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Challenge problems: Inscribed shapes

Solve three challenging problems that apply properties of inscribed shapes to find an arc length or a missing angle.

Problem 1

In the figure below, the length of PB is 15 units.
What is the exact length of AB on circle P?
Choose 1 answer:

Problem 2

In the figure below, quadrilateral ABCD is inscribed in circle P. The radius of circle P is 5 units.
What is the length of BCD?
Either enter an exact answer in terms of π or use 3.14 for π and enter your answer as a decimal.​

Problem 3

In the figure below, angles CBD and BCA are inscribed in circle P. The radius of circle P is 5 units. The arc length of BA is π, and the arc length of CD is 3π.
What is the exact measure of AED, in radians?
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.​
radians

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  • leaf green style avatar for user Rizzy
    I don't get the last one. Where did the 1/10 pi and 3/10 pi come from? I'm so confused.
    (12 votes)
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    • aqualine tree style avatar for user Judith Gibson
      Before going into the detail of the problem, here's an overview of how to approach solving it..
      We want to find the measure of angle AED.
      That angle is a vertical angle with angle BEC (so they're equal).
      And angle BEC is one of the 3 angles in triangle BEC.
      The other 2 angles are inscribed angles for which we're given their arc lengths.
      So we can find the measure of those 2 angles, then use the fact that the sum of the angles of any triangle is 180 degrees ( pi radians) to find angle BEC which equals angle AED (because they're vertical angles).
      Now, for the detail!
      1. We'll use the fact that, because the radius is given as 5, the circumference (2pi*r) can be calculated to be 10pi
      2. So, now ask yourself : the length of arc BA (given as pi) is what part of the circumference of 10pi?
      This gives the fraction of pi / 10pi which equals 1 / 10 .
      3. Do the same for arc CD (given as 3pi).
      This gives the fraction of 3pi / 10pi which equals 3 / 10.
      4. Now that we have the "arc length to circumference" ratios, we use those same ratios to proportion the inscribed angles (which by definition are half the arc length)
      So, dividing 1 / 10 by 2 gives 1 / 20 for angle BCA and dividing 3 / 10 by 2 gives 3 / 20 for angle CBD.
      In other words, angle BCA has 1 / 20 of the angles in a circle (360 degrees or 2pi)
      and angle CBD has 3 / 20 of them.
      So, angle BCA = ( 1 / 20 ) * 2pi = ( 1 / 10 ) * pi
      and angle CBD = ( 3 / s0 ) * 2pi = ( 3 / 10 ) * pi
      5. The 3 angles in triangle BCE add up to pi radians ( = 10/10 pi),
      ( 1 / 10 ) * pi + ( 3 / 10 ) * pi + angle BEC = ( 10 / 10 ) * pi
      so, the missing angle (which angle AED equals) must be ( 6 / 10 ) * pi = 3pi /5
      This is a lengthy reply, but I hope it helps you to be less confused (instead of more confused!).
      (39 votes)
  • primosaur ultimate style avatar for user jiceprice333
    Can someone explain how I can intuitively know when I read a problem like these to know whether I should be looking for a result in Radians or in Degrees? I keep trying to answers these by what I think should be the approach and keep using the wrong one. I must be missing something obvious. Thank you.
    (6 votes)
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    • male robot hal style avatar for user KEVIN
      Let's look at question 3 as an example. While we could convert our radian measurements to degrees, we are asked to find the answer in radians. If they had mixed/matched some of the information in degrees and radians, then we would have to consider them both in our computation. Also, again in question 3, we are given measurements that reference pi; that is a dead give away that we will at least have to consider computing with radians. Hope this helps a bit.
      (13 votes)
  • male robot hal style avatar for user Sarvesh Babu
    In problem 2 I set up the problem in such a way that the: Total circumference - BAD = BCD. So I got 10"pi"-BAD = BCD. Then I solved for BCD using Arc length method that sal taught us with the proportions a few videos back.

    BAD/10"pi"= 9"pi"/20/2"pi"

    BAD/10"pi"= 9/40

    (I cross multiplied)

    90"pi"= 40BAD

    BAD = 90"pi"/40

    BAD=9"pi"/4

    Then I subtracted that from the total circumference

    10"pi"-(9"pi")/4=31"pi"/4

    So BCD is 31"pi"/4

    Could you please tell me where i went wrong?
    (4 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      You went wrong in the second paragraph: BAD/10"pi"= 9"pi"/20/2"pi".

      9𝜋 ∕ 20 is the angle at 𝐶, but we want the angle at 𝑃, which is 2 ∙ 9𝜋 ∕ 20 = 9𝜋 ∕ 10.

      So, the equation should read:
      (Arc 𝐵𝐴𝐷) ∕ (10𝜋) = (9𝜋 ∕ 10) ∕ (2𝜋) ⇔ (Arc 𝐵𝐴𝐷) = 9𝜋 ∕ 2 ⇒ (Arc 𝐵𝐶𝐷) = 11𝜋 ∕ 2.
      (9 votes)
  • blobby green style avatar for user Liam Pirie
    I dont understand how the angles in the last problem could possibly add up to 180 they are all 2 or 3 when multiplied out
    (4 votes)
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  • blobby green style avatar for user Mary
    I do not understand a part of the explanation on problem 1, which reads, "The sides line segment AP and line segment PB are congruent because they are radii of the same circle."


    Also, is the measure of an angle the same as its arc measure?
    (3 votes)
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    • leaf green style avatar for user kubleeka
      A radius of a circle is a straight line drawn from the center to the edge. All radii of a given circle are the same length (this is what it means to be a circle).

      So because AP and PB are both radii of circle P, they must be the same length (congruent).

      Also, yes, the measure of an arc is the same as the measure of the central angle that contains it. However, this problem asks for the length of the arc, not the measure.
      (5 votes)
  • blobby green style avatar for user shouidi
    How do we know in question 2 that m< MBD + 9/20 pi equals pi and not 180?
    (3 votes)
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  • stelly yellow style avatar for user Suresh123.Ani
    what is the difference between arc length and arc measure ,given one how do you find the other
    (1 vote)
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    • mr pink green style avatar for user David Severin
      The words give a hint. Arc length is a distance around a part of the circle, so this is comparable to the total length around a circle C = 2πr. The arc measure is an angle measure which is compared to the total angle around a circle 360 degrees or 2π radians. Set up a proportion between these two and solve.
      (6 votes)
  • blobby green style avatar for user Mary
    I thought the arc measure was supposed to be equal to the angle that intercepts it or in other words the central angle is equal to the arc it intercepts? However, in the second problem, the explanation states, "The measure of an inscribed angle is half of the measure of the arc it intercepts."
    (2 votes)
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    • sneak peak green style avatar for user Isabelle Thurmond
      In problem 2, the vertex of the angle is not positioned at the center of the circle. If it were, the arc that it intercepts would have the same measurement.

      However, the vertex of the angle is positioned on the circle itself, making the arc that the inscribed angle intercepts twice as big as the angle itself.
      (3 votes)
  • starky sapling style avatar for user Elaine
    What do you people prefer - Radians or Degrees? I actually love radians...
    (3 votes)
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  • blobby green style avatar for user stewart.joel
    I saw the circle proportions equated when I thought the problem should have been solved, but was missing that arcMeasure/radiansInCircle is all in radians. Overall, I kept missing that I am figuring in radians and converting to units. "==" is to be read as "is equivalent to". For future me:

    (arc length in units/ circumference in units) represents arc length as a proportion of the entire circle's arc length,
    (arc measure in radians / circle measure in radians) represents arc measure in radians as a proportion of the entire circle in radians,
    (arc measure in degrees / circle measure in degrees) represents arc measure in degrees as a proportion of the entire circle in degrees.
    Each proportion is equivalent to the other.

    In problem 2: the radius is 5 units. Opposite angles in a quadrilateral add up to 180° (which is equivalent to π radians).
    As the measure of angle BCD is (9π/20) radians, the measure of angle BAD must be π radians - (9π/20) radians == (11π/20) radians.
    The radian measure of arc BCD is twice its angle: 2*(11π/20) radians == (11π/10) radians.
    Convert the radians to the units: (arcLength) Units / (10π) Units == (11π/10) radians / (2π) radians == (11π/2) units.

    In problem 3: the radius is 5 units. length of arc BA is π units -- use the proportions to find its equivalency in radians: (π/5) radians; the angle in radians is half the arc length in radians: therefore angle BCA is (π/10) radians.
    Length of arc CD is 3π units -- use the proportions to find its equivalency in radians: (3π/5) radians; the angle in radians is half the arc length in radians: therefore angle CBD is (3π/10) radians.
    Therefore angle BEC is π radians - (3π/10) radians - (1π/5) radians == (6π/10) radians == (3π/5) radians.
    Therefore angle AED is (3π/5) radians.

    I think having the units in the answer-explanations would have sped up debugging what I was failing to observe.
    (2 votes)
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