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### Course: Geometry (all content)>Unit 14

Lesson 7: Inscribed angles

# Challenge problems: Inscribed angles

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, $\mathrm{\angle }ABC$ is inscribed in circle $P$. The length of $\stackrel{―}{PC}$ is $12$ units. The arc length of $\stackrel{⌢}{AC}$ is $\frac{68}{15}\pi$.
What is the measure of $\mathrm{\angle }ABC$ in degrees?
${}^{\circ }$

## Problem 2

In the figure below, $\mathrm{\angle }ABC$ is inscribed in circle $P$. The length of $\stackrel{―}{PC}$ is $4$ units.
What is the length of $\stackrel{⌢}{AC}$?
Either enter an exact answer in terms of $\pi$ or use $3.14$ for $\pi$ and enter your answer as a decimal rounded to the nearest hundredth.​
units

## Want to join the conversation?

• 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
= 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.
• shouldn't there be more problems on this stuff?
• 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.
• 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,

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.
• The question is I still don't understand math!
• 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.
• How do you get 2 radians?
• 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.