Inscribed and Central Angles Showing that an inscribed angle is half of a central angle that subtends the same arc
Inscribed and Central Angles
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- What I want to do in this video is to prove one of the more
- useful results in geometry, and that's that an inscribed angle
- is just an angle whose vertex sits on the circumference
- of the circle.
- So that is our inscribed angle.
- I'll denote it by psi -- I'll use the psi for inscribed angle
- and angles in this video.
- That psi, the inscribed angle, is going to be exactly 1/2 of
- the central angle that subtends the same arc.
- So I just used a lot a fancy words, but I think you'll
- get what I'm saying.
- So this is psi.
- It is an inscribed angle.
- It sits, its vertex sits on the circumference.
- And if you draw out the two rays that come out from this angle
- or the two cords that define this angle, it intersects the
- circle at the other end.
- And if you look at the part of the circumference of the circle
- that's inside of it, that is the arc that is
- subtended by psi.
- It's all very fancy words, but I think the idea is
- pretty straightforward.
- This right here is the arc subtended by psi, where psi is
- that inscribed angle right over there, the vertex sitting
- on the circumference.
- Now, a central angle is an angle where the vertex is
- sitting at the center of the circle.
- So let's say that this right here -- I'll try to eyeball
- it -- that right there is the center of the circle.
- So let me draw a central angle that subtends this same arc.
- So that looks like a central angle subtending that same arc.
- Just like that.
- Let's call this theta.
- So this angle is psi, this angle right here is theta.
- What I'm going to prove in this video is that psi is always
- going to be equal to 1/2 of theta.
- So if I were to tell you that psi is equal to, I don't know,
- 25 degrees, then you would immediately know that theta
- must be equal to 50 degrees.
- Or if I told you that theta was 80 degrees, then you would
- immediately know that psi was 40 degrees.
- So let's actually proved this.
- So let me clear this.
- So a good place to start, or the place I'm going to
- start, is a special case.
- I'm going to draw an inscribed angle, but one of the chords
- that define it is going to be the diameter of the circle.
- So this isn't going to be the general case, this is going
- to be a special case.
- So let me see, this is the center right here of my circle.
- I'm trying to eyeball it.
- Center looks like that.
- So let me draw a diameter.
- So the diameter looks like that.
- Then let me define my inscribed angle.
- This diameter is one side of it.
- And then the other side maybe is just like that.
- So let me call this right here psi.
- If that's psi, this length right here is a radius -- that's
- our radius of our circle.
- Then this length right here is also going to be the radius of
- our circle going from the center to the circumference.
- Your circumference is defined by all of the points that are
- exactly a radius away from the center.
- So that's also a radius.
- Now, this triangle right here is an isosceles triangle.
- It has two sides that are equal.
- Two sides that are definitely equal.
- We know that when we have two sides being equal, their
- base angles are also equal.
- So this will also be equal to psi.
- You might not recognize it because it's
- tilted up like that.
- But I think many of us when we see a triangle that looks like
- this, if I told you this is r and that is r, that these two
- sides are equal, and if this is psi, then you would also
- know that this angle is also going to be psi.
- Base angles are equivalent on an isosceles triangle.
- So this is psi, that is also psi.
- Now, let me look at the central angle.
- This is the central angle subtending the same arc.
- Let's highlight the arc that they're both subtending.
- This right here is the arc that they're both going to subtend.
- So this is my central angle right there, theta.
- Now if this angle is theta, what's this angle going to be?
- This angle right here.
- Well, this angle is supplementary to theta,
- so it's 180 minus theta.
- When you add these two angles together you go 180 degrees
- around or they kind of form a line.
- They're supplementary to each other.
- Now we also know that these three angles are sitting
- inside of the same triangle.
- So they must add up to 180 degrees.
- So we get psi -- this psi plus that psi plus psi plus this
- angle, which is 180 minus theta plus 180 minus theta.
- These three angles must add up to 180 degrees.
- They're the three angles of a triangle.
- Now we could subtract 180 from both sides.
- psi plus psi is 2 psi minus theta is equal to 0.
- Add theta to both sides.
- You get 2 psi is equal to theta.
- Multiply both sides by 1/2 or divide both sides by 2.
- You get psi is equal to 1/2 of theta.
- So we just proved what we set out to prove for the special
- case where our inscribed angle is defined, where one of the
- rays, if you want to view these lines as rays, where one of the
- rays that defines this inscribed angle is
- along the diameter.
- The diameter forms part of that ray.
- So this is a special case where one edge is
- sitting on the diameter.
- So already we could generalize this.
- So now that we know that if this is 50 that this is
- going to be 100 degrees and likewise, right?
- Whatever psi is or whatever theta is, psi's going to be 1/2
- of that, or whatever psi is, theta is going to
- be 2 times that.
- And now this will apply for any time.
- We could use this notion any time that -- so just using that
- result we just got, we can now generalize it a little bit,
- although this won't apply to all inscribed angles.
- Let's have an inscribed angle that looks like this.
- So this situation, the center, you can kind of view it as
- it's inside of the angle.
- That's my inscribed angle.
- And I want to find a relationship between this
- inscribed angle and the central angle that's subtending
- to same arc.
- So that's my central angle subtending the same arc.
- Well, you might say, hey, gee, none of these ends or these
- chords that define this angle, neither of these are diameters,
- but what we can do is we can draw a diameter.
- If the center is within these two chords we
- can draw a diameter.
- We can draw a diameter just like that.
- If we draw a diameter just like that, if we define this angle
- as psi 1, that angle as psi 2.
- Clearly psi is the sum of those two angles.
- And we call this angle theta 1, and this angle theta 2.
- We immediately you know that, just using the result I just
- got, since we have one side of our angles in both cases being
- a diameter now, we know that psi 1 is going to be
- equal to 1/2 theta 1.
- And we know that psi 2 is going to be 1/2 theta 2.
- Psi 2 is going to be 1/2 theta 2.
- So psi, which is psi 1 plus psi 2, so psi 1 plus psi 2 is going to
- be equal to these two things.
- 1/2 theta 1 plus 1/2 theta 2.
- psi 1 plus psi 2, this is equal to the first inscribed
- angle that we want to deal with, just regular psi.
- That's psi.
- And this right here, this is equal to 1/2 times
- theta 1 plus theta 2.
- What's theta 1 plus theta 2?
- Well that's just our original theta that
- we were dealing with.
- So now we see that psi is equal to 1/2 theta.
- So now we've proved it for a slightly more general case
- where our center is inside of the two rays that
- define that angle.
- Now, we still haven't addressed a slightly harder situation or
- a more general situation where if this is the center of our
- circle and I have an inscribed angle where the center isn't
- sitting inside of the two chords.
- Let me draw that.
- So that's going to be my vertex, and I'll switch colors,
- so let's say that is one of the chords that defines the
- angle, just like that.
- And let's say that is the other chord that defines
- the angle just like that.
- So how do we find the relationship between, let's
- call, this angle right here, let's call it psi 1.
- How do we find the relationship between psi 1 and the central
- angle that subtends this same arc?
- So when I talk about the same arc, that's that right there.
- So the central angle that subtends the same arc
- will look like this.
- Let's call that theta 1.
- What we can do is use what we just learned when one side of
- our inscribed angle is a diameter.
- So let's construct that.
- So let me draw a diameter here.
- The result we want still is that this should be 1/2 of
- this, but let's prove it.
- Let's draw a diameter just like that.
- Let me call this angle right here, let me call that psi 2.
- And it is subtending this arc right there -- let me do
- that in a darker color.
- It is subtending this arc right there.
- So the central angle that subtends that same arc,
- let me call that theta 2.
- Now, we know from the earlier part of this video that psi
- 2 is going to be equal to 1/2 theta 2, right?
- They share -- the diameter is right there.
- The diameter is one of the chords that forms the angle.
- So psi 2 is going to be equal to 1/2 theta 2.
- This is exactly what we've been doing in the last video, right?
- This is an inscribed angle.
- One of the chords that define is sitting on the diameter.
- So this is going to be 1/2 of this angle, of the central
- angle that subtends the same arc.
- Now, let's look at this larger angle.
- This larger angle right here.
- Psi 1 plus psi 2.
- Right, that larger angle is psi 1 plus psi 2.
- Once again, this subtends this entire arc right here, and it
- has a diameter as one of the chords that defines
- this huge angle.
- So this is going to be 1/2 of the central angle that
- subtends the same arc.
- We're just using what we've already shown in this video.
- So this is going to be equal to 1/2 of this huge central angle
- of theta 1 plus theta 2.
- So far we've just used everything that we've learned
- earlier in this video.
- Now, we already know that psi 2 is equal to 1/2 theta 2.
- So let me make that substitution.
- This is equal to that.
- So we can say that si 1 plus -- instead of si 2 I'll write
- 1/2 theta 2 is equal to 1/2 theta 1 plus 1/2 theta 2.
- We can subtract 1/2 theta 2 from both sides, and
- we get our result.
- Si 1 is equal to 1/2 theta one.
- And now we're done.
- We have proven the situation that the inscribed angle is
- always 1/2 of the central angle that subtends the same arc,
- regardless of whether the center of the circle is inside
- of the angle, outside of the angle, whether we have a
- diameter on one side.
- So any other angle can be constructed as a sum of
- any or all of these that we've already done.
- So hopefully you found this useful and now we can actually
- build on this result to do some more interesting
- geometry proofs.
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