Couldn't you have just used the formula S=r*theta ?

Problem 2 : ∠APC = 2 * ∠ABC = 2 * 2/5 π radians = 4/5 π radians Arc AC = 4/5 π radians * 4 units per radian = 16/5 π units

In regards to the 2nd problem. I calculated the "length of AC" to be 7.61... Isn't the term "length of AC" meaning the direct measurement from A to C (forming a triangle APC)? Shouldn't the 2nd problem say, "what is the arc length of AC"?

I don't know if anything's changed since you were here last, but for me it says, "What is the length of AC," and there's a little curve over the letters AC. That means they're talking about arc AC.

just a question of comprehension, in problem 1, using the ratio problem arc length/circumference=arc measure/360, why are you able to shift a denominator dividing a fraction to another number, or in relation to the equation, when you multiply 360 to both sides, and then shift the 24pi under it?

Hi Joshua, To answer the "why" part of your question- Whatever you multiply to one side of the equation, you need to multiply it to the other side of the equation too. When you do so, are are not changing the value of the equation. This helps us to isolate the unknown term (arc measure in this case) to one side of the equation. So, this is what your equation will look like- 68/15 pi/ 24 pi = arc measure/ 360 we can write the above equation in a simplified form too and it would look like- 68 pi/ 15 X 24 pi = arc measure/ 360 When we multiply both sides by 360, the equation will be- 360 X 68 pi/ 15 X 24 pi = arc measure X 360/ 360 (360/ 360 = 1) you can see that multiplying by 360 on both sides does not change the equation but will help us to isolate "arc measure" to one side of the equation. When you simplify, you will get- 360 X 68 pi/ 15 X 24 pi = arc measure arc measure = 68. Hope this helps. Regards, Aiena.

couldn't we just transform the radians in degree before using the proportion?

Whenever you convert radians to degrees there will be most likely be rounding involved, meaning the results of your calculations will not be as exact but rather a close approximation. This means, when you enter the answer it will come up wrong, as it is looking for the exact answer based on the radians, not the approximated answer of degrees.

On the second problem is the angle measure in degrees or radians?

If you see a π in the angle, you can assume it is in radians.

The question is I still don't understand math!

i believe that your problem is a skill issue

How do I find the angle of a circle that is inscribed with fractions

Hi 19mwebb21c, Below are the two ratios that will help you find the arc measure: 1. arc measure/ degrees or radians in a circle = arc length/ circumference of the circle 2. arc measure/ degrees or radians in a circle = arc area/ area of the circle It does not matter that the arc length or the area of the circle is a fraction or an integer. Hope this helps. Feel free to ask any specific question. Regards, Aiena.

What is 2/5 π in the problem 2? I guess this is the angle in radians but there is no information about it.

It is the measure of angle ABC. You can use it to find the measure of angle APC, which can then help you find the arc length

Main content

Course: Geometry (all content) > Unit 14

Lesson 7: Inscribed angles

Challenge problems: Inscribed angles

Google Classroom

Solve two challenging problems that apply the inscribed angle theorem to find an arc measure or an arc length.

Problem 1

In the figure below,

∠ A B C

is inscribed in circle

P

. The length of

\overset{―}{P C}

12

units. The arc length of

\overset{⌢}{A C}

\frac{68}{15} π

What is the measure of $∠ A B C$ ‍ in degrees?

^{\circ}

We need to know the total circumference in order to determine the arc measure for

\overset{⌢}{A C}

$\begin{aligned} Circumference & = 2 π r \\ = 2 π (12) \\ = 24 π \end{aligned}$ ‍

We know the length of

\overset{⌢}{A C}

, so we set up a proportion to figure out its arc measure.

$\begin{aligned} \frac{arc length}{circumference} & = \frac{arc measure}{degrees in a circle} \\ \frac{(\frac{68}{15} π)}{24 π} & = \frac{arc measure}{360^{\circ}} \\ arc measure & = \frac{68}{15} π \cdot \frac{360^{\circ}}{24 π} \\ = 68^{\circ} \end{aligned}$ ‍

The measure of an inscribed angle is half of the measure of the arc it intercepts. If

m \overset{⌢}{A C} = 68^{\circ}

, then

m ∠ A B C = 34^{\circ}

The measure of $∠ A B C$ ‍ is $34^{\circ}$ ‍.

Problem 2

In the figure below,

∠ A B C

is inscribed in circle

P

. The length of

\overset{―}{P C}

4

units.

What is the length of $\overset{⌢}{A C}$ ‍?
Either enter an exact answer in terms of $π$ ‍ or use $3.14$ ‍ for $π$ ‍ and enter your answer as a decimal rounded to the nearest hundredth.

units

We are trying to figure out the length of the part of the circumference of circle

P

that begins at point

A

and ends at point

C

Let's calculate the full circumference first, then solve proportionally for the arc length based on the radians in the arc measure.

We know that circle

P

has a radius of

4

units because

\overset{―}{P C}

is a radius.

$\begin{aligned} Circumference & = 2 π r \\ = 2 π (4) \\ = 8 π \end{aligned}$ ‍

The measure of an inscribed angle is half of the measure of the arc it intercepts. If

m ∠ A B C = \frac{2}{5} π

, then

m \overset{⌢}{A C} = \frac{4}{5} π

Finally, we can set up a proportion to solve for the part of the circumference intercepted by that angle.

$\begin{aligned} \frac{arc length}{circumference} & = \frac{arc measure}{radians in a circle} \\ \frac{arc length}{8 π} & = \frac{(\frac{4}{5} π)}{2 π} \\ arc length & = \frac{4 π}{5} \cdot \frac{8 π}{2 π} \\ = \frac{16}{5} π \end{aligned}$ ‍

The length of $\overset{⌢}{A C}$ ‍ is $\frac{16}{5} π$ ‍.

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jk0169
Posted 8 years ago. Direct link to jk0169's post “Couldn't you have just us...”
Couldn't you have just used the formula S=r*theta ?
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(21 votes)
Answer
- Deep Bhatia
  Posted 8 years ago. Direct link to Deep Bhatia's post “Problem 2 : ∠APC = 2 * ∠A...”
  Problem 2 : ∠APC = 2 * ∠ABC = 2 * 2/5 π radians = 4/5 π radians
  Arc AC = 4/5 π radians * 4 units per radian
  = 16/5 π units
  Comment on Deep Bhatia's post “Problem 2 : ∠APC = 2 * ∠A...”
  (17 votes)
Yames Yamb
Posted 8 years ago. Direct link to Yames Yamb's post “In regards to the 2nd pro...”
In regards to the 2nd problem.
I calculated the "length of AC" to be 7.61... Isn't the term "length of AC" meaning the direct measurement from A to C (forming a triangle APC)? Shouldn't the 2nd problem say, "what is the arc length of AC"?
Button navigates to signup pageComment on Yames Yamb's post “In regards to the 2nd pro...”
(10 votes)
Answer
- Thessalonika
  Posted 8 years ago. Direct link to Thessalonika's post “I don't know if anything'...”
  I don't know if anything's changed since you were here last, but for me it says, "What is the length of AC," and there's a little curve over the letters AC. That means they're talking about arc AC.
  Button navigates to signup page
  (24 votes)
Priya
Posted 7 years ago. Direct link to Priya's post “shouldn't there be more p...”
shouldn't there be more problems on this stuff?
Button navigates to signup pageComment on Priya's post “shouldn't there be more p...”
(16 votes)
Answer
Abderrahim Jami
Posted 8 years ago. Direct link to Abderrahim Jami's post “couldn't we just transfor...”
couldn't we just transform the radians in degree before using the proportion?
Button navigates to signup pageButton navigates to signup page
(7 votes)
Answer
- Ina
  Posted 6 years ago. Direct link to Ina's post “Whenever you convert radi...”
  Whenever you convert radians to degrees there will be most likely be rounding involved, meaning the results of your calculations will not be as exact but rather a close approximation.
  
  This means, when you enter the answer it will come up wrong, as it is looking for the exact answer based on the radians, not the approximated answer of degrees.
  Button navigates to signup page
  (5 votes)
Joshua DuWors
Posted 7 years ago. Direct link to Joshua DuWors's post “just a question of compre...”
just a question of comprehension, in problem 1, using the ratio problem arc length/circumference=arc measure/360, why are you able to shift a denominator dividing a fraction to another number, or in relation to the equation, when you multiply 360 to both sides, and then shift the 24pi under it?
Button navigates to signup pageButton navigates to signup page
(9 votes)
Answer
- Aiena
  Posted 7 years ago. Direct link to Aiena's post “Hi Joshua, To answer the...”
  Hi Joshua,
  
  To answer the "why" part of your question-
  Whatever you multiply to one side of the equation, you need to multiply it to the other side of the equation too. When you do so, are are not changing the value of the equation. This helps us to isolate the unknown term (arc measure in this case) to one side of the equation.
  
  So, this is what your equation will look like-
  68/15 pi/ 24 pi = arc measure/ 360
  
  we can write the above equation in a simplified form too and it would look like-
  68 pi/ 15 X 24 pi = arc measure/ 360
  
  When we multiply both sides by 360, the equation will be-
  360 X 68 pi/ 15 X 24 pi = arc measure X 360/ 360 (360/ 360 = 1)
  you can see that multiplying by 360 on both sides does not change the equation but will help us to isolate "arc measure" to one side of the equation.
  
  When you simplify, you will get-
  360 X 68 pi/ 15 X 24 pi = arc measure
  arc measure = 68.
  
  Hope this helps.
  
  Regards,
  
  Aiena.
  Button navigates to signup page
  (6 votes)
Kaystis Cook
Posted a year ago. Direct link to Kaystis Cook's post “The question is I still d...”
The question is I still don't understand math!
Button navigates to signup pageComment on Kaystis Cook's post “The question is I still d...”
(3 votes)
Answer
- DerekG
  Posted a year ago. Direct link to DerekG's post “i believe that your probl...”
  i believe that your problem is a skill issue
  Comment on DerekG's post “i believe that your probl...”
  (16 votes)
Aaroh Gokhale
Posted 6 years ago. Direct link to Aaroh Gokhale's post “On the second problem is ...”
On the second problem is the angle measure in degrees or radians?
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(4 votes)
Answer
- David Severin
  Posted 6 years ago. Direct link to David Severin's post “If you see a π in the ang...”
  If you see a π in the angle, you can assume it is in radians.
  Comment on David Severin's post “If you see a π in the ang...”
  (11 votes)
Connor Hinkel
Posted 4 years ago. Direct link to Connor Hinkel's post “How do you get 2 radians?”
How do you get 2 radians?
Button navigates to signup pageComment on Connor Hinkel's post “How do you get 2 radians?”
(5 votes)
Answer
19mwebb21c
Posted 7 years ago. Direct link to 19mwebb21c's post “How do I find the angle o...”
How do I find the angle of a circle that is inscribed with fractions
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Aiena
  Posted 7 years ago. Direct link to Aiena's post “Hi 19mwebb21c, Below are...”
  Hi 19mwebb21c,
  
  Below are the two ratios that will help you find the arc measure:
  
  1. arc measure/ degrees or radians in a circle = arc length/ circumference of the circle
  2. arc measure/ degrees or radians in a circle = arc area/ area of the circle
  
  It does not matter that the arc length or the area of the circle is a fraction or an integer.
  
  Hope this helps. Feel free to ask any specific question.
  
  Regards,
  
  Aiena.
  Button navigates to signup page
  (4 votes)
Remedy
Posted 10 months ago. Direct link to Remedy's post “What is 2/5 π in the prob...”
What is 2/5 π in the problem 2? I guess this is the angle in radians but there is no information about it.
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Deniss Fjodorovs
  Posted 10 months ago. Direct link to Deniss Fjodorovs's post “It is the measure of angl...”
  It is the measure of angle ABC. You can use it to find the measure of angle APC, which can then help you find the arc length
  Button navigates to signup page
  (3 votes)