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## SAT

### Course: SAT > Unit 10

Lesson 4: Additional topics in math- Volume word problems — Basic example
- Volume word problems — Harder example
- Right triangle word problems — Basic example
- Right triangle word problems — Harder example
- Congruence and similarity — Basic example
- Congruence and similarity — Harder example
- Right triangle trigonometry — Basic example
- Right triangle trigonometry — Harder example
- Angles, arc lengths, and trig functions — Basic example
- Angles, arc lengths, and trig functions — Harder example
- Circle theorems — Basic example
- Circle theorems — Harder example
- Circle equations — Basic example
- Circle equations — Harder example
- Complex numbers — Basic example
- Complex numbers — Harder example

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# Circle theorems — Basic example

Watch Sal work through a basic Circle theorems problem.

## Want to join the conversation?

- Couldn't you find the side of tiangle using the probeties of a 3-4-5 triangle?(12 votes)
- Yes, you can always do that if you encounter a right triangle with a hypotenuse of 5 and one leg measuring 3. Here we DON'T know that the small leg is 3 at first. The reason we can use the 3, 4, 5-triangle AFTER we know for sure that BC = 3 is that we know from using the Pythagorean Theorem (once or dozens of times) that our result will be 4 IF the hypotenuse is 5 and the other leg is 3. Then we learn that 3, 4, 5 is a Pythagorean triplet like 12, 13 , 5 and 24, 7, 25 and 6, 8, 10

HOWEVER, the point with this question is that we**don't**know without using some other geometry that the small leg is actually 3.

In geometry, we cannot use the fact that it`seems like`

about half of 6. We need to use the fact that the radius is the hypotenuse of both right triangles. Then we can use the HL Congruency to show that the triangles are congruent and then using CPCTC that the AC and BC are congruent. Then we know that OC bisects AB, and BC is 3 units long. WHEW!

OR, as Sal did here, we can use the`great shortcut--thanks to one of the circle theorems--`

that a radius bisects chord AB if it is perpendicular to it, which is**given**.

BOOM! We then can be confident that the leg BC is 3 units long and use the other shortcut of the Pythagorean Triple 3, 4, 5 to answer.(32 votes)

- Is the Principal Root just another name for the square root?(4 votes)
- No, a principal root is only the positive answer of the square root. Example: the square root of 25 is 5 and -5 but the principal root is only 5.(17 votes)

- Why can't we use the formula: arc length = Theta * radius(7 votes)
- You could, and that would be a lot faster than the process Sal uses. I think he didn't because to apply that formula your angle value has to be in radians, which is very, very easy to just forget about in the heat of the moment, and so using a simpler slightly more intuitive formula would be better for most students. But if you're nailing questions with arc length = theta * radius, go for it. It's every bit as valid, and all you have to do is convert to radians every now and then.(7 votes)

- As soon as you found the 3 you would notice that its a 3 4 5 right traiangle and done(9 votes)
- At first I thought that the arc AB equals 6 , because it said chord.(2 votes)
- As far as I know, "chord" refers to the line, not the arc.(8 votes)

- I thought the answer would be 135 because I read (in the PWN the SAT book) that the central angle of a circle is equivalent to the arc length...can someone explain in what cases this rule/theorem is valid? is it cause we don't know if O is the center or the circle or not?(2 votes)
- You might have misunderstood what you were reading. The central angle of a circle is equivalent to the
*angle measure*of the arc, not its length. Think about it. Angle by itself cannot be the only factor that determines length, because bigger circles will have bigger arcs than smaller circles would, even if the central angle is the same.

It's weird that the problem doesn't explicitly say that the circle's center is O, probably because it's just for practice. On the actual SAT, you'd be right to be careful of assuming that a given point is the center of a circle, or that two lines are parallel, for example, unless the problem tells you.(2 votes)

- What is a sector?(2 votes)
- but doest 9/4 pi equal 405? why cant 405 be the right answer?(2 votes)
- 135 divided by 6 is 22.5, not 27. I’m confused.(1 vote)
- But alas, he is not saying 135 / 6. At approximately2:57, he says 135 / 5, which is indeed 27.(2 votes)

- where did he get the 25 from?(1 vote)
- 5(the length of the radius)squared is 25(2 votes)

## Video transcript

- [Instructor] What is
the length of arc ABC? So arc ABC is this arc right over here and the reason why they
have to give us three points is because if they just said arc AC, it would still ambiguous
because there's two arc ACs. It could be this arc right over here or it could be this arc over here. And so when they tell us B we know that we're going from A to C through B, so that's why this kind of, by
giving us that third letter, it resolves the ambiguity on
which arc we're talking about. What is the length of that arc? Well the first thing we could do is well, let's just
think about the entire, what's the circumference of this circle? And then we can say well, what fraction of the entire circumference is this arc? And a big hint here is if we were to go all the way around the circle,
that would be 360 degrees, but we're only going 135
of those 360 degrees. But let's first think
about the circumference. So the circumference of
a circle is equal to two, is equal to two pi times
the radius of the circle, and they give us the
radius as three inches. So the circumference is going to be two pi times three inches, so times three inches, or two times three is six, times pi, so it's going to be six pi, six pi inches. Now, and actually if
you look at the choices, if you look at the choices over here, the entire circumference of
the circle is six pi inches. Remember, pi is 3.14159,
so this is going to be something in the, you
know, it's going to be a little bit over 18 or 19 inches. So you could immediately,
if the entire circumference is a little over 18 or 19 inches, this is only a fraction of
that entire circumference, so actually if you even
were looking at the choices, you'd say, well hey,
these are way too large. But actually, let's see if we can get to the exact right answer. So the entire circumference
is six pi inches, but this arc length isn't
the entire circumference. It only goes 135 degrees
out of 360 degrees, so we could say, so it is 135/360 of the entire circumference. Remember, if we were to
go all the way around, that's 360 degrees, while
this angle right over here is 135 degrees, the angle
that is subtended by this arc. So this arc length is going to be 135/360 of the entire circumference, so times six pi, six pi inches. So let's see if we can
simplify this a little bit. So let's see, we could
divide the numerator and the denominator by six. Six divided by six is one. 360 divided by six is 60. Let's see, if we divide 135 by five, that is going to be 27. Yeah, 100 divided by five is 20, and then 35 divided by five is seven, so this is going to be 27, and then this would be 12,
if we divide 60 by five. Let's see, we could then
divide the numerator and the denominator by three. 27 divided by three is nine, and 12 divided by three is four, so we're left with 9/4 times pi inches, so 9/4 times pi inches, or we could multiply the
numerator times the pi. We would say nine pi over four inches. Now when you look at the choices, that is this one, that is
this one right over there.