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Challenge problems: Arc length (radians) 2

Solve two challenging problems that ask you to find an arc measure using the arc length.

Problem 1

In the figure below, AC is a diameter of circle P. The radius of circle P is 30 units. The arc length of BC is 13π.
A circle that is centered around point P. Points A, B, and C are on the circle and line segment A C is a diameter. Line segments A P, B P and C P are radii of the circle that are thirty units long. Minor arc B C is thirteen pi units long.
What is the arc measure of AB, in radians?
Choose 1 answer:

Problem 2

In the figure below, AD is a diameter of circle P. The radius of circle P is 27 units. The arc length of BC is 8110π.
A circle centered at point P. Points A, B, C, and D all lie on the circle in a clockwise direction. Line segment A D is a diameter of circle P. Angle A P B is a right angle. Segment B P is a radius, and it's twenty seven units. Segment C P is a radius, and the arc length of B C is eighty-one pi divided by ten radians.
What is the arc measure of CD, in radians?
Choose 1 answer:

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  • duskpin sapling style avatar for user Natsu and Happy
    With the first qs, why would 17pi be incorrect? All I did was find the circumference of the circle which is 60pi then divided it by 2 because the diameter indicated to me that we were only focusing on the right side of the circle. Then I subtracted 13pi from 30pi to get 17pi; and if you were to place 17pi into the missing length, you would still get a circumference of 60pi....I still don't get why this method is wrong...
    (15 votes)
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  • mr pink red style avatar for user Abbey
    Why do the explanations have you find the circumference and set up a ratio when you can do the problem so much simpler? A couple videos back Sal taught that the arc measure times the radius equals the arc length. In problem 1, just divide the arc length (13pi) by the radius (30) to get the arc measure of BC (13pi/30). Then subtract that from 2pi and you get 17pi/30, which is the correct answer.

    Is it just me or does the whole 'finding circumference and setting up ratio' thing seem a bit unnecessary?
    (8 votes)
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  • leaf blue style avatar for user Vinit Srivastava
    why so long?
    there is more simple way to do both question.for example 1st question:
    arc measure =arc length/radius(13pi/30)
    then pi-(13pi/30)=17pi/30
    DONE! in 2 steps
    (5 votes)
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  • aqualine seed style avatar for user Wei  Chen
    Hey guys, I have a question for problem 2. According to the definition of radian, arc length divided by radius equals to arc measure(in radians) ( e.g. arc length r / radius r = arc measure 1 radian). If this is true, since angle BPD = pi/2, I can know the arc length of BCD is pi/2 * 27 = 27pi/2 ( arc measure * radius r = arc length). To find the arc length of arc length CD, I can just deduct the length of arc BC from the arc length BCD, 27pi/2 - 81pi/10 = 135pi/10 - 81pi/10 = 54pi/10 = 27pi/5, which is clearly the wrong answer for this problem. Can anyone tell me what I did wrong here? Is it the assumption that arc length divided by radius equals to arc measure(in radians) is wrong? Or is it something else? Thanks!
    (2 votes)
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  • blobby green style avatar for user shaanth21
    Can someone explain the proportion to me in number 1?
    (2 votes)
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    • male robot hal style avatar for user Art
      the arc LENGTH to circumference (a length) = arc measure (angle measurement) to radians in the circle (angle measurement) It is like a percentage, part divided by the whole. The parts are arc length and arc measure; the whole are circumference and radians in a circle. 2pi
      (4 votes)
  • hopper cool style avatar for user Uddip Kashyap
    why did you multiply by 2pi in the second problem?
    (3 votes)
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  • leaf green style avatar for user Nicolo-72
    Can somebody please tell me where pi/2 comes from in the question
    (2 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      ∠𝐴𝑃𝐵 is a right angle, so its measure is 𝜋/2 radians.

      It's a bit unnecessary to include it in the calculations, though, because if three angles are supplementary and one of the angles is a right angle, then the other two angles are complementary, and thus:
      3𝜋/10 + 𝑚∠𝐶𝑃𝐷 = 𝜋/2 ⇔ 𝑚∠𝐶𝑃𝐷 = 𝜋/5
      (3 votes)
  • blobby green style avatar for user laddhanishtha
    can we use radians for other figures like spheres, cylinders, etc? if so how? if not, why not? much appreciated!
    (2 votes)
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  • blobby green style avatar for user Gregory  Gentry
    i did all first try correct
    (2 votes)
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  • blobby green style avatar for user Bipin Dhakal
    points A and B lies on a circlr with radius 1 and arc AB has length π/3. what fraction of circumferenceof the circle is the length of arc AB?
    (1 vote)
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