WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For the next lesson, we are going to go over arcs and chords of circles.
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An arc of the chord: first of all, we know that chords are segments within a circle whose endpoints are on the circle.
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So then, here, this is a chord, AB; remember: it has to be a line segment.
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There have to be two endpoints, and those endpoints have to be on the circle; that is a chord.
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Now, the arc of the chord would be the intercepted arc, and it would be the minor arc.
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It wouldn't be this major arc over here; it would be the minor arc, AB.
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This is a chord, and this would be the arc of the chord; it shares the same endpoints.
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In a circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
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This is the first theorem; we are actually going to go over a few different theorems today.
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The first theorem is saying that it could be within one circle or congruent circles.
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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.
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So, by saying "if and only if," it is saying it could work both ways, including vice versa.
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If the arcs are congruent, then their corresponding chords will be congruent.
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Since these chords are congruent, I know that these arcs are congruent.
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An **inscribed polygon** is when you have a polygon within a circle with all of the vertices lying on the circle.
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It could be this right here; this is a quadrilateral; but it could be a triangle; it could be any type of polygon
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that is inside the circle, with all of the vertices touching.
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So, if you have something like this, that would not be an inscribed polygon, because the vertices are not on the circle.
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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."
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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.
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Arcs and chords: this next theorem is saying that, if you have a diameter (you know that AC is a diameter,
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because it is passing through the center) that is perpendicular to a chord
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(here is chord BD; they are perpendicular), then the diameter will bisect that chord.
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Remember: "bisect" means to cut it in half, so the diameter is cutting this chord in half, into two equal segments.
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BE then will be congruent to DE; this little piece and this little piece will be congruent.
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And then, these intercepted arcs will also be congruent.
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Then, if this BE is congruent to DE, then this arc BC is going to be congruent to DC.
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So again, if the diameter is perpendicular to a chord, then the chord, along with the intercepted arcs, are bisected.
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But it has to be perpendicular, and it has to be a diameter; a diameter has to be perpendicular with a chord.
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If I just have any other chord (even the diameter is also known as a chord--it has to be the diameter)
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that is bisecting a chord, then that is not part of the theorem.
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It has to be a diameter perpendicular to a chord; then it is bisected.
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The next theorem: In a circle, or in congruent circles, two chords (here is one and here is the other)
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are congruent, if and only if they are equidistant from the center.
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Here, we talked about this in the beginning of the course.
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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--
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well, let's say that this is you, standing in a room, and this is a wall.
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If you want to find the distance, how far away you are standing from the wall--you want to measure the distance--
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you would have to measure the distance from where you are standing to the wall so that it is perpendicular.
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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.
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Any time you want to find the distance between a line and a point, it has to be perpendicular.
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Here, to find the distance between this chord and this point, I am going to draw the distance so that it is perpendicular.
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I know that it doesn't look like it, because it is slightly angled; but that would be the perpendicular distance.
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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.
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This represents the distance of this chord from the point, and this represents the distance from the point to the chord.
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If they have the same distance--if they are equally distant from the center--this chord and this chord
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(that means if this is congruent to this), then the chords are congruent;
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this chord will be congruent to this chord, but only if they are equidistant.
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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,
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and we have a chord here; there is the center; now, they draw lines like this to show the distance;
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and they are asking if this chord is congruent to this chord: is AB congruent to CD--are they congruent?
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Well, if this is all they give you, then you would have to say either "no" or "not enough information,"
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because even though this is representing the distance, you don't know if they are congruent.
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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?
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You don't know; so then here, you would probably have to say either "no" or "not enough information."
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Now, how about this? Let's say you get a problem like that.
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Are these chords, AB and CD, congruent?
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Well, again, you would have to say either "no" or "not enough information."
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Why is that? They show you the two chords, and they are showing you that it is the perpendicular distance.
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But they don't give you that they are equidistant from each other.
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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.
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Only if they show you that, then you can say that AB is congruent to CD.
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If it doesn't show you that it is perpendicular, and they just do that; then this is not enough information.
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You can't assume that this distance is equal to this distance; so then, you don't know.
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This could be 6 centimeters, and this could be 7 centimeters; you don't know, so in that case, you can't say yes.
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Also, if they give you this, showing you the distance, and this: well, it looks like it is equidistant,
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because they are saying that this segment and this segment are congruent.
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But again, you can't say "yes" to this, because you don't know if it is perpendicular.
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They could have just found the distance from here to here, like that, instead of finding it perpendicularly.
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Here, again, you can't say "yes"; you can't say that AB is congruent to CD.
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It has to be congruent, but it has to be showing you that it is the distance away from them; this one is "yes."
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And then, now let's go over examples: Name two pairs of congruent arcs.
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If we look at the circle, we have a whole bunch of stuff; I see a whole bunch of chords here.
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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?
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Let's see, I have a diameter here; here is my center, diameter, center, diameter;
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and then, I have a chord that is perpendicular to this diameter, and then I have this chord that is perpendicular to this diameter.
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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.
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I can say that AH (because BD was the diameter, and BC is the chord, so AC is being bisected) is congruent to HC,
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and then that means that the intercepted arc (this) is congruent to this.
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I can also say, since I have another diameter perpendicular to another chord, that this is going to be congruent to this,
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which means that this arc is congruent to this arc.
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Name two pairs of congruent arcs: the first pair would be arc AB, which is congruent to arc CB.
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And the next pair would be arc AF, which is congruent to arc EF; those are my two pairs.
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The next one: If AJ equals 6, then find EJ.
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Well, we already wrote that it is congruent; so if AJ is 6, then EJ has to also be 6.
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And then, we know that AE is going to be 12.
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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.
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What is congruent to BD? CF is the segment congruent to BD.
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And if you were to explain why, then you would say, "Well, all of the diameters within the same circle are congruent."
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The next example: Find the measure of the given arc.
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They are asking for the measure of arc AC, right here; you know that they are not talking about AC here,
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the major arc, because it is just using two variables; if they wanted you to find this major arc,
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then it would have to be measure of arc ABC, using three variables.
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Here, the measure of this arc is 130; we know that, since BC is congruent to AC, that theorem says that,
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if two chords are within the same circle (or congruent circles), the intercepted arcs of the chords are also congruent.
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So then, this arc will be congruent to this arc; this is 130; this is congruent to this.
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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,
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so it is 2x, is going to equal 360; so then, 2x is going to equal 230, so x is going to be 115.
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This is 115 degrees, and this is 115 degrees; and that is the measure of arc AC, 115.
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The measure of arc BC: they don't give us any measures for this one, but since they give us
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that all of these four chords are congruent, you can say that their arcs are also congruent.
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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.
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So, the measure of arc BC is 90.
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The third example: Determine if AB, this chord, and CD are congruent.
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Here, are they congruent? Well, no, we can't say that they are congruent,
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because (number 1) we don't know if these chords are equidistant from the center.
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Here is the center; what is the distance of this, and what is the distance of that?
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We don't know, so this one would be "no," or you don't know--"not enough information."
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What if they did that?--there is still not enough information; it has to show you that the chord is perpendicular,
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that this has a perpendicular distance, because distance from this line to this point has to be perpendicular distance; and it is the same.
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If it is like that, then it would be "yes"; so in this case, it would be "AB is congruent to CD."
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And this one here: they give you that arc AB is congruent to arc BC.
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Well, if these arcs are congruent, then their corresponding chords have to be congruent; so if this is 3, then this will be 3.
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And then, BC is congruent to CD, which means that this intercepted arc is also congruent to this intercepted arc.
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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."
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And then, with the arcs and the chords that are corresponding, if the arcs are congruent, then their chords are congruent;
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if the chords are congruent, then their intercepted arcs are congruent.
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In this case, it is "yes"; AB is congruent to CD.
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And then, the last one: Find the value of x.
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Here, you can't say that these chords are congruent; you would have to say that they are equidistant, like that.
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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.
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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.
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So, I just made them equal to each other: 2x - 10 = x + 2; and then, you get x = 12.
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And then, the next one: here, this is the diameter, because it is passing through the center (and that is the center, right there).
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And it is perpendicular to this chord, so I know that this segment and this segment are congruent; the chord is bisected.
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So then, 4x - 1 is going to be equal to 6x - 9; so I am going to add the 9 here;
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that is going to give me 8 =...subtract the 4x; it is 2x there; divide the 2; so x is going to equal 4.
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There is my value of x; and again, the diameter is perpendicular to the chord; therefore, the chord is bisected.
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And these little arcs are also congruent.
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That is it for this lesson; thank you for watching Educator.com.