anonymous
  • anonymous
https://media.education2020.com/evresources/2092539_2f57b746-c73f-4a4b-8c7e-ce9ee207b927.png Find the length of the radius for circle C.
Geometry
jamiebookeater
  • jamiebookeater
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mathstudent55
  • mathstudent55
When the radius of a circle is r, what is the circumference of the circle?
anonymous
  • anonymous
C= 2 pi r
mathstudent55
  • mathstudent55
Great. The circumference of a circle is an arc that corresponds to a central angle of 360 degrees or 2 pi radians.

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mathstudent55
  • mathstudent55
Now look in your figure.
mathstudent55
  • mathstudent55
You have a central angle of 4pi/9 radians corresponding to an arc length of 20pi/3 units.
mathstudent55
  • mathstudent55
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nincompoop
  • nincompoop
maybe explain what \(\pi \) corresponds to as in a unit circle
mathstudent55
  • mathstudent55
If we divide the circumference by the central angle, we get the radius.
anonymous
  • anonymous
So how do we get the circumference from what we have now
anonymous
  • anonymous
Since we don't have the radius I'm confused 😕
mathstudent55
  • mathstudent55
In a circle, the central angle is \(2 \pi \) rad. Dividing the circumference by the central angle gives us the radius. \(\dfrac{2 \pi r}{2 \pi} =r \)
mathstudent55
  • mathstudent55
Let me explain more clearly.
mathstudent55
  • mathstudent55
In this problem we are dealing with an arc length, and a radius, and a central angle.
anonymous
  • anonymous
But how can we use r when we don't have the radius
nincompoop
  • nincompoop
we can skip the circumference since we are only dealing with an arc the length of an arc is calculated by the formula, \(\large s = r \times \theta \) you are given the measurements of s and angle
mathstudent55
  • mathstudent55
When the central angle is given is radians, the length of the arc is the central angle multiplied by the radius.
anonymous
  • anonymous
Ok so
anonymous
  • anonymous
What would the circumference be
mathstudent55
  • mathstudent55
Since you are given the length of the arc and the central angle, and we are dealing with radian measures, divide the length of the arc by the central angle and you get the radius.
nincompoop
  • nincompoop
then we know how to approach how to calculate r \(\large \ r = \frac{s}{\theta} \rightarrow \huge \frac{20 \pi}{3} \div \frac{4 \pi}{9} \) follow the rules when dividing two fractions
mathstudent55
  • mathstudent55
\(s = r \theta\) \(r = \dfrac{s}{\theta} \) You have s, the arc length, and theta, the central angle. Plug them is and find r.
anonymous
  • anonymous
I got it thanks so much
anonymous
  • anonymous
I have one more question.
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nincompoop
  • nincompoop
have you been paying attention or not? now you are being asked to solve for s the arc length, \(s = \large r \times \theta \)
mathstudent55
  • mathstudent55
In the second problem you use the arc length formula directly. \(s = r \theta\) You are given r, the radius, and theta, the central angle. just multiply them together.
anonymous
  • anonymous
Last question.
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mathstudent55
  • mathstudent55
You mean the red section of the circle?

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