Circles

# Central, inscribed and circumscribed angles

## Inscribed and central angles

Showing that an inscribed angle is half of a central angle that subtends the same arc
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## Inscribed and central angles

Discussion and questions for this video
how would it be use for this equation for an acual problem? i remember things applied in real situation.

There are a host of phenomena in physics that vary with distance in a manner which is proportional to the inverse of the square of the distance.

If you were to measure the diameter of the sun when viewed at a distance, and were then to double that distance the observed diameter would be one half that of the original observation. Because the observed surface area of the suns disk is related to its radius which varies in the same manner, the surface area of the image will be 1/4 of the original image. To be more explicit A=Pi*r^2, and so if you substitute (1/2)r for r you will find that A=(1/4)*Pi*r^2. It turns out this result can be generalized and so is applicable to all distances. This notion also is implicit in perspective drawings and in optical phenomena. Hold your hand one foot from your eye, and then increase the distance to two feet. The size of the image of your hand in your visual field is owing to the angle subtended.

The gravitational force between two objects also varies in this manner. Hence the formula F=G(m1*m2)/R^2 where R=distance between two objects.

So yeah, this is actually something that many people find both interesting and useful. To understand all sorts of physics and engineering topics an understanding geometry is needed. The lecture that Sal gave is a part of this body of knowledge.

It might also be said that geometry is "a real situation."
At 0:25, is there a special meaning si has? Oh, and theta too. (2:01.)
psi and theta are just like using x in algebra they are just variables
what is sy and data?
PSY is a Korean music artist known for his songs "Gangnam Style" and "Gentleman" and data is a term for information used usually in context with computers and the internet.

But you are probobly wondering about PSI (Ψ) and THETA (θ) which are the 23 and eighth letters in the greek alphabet, respectively. Sal sometimes uses them as variables, instead of x or y or whatever.
At 8:22 how do you know psi 1 is = to 1/2 beta 1, and same for psi 2 and beta 2?
At 6:00 minutes of this video Sal proves psi = 1/2 beta when one ray is a diameter, in the example you're asking about he creates a diameter so he can use this result to know Psi 1 = 1/2 beta 1 and the same for psi 2 and beta 2.
at 00:38 you mentioned si for the first time. What does si signify?
Psi (ψ) (not si) is a Greek letter. It's just a variable, like theta (θ). He's using it because there are two important angles so he needs a different letter for each.
Could you do a problem like this with different variables?
Like, instead of Psi and Theta, could you use x and y?
Yes, you can use whichever variables you like. But x and y are commonly used to indicate lengths, and psi and theta are commonly used to indicate angles, so that's what Sal has done.
Can the rays of Si be outside of the circle and not inside?
The answer is yes. The rays forming si can intersect outside the circle. In this case, the the angle will sustend two arcs on the circle. The measure of angle si will then be 1/2(arc2 - arc1). Arc2 being the larger arc and Arc1 being the smaller arc.
1 Comment
At 0:11 and 0:45, Sal says "vertex", what does that mean?
The Vertex is the point where the two line segments meet. So at 00:11 when he points to the vertex he is staying, the place where the two line segments meet, which is on the circumference of the circle
I had a question about theta. If someone draws the symbol theta would you draw the line that goes in the center only as long as the radus or to the diameter? Thank you very much.
Theta is simply a variable that he uses to describe one of the two types of angles used in the above video: central angles. The symbol theta has no meaning specific to the concept discussed in the above video; like Psi, which he uses to indicate the inscribed angles, the Theta symbol is used to represent something, therefore it is not how the symbol is drawn that matters--it is what the symbol represents that matters.
Wait, in that third situation, what garuntee do you have that si 2 = theta 2? There are three triangles in that part of the video (at 11:25) and in the other parts, then there are two triangles? It doesn't really make much sense.
he's taking from the first example- if one of the chords of the angle is the diameter then we already know that psi= 1/2theta, we already proved that. so using that we know that psi2=½(theta2) and we can and since the huge angle we made also has one of the chords on the diameter, we can state that psi1+psi2=½(theta1+theta2). Since we know that psi2=1/2theta1, we can plug it in and now we have psi1+1/2theta1=½theta1 +½ theta2 and the term 1/2theta1 cancels from both sides and you are left with psi1=1/2theta1, which are the two measurements we were looking for.Hope this helps.
At 10:05 What is "si"?
Actually, it is Psi, don't do Psy because he is a K-pop star.
Psi is a Greek letter used to represent an unknown value of a angle formed by 2 chords which have a common endpoint. (A inscribed angle)
I don't get it... why not just use X or Z?
Because X and Z are for algebra. psi (but the p is silent) and theta. They're just variable traditionally used for angles( they are Greek letters)
Theta and Psi are letters in the Greek alphabet. They are commonly used to represent angles, the same way letters like a, b, c, or x, y, or z are commonly used as variables to represent unknowns in algebraic expressions. He could have used any letter to represent the angles. Hope this helps. Good Luck.
what are planes
A plane is a flat surface with a thickness of 0. Any three points not on the same line will make a plane.
what's psi annd what are its uses ?
Although the definition for p.s.i above is correct, what is used in the video is Psi, a Greek letter. It is used to represent an unknown angle, like a more traditional x or y.
Geometry can be taken in 8th grade, I don't think any 7th graders take it but maybe, and of course any high schooler can take it, 9-12th, but its usually 10th grade (:
how does he know psi 1 equal 1/2 theta 1 at 8:15 in the vidio? i didn't undrstand the proof. i don't think i saw a proof. like wise for scy 2 and thaita 2 at 8:20
Sal wasnt really proving anything in particular. All he wanted to show was that the centre neednt be within the arc being subtended.
And he just did that using stuff that he had taught before
He clearly shows that an inscribed angle is one half of the central angle that subtends the same arc but does that mean that it is one half of the arc that it subtends as well?For example if an arc is 90 degrees does that mean that the inscribed angle subtended by that arc is 45 degrees?
I think that when you're measuring an arc in degrees, you're basically finding the central angle. So if I understood both questions correctly, it's both yes.
Are Psi and Theta used as placeholder for a value?
Yes. Just like x and y, they stand in for a value you don't know at the moment or else as a variable that might change. θ and ψ are nearly always used to represent angles, while x and y are used for points, lines, and curves.

For reference sake, just as x is the standard variable to use for an unknown, and if you need a second one it is standard to use y; with angles it is standard to use θ as the first unknown angle's variable and ψ for the second unknown angle. However, some people (myself included) tend to use α and β after having used θ.
But, really, it is all just arbitrary placeholders. You can call it anything you like, but math teachers tend to like for you to stick to traditional variables.
1 Comment
Why doesn't the question page show up? I can't see the answer to my own question!
Go to your profile and click on discussion to see questions you've asked and answers you've given. There's a lot of people on here, so it can be hard to find yourself without that.
Since si is an inscribed circle and theta is the central angle that subtends the same arc as si, shouldn't theta be 2 times si and not 1/2 of si?
why psi and theta instead of a or b? Do they stand for something or are they just variables?
What does Sai or Si, whatever you call it, mean?
do you mean psi?
if so, its a Greek letter used to denote angles
You have explained this much better than my teacher! Can you please make more geometry videos. Especially using the Common Core.
What does Subtended mean at 1:32
It's the amount of arc that's 'cut out' by the two lines extended from the angle we're talking about. - that should be clear from the vid @1:32
Is Si a mathematical expression like theta or is it representing a number?
It is spelt Psi. See http://en.wikipedia.org/wiki/Psi_(letter). It is a greek letter, like alpha, beta, gamma, pi. For a full list of greek letters and how they're used in mathematics and science see http://en.wikipedia.org/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering. Sal could've picked another letter if he wanted. Perhaps he wanted to have a change for this video.
If the interior angles of a circle add up to 360 degrees, then what about an ellipse? Would the angles of an ellipse add up to 360 degrees as well?
Yes. You know how all squares are rectangles, but not all rectangles are squares? The same is true for circles. All circles are ellipses, but not all ellipses are circles.
Since a circle is just a special ellipse, then if a circle = 360, then so does an ellipse.
1 Comment
I have always wondered is it Theta or Beta?
Theta and Beta are just traditionally used variables for angles.
College Board Sat Official Practice Test Question: #8. Square RSTU is inscribe in the circle. What is the degree measure of arc ST?
can i get a credit for watching these video taking notes
at 0:35 to 1:00 can someone help me understand this its verry confusing
Just keep watching. It'll become more clear.
what does si mean or stand for? i dont get it:(
psi is a greek character, he's just using it to denote the angle.
HEY guys(gals) I have a question... what is the meaning of si(the pitchfork looking thing) and beta? (the circle with the line through it?)
Psi and theta are Greek letters. They are used many times as variables for unknown angles.
Is si/psi spelled si or psi? I've seen both spellings this video.
It is spelled "Psi" and the sign Ψ is the 23 letter of the greek alphabet which means Ps
What is subtending?
Subtending of an angle is the angle created when two lines intersect in a circle at the center.
What is "theda"?
It's "theta". It is a Greek letter that is used mostly to represent unknown angles.
how to find lettered angles in a circle?
The lettered angles are just variables, like x or y. Think nothing differently of them then you would if you were dealing with a regular a,b,c variable.
is SI a greek character??
yes and it's psi not si. =)
Why does Sal use psi? What is the purpose of Psi? I dont use psi and i also didnt know what it was.
It is standard practice in mathematics to use Greek letters to represent angles. Among the more commonly used are θ, ψ, φ , α, β, γ (Theta, psi, phi, alpha, beta, and gamma). Sometimes you'll see people use omega (ω).
It would be a good idea to learn the Greek alphabet. You don't necessarily need to know the names of all the letters, but just be able to recognize and draw them (though I would recommend learning the names of the more commonly used Greek letters)
okay this is a good video but my school isn't learning si or data so now I'm even more confused what do they stand for?
Psi and Theta are just variables like "x" and "y" that stand for angles. They are greek letters.
what does subtending mean?
Subtending an arc means that the arms of the angle are separating a part of the circle. They are creating a new arc of the circle.

It basically is saying that there are two (or more) angles whose arms end in the same place.
how does sal gets 1/2 at 5:50 - 5:56
He divided both sides of the equation by 2.
i.e. if 2 pens are worth 1 dollar, then 1 pen is worth 1/2 a dollar.
What is the biggest and smallest possible angle
Technically, the smallest angle could be 0 degrees, and the largest, 360. But, with both of those the angle is practically either ending, or just beginning, and is really non-existent. So, the smallest angle possible would be 1 degree large (or a decimal value of that, such as 0.000001), and the largest angle would be 359 (or again, a decimal value, such as 359.99999). Any smaller or larger (depending on what we are considering) and the angle starts all over again, or becomes nothing at all. Does that make sense?
find RVT if <RST is a central angle of circle S.
S = 118
RST is because if the circle is called circle S, than <S = <RST.
what are these ideas useful for?
Does 'subtend' just mean that it goes through an arc of the circle?
am i spelling this right?: si and data
its psi; theta
Why is it the only actual number I see in this entire video is 180? Don't the other angles have to be numbers?

I've only watched this video five times now. Having trouble conceptualizing this.
its all unknowns- the only thing you know for sure is that the angle of a straight line is 180. the ffigures are only there to help, they dont show the actual angle, therefore they are represented by variables. in geometry, greek letters are used as variables just as english letters are used in algebra. Yes, the other angles have to be numbers, but we dont know exactly what numbers they are.
Whhat does he mean when he says, when we have two sides being equal (isosceles) then the base is equal? What is the side and what is the base?
Could someone tell me what subtend means? Thanks.
Subtend means "intersect" or "encompass." For example, at about 2:00, the purple arc is subtended by psi and theta, that is, it is "inside" the angle.
What does 'si' (that cactus-shaped thing that Sal always uses) mean? It's the first time I've ever heard anyone use it in my whole life.

-mdanivas, a.k.a. Pants
This is Theta: ϴ and this is psi: ψ and they just are GrΣΣk letters used to mean unknown in Geometry.
how would any non- physicist use this in their life?
What is Si/Psi?
A greek letter used to note the inscribed angle...and is it psi
what is sy and data?
Psi and Theta are both letters in the Greek alphabet. Psi is the 23rd letter of the Greek alphabet while Theta is the 8th.
what is sy and data? please define and explain. THANK YOU!
Psi and theta and Greek letters. They are used in much the same way as x and y. They are variables.
at 2:01 what does si has mean?
Psi (not si) is a letter in Greek alphabet (Ψ). Mathematicians use Greek letters to write angles. They often use other letters : α(alpha), β(beta), γ(gamma), θ(theta)
How would you solve PSQ=3y-15 and PRQ=2y +25?
The problem is: Find m<PSQ if M<PSQ=3y-15 and m<PRQ= 2y+25
< = angle sign
A circle with an area of 49π square centimeters is inscribed (tightly inside of) a square. What is the area of the square?
could u tell me the proof of the inscirbed amgle theroem, case 2?
When the angles form a triangle or a quadrilateral, how would you find the missing angles of the polygon inscribed in the circle?
what is the angle called when it is neither central nor inscribed? and what is the formula for it. does the arc equal the angle or is it the angle half of the arc? help!
would the central angle also show the measure of the arc (number of degrees)?
Only thing to do in exercise: blue angle divided by 2 = orange angle and orange angle times 2 = blue angle. Do that for all the 3 exercise on this and you get 100% :) logic
what does si represent
its psi not si , and its a greek character. its just used to represent a random number , just like we use x , y etc.
at 0:35 can somebody please explain this to me in ways that are easier to understand just trying to comprehend what he had said is making my head hurt XD
@$#^%^$^%*&*&%$&&%%*&%%$#^%#%#@&^$#&%$#*&^!
Sal proved that theta is twice psi in the first case, but it seems (at least to me) that he used the fact that theta is "supposed" to be twice as much as psi to prove the other two cases instead of proving them all from scratch. Can anyone explain if I'm missed something?
Well, in 2nd case you can see that psi1 have same arc as teta1 and that exactly same as the first case
that's why Sal write that psi1 = 1/2 teta1
that's because we already learning that in 1st case!
is the diameter the line that goes across the circle?
Yes, the diameter is the line across the entire circle and it can be confused with the radius which only goes halfway across the circle...both go throuch the exact center of the circle.
What is psi?
Psi is simply a greek letter that Sal has chosen to represent an angle. It is a variable, just like x, y, z or theta.
what value does SI have i can not find a video about it
So how do you work out trying to find inscribed angles with a quadrilaterals inside of circles? Do you just separate the angles?
how much rays are on this circle?
A state park decided to keep track of how many people use each of its two hiking trails each year.