that this chord is congruent to this chord, and then you are going to make them equal to each other: 2x - 10 = x + 2.1200

And this will be x = 12; so here, again, since we know that the chords are equidistant from the center, the chords will be congruent.1214

So, I just made them equal to each other: 2x - 10 = x + 2; and then, you get x = 12.1232

And then, the next one: here, this is the diameter, because it is passing through the center (and that is the center, right there).1241

And it is perpendicular to this chord, so I know that this segment and this segment are congruent; the chord is bisected.1254

So then, 4x - 1 is going to be equal to 6x - 9; so I am going to add the 9 here;1267

that is going to give me 8 =...subtract the 4x; it is 2x there; divide the 2; so x is going to equal 4.1277

There is my value of x; and again, the diameter is perpendicular to the chord; therefore, the chord is bisected.1293

And these little arcs are also congruent.1301

That is it for this lesson; thank you for watching Educator.com.1308