### Arcs and Chords

- Arc of the chord: An arc that shares the same endpoints of the chord
- In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent
- Inscribed polygon: An inscribed polygon is a polygon inside the circle with all of its vertices on the circle
- In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc
- In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center

### Arcs and Chords

―CD ≅ ―BE , determine whether ⁀CD ≅ ⁀BE

A polygon inside a circle is called an inscribed polygon.

―BC ⊥―DE , write a pair of congruent segments and a pair of congruent arcs.

- ―AB ≅ ―AC
- ⁀CE ≅ ⁀BE

⁀CE ≅ ⁀BE

Determine whether the following statement is true or false:

If ―BC ⊥―DE , ―MN ⊥―DE , ―AD ≅ ―AE , then ⁀MN ≅ ⁀BC

A diameter divides a circle into two congruent arcs.

―BC ⊥―DE , write 2 pairs of congruent arcs.

∆BCD is an equilateral triangle, find m⁀BC.

- m⁀BC = m⁀DC = m⁀BD
- m⁀BC + m⁀DC + m⁀BD = 360
^{o}

^{o}

- ―AB ≅ ―AC
- AB = AC
- 5x + 7 = 3x + 15
- 2x = 8

Name a segment congruent to ―AC.

In a circle, if two chords ―MN and ―PQ are congruent, then ⁀MN and ⁀PQ are congruent.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Arcs and Chords

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Arcs and Chords 0:07
- Arc of the Chord
- Theorem 1: Congruent Minor Arcs
- Inscribed Polygon 2:10
- Inscribed Polygon
- Arcs and Chords 3:18
- Theorem 2: When a Diameter is Perpendicular to a Chord
- Arcs and Chords 5:05
- Theorem 3: Congruent Chords
- Extra Example 1: Congruent Arcs 10:35
- Extra Example 2: Length of Arc 13:50
- Extra Example 3: Arcs and Chords 17:09
- Extra Example 4: Arcs and Chords 19:45

### Geometry Online Course

### Transcription: Arcs and Chords

*Welcome back to Educator.com.*0000

*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 polygon*0143

*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 chord*0209

*(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 chord*0430

*(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 us*0960

*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

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