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Challenge problems: Arc length 2

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

Problem 1

In the figure below, the radius of circle P is 10 units. The arc length of ABC is 16π.
A circle that is centered around point P. Points A, B, and C Line segments A P and C P are radii of the circle that are ten units long. Major Arc A B C is sixteen pi units.
What is the arc measure of AC, in degrees?
Choose 1 answer:

Problem 2

In the figure below, the radius of circle P is 18 units. The arc length of BA is 14π.
What is the arc measure of BC, in degrees?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
A circle that is centered around point P. Points A, B, and C Line segments A P, B P and C P are radii of the circle that are eighteen units long. Angle A P C is seventy-six degrees.

Want to join the conversation?

  • primosaur ultimate style avatar for user Lachesis
    In problem 2, why would arcAC + arcCB = arcAB? If the arc were labeled arcACB, I would understand, but the designation arcAB suggests that it refers to the arc on the other side of the circle, so that arcAC, arcAB and arcCB are three separate arcs that together form the full circle. There is no statement that the figure is drawn to scale, so there is no reason to assume that there are not three angles with measures 76º, 216º and 144º adding up to the full 360º of the circle.
    (71 votes)
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  • blobby green style avatar for user nroy
    in the 2nd question, why isn't arc AB called ACB for clarity?
    (8 votes)
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    • mr pants teal style avatar for user Mike G
      I agree, this problem is very tricky. I reviewed several times and realized most of my confusion was getting the distance between AB and BA confused. (AB is the longer of the two) What helped me understand was that the distance was given as 18PI. This made sense when I figured out the circumference of the whole circle was 36PI.
      The distance of the section they are asking for is the smaller of the two (BCA) vs BA
      After I figured that out, the hints make sense.
      (5 votes)
  • blobby green style avatar for user 😊
    well, I have a question in problem number 2... after we get 140, why did we subtract it from 76! it doesn't make sense at all! the two angles are not equal to the other (larger) and when we add 76,140,and 64, it gives us 280 not 360.. this is driving me crazy!! I spent 2 hours trying to understand this problem..
    (5 votes)
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    • duskpin ultimate style avatar for user Gray
      BC is 64 degrees. AC is 76 degrees. They both add up to 140 degrees. I think what you did was double that and likely got confused in the shown work.
      Conveniently in this problem, every pi length is 10 degrees, as 36pi is a full circle.
      So we have 14pi, which is the length of AB. that means, the diameter created by B going through the center to A is 4pi, which makes one side with the length 18pi. To get the full circle, we double that, which gets us to 36pi.
      Hopefully this helps!
      (4 votes)
  • winston baby style avatar for user Samuel Lee
    There is a lot of confusion in the comment section and I am baffled at what all of you are saying. Where did you get all these other numbers like "144" from? Can someone please explain to me in a lucid manner?
    (4 votes)
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  • piceratops ultimate style avatar for user Jukka Kuivikko
    I do not understand that how does, the angle degrees add to 360? 140 degrees+76degrees+64degree = 280 degrees, not 360degrees as a full circle?
    (3 votes)
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    • hopper cool style avatar for user HZWang
      Hi Jukka, I'm afraid you understood the explanation wrong. The 140 degrees is the combination of the 76 and 64 degree angle. If you wanted to check if the angles add up to 360, you would either add 76 and 64 and the major arc measure of BC or add 140 and the major arc BC. Of course, you would need another point on the circle to properly use the term 'major arc'. The problem explanation does not go into solving for the obtuse angle in the diagram. You could solve for it though: 360-140=220 degrees.
      I hope that cleared any confusion, and feel free to ask if you have any more questions!
      All in all, great question!
      ~Hannah
      (6 votes)
  • purple pi purple style avatar for user Minvy
    Q2.
    AC+CB=AB?? So all that theory about circles adding up to 360 degrees is wrong?
    (3 votes)
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  • blobby green style avatar for user Kharla Mendez
    Label in problem 2 is confusing. Better say 14pi is the arclength of ACB rather than AB.
    (2 votes)
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  • blobby green style avatar for user michelle perez
    mr the problem 2 from arc length 2 assignment the answer is wrong. the answer is 144 degree
    (3 votes)
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    • mr pink green style avatar for user David Severin
      With a radius of 18, the circumference is C=2πr=36π. So the arc is 14π/36π of the circumference or 7/18 of a circle. In degrees, 7/18*360 = 140 degrees for arc BA, and since you already know 76 degrees, 140-76=64 degrees. How are you getting 144 degrees?
      (2 votes)
  • aqualine seed style avatar for user murarisettigayatri20
    In the figure above, the circle has center O and angle ∠XOZ = 40°. If the length of arc Arc XZ (shown in bold) is between 3 and 5, what is one possible integer value for the length of the radius?
    (3 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      We have the arc length (at least a range of values for the arc length) and the angle, and we want to find the radius. To do this, we can set up our usual proportion, where the fraction of 360 degrees that is the arc's angle is the same as the fraction of the circumference that is the arc length:
      40 / 360 = arc length / (2pi*r)
      We know the arc length is between 3 and 5, so the first thing that comes to mind is to test both 3 and 5, and see what our possible values of radius come out to be:
      40/360 = 3/(2pi*r)
      1/9 = 3/2pi*r
      2pi*r = 27
      r = 27 / 2pi = 4.29...
      40/360 = 5 / (2pi*r)
      1/9 = 5 / 2pi*r
      2pi*r = 45
      r = 45 / 2pi = 7.16...
      So, if we know that the length is between 3 and 5 for an arc that stretches over 40 degrees, the radius (integer values) has to be between 5 and 7, inclusive. Does this help?
      (2 votes)
  • leaf blue style avatar for user KAMRYNW
    It seems that it is saying arc ba is = to arc bca ..how is that so?
    (1 vote)
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