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Challenge problems: circumscribing shapes

Solve two challenging problems that apply properties of tangents to find the perimeter of a circumscribing shape.

Problem 1

All sides of triangle, A, B, C are tangent to circle P.
A circle centered around point P. The circle is inscribed inside triangle A B C so that each side is tangent to the circle. Side A C is fourteen units. From the tangent point on side A B to point B is sixteen units.
What is the perimeter of triangle A, B, C?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units

Problem 2

All sides of quadrilateral A, B, C, D are tangent to circle P.
A circle centered around point P. The circle is inscribed inside quadrilateral A B C D so that each side is tangent to the circle. Side C D is twelve units. From the tangent point on side A B to point B is nine point six units. From the tangent point on side A D to point A is three point seven units.
What is the perimeter of quadrilateral A, B, C, D?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
units

Want to join the conversation?

  • blobby green style avatar for user rvillard
    I am confused. How did you solve when you still haven't found x?
    (5 votes)
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    • starky ultimate style avatar for user Felicia L.
      In both cases, the x cancels itself out. Take question 2: the perimeter is a whole bunch of numbers which I am too lazy to type out, and then there is an unknown length which I will call x. To find the other unknown length, you can just take 12-x. The perimeter will be all those numbers + x + 12 - x. The solution to x+12-x is 12, as you can rearrange it as x-x+12.
      (32 votes)
  • ohnoes default style avatar for user Harmon Byerly
    what are the answer to number 1 and/or 2. Im very confused on this subject matter.
    (4 votes)
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    • duskpin tree style avatar for user Arohee Bhoja
      Hi! I think I might be able to help :)

      So basically for #1, the main theorem you want to apply to the problem is the theorem that "2 tangent lines drawn to a circle from the same point are congruent." Let's call the point between AC "x", the point between BC "y" and the point between AB "z".
      Because of the theorem, you can say that YB is congruent to ZB (YB = ZB = 16). You have one piece of the puzzle right there!

      Now you can move on to the other pairs of tangent lines. Based on the same theorem we can say that XC is congruent to YC, and that XA is congruent to ZA. Let's set XC equal to "x" (XC = YC = "x") Because AC is equal to 14, we can call XA "14-x." (XA = ZA = "14-x")

      Now, let's add it all up! 2x + 2(14-x) + 2(16) = 28 + 32 = 60.

      #2 is extremely similar. If you want more help, don't hesitate to ask me or anyone else on here! :)
      (15 votes)
  • blobby green style avatar for user Abhi Reddy Mulukuri
    i did not understand the way to find perimiter of quadrilaterals and triangles
    (6 votes)
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  • blobby green style avatar for user b fish
    Challenge problems: circumscribing shapes Problem 1
    What logic tells us that we can use the same variable x on line AB versus line AC? They visually look to be very different lengths.
    (3 votes)
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  • blobby green style avatar for user shaikh sharfuddin
    Interior and exterior bisector of angle A of triangle ABC met BC and BC produce point P&Q result.if point O is mid point of PQ .show that OA is tangent to the circle circumscribed about triangle ABC
    (4 votes)
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  • spunky sam blue style avatar for user 💎Chυcκ Lørrε💎
    Hey is this sentence true:"Any two segments tangent to a circle from a common endpoint are congruent." Is that because of they both have a 90° angle?
    And do you know why I can't sending messages nor asking questions but I can only edit something?
    (3 votes)
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  • blobby green style avatar for user fishthe000
    how are we suposed to know which one is the length of the dius
    (2 votes)
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  • blobby green style avatar for user Pratik Chikhali
    A jogging track of uniform width is to be constructed circumscribing a rectangular garden of dimensions 20m * 10 m. if the width of the track is 5 m, What is the area?
    (2 votes)
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  • blobby green style avatar for user Will Crosswhite
    I noticed a pattern where the solution was 2*each side. Is that applicable to all problems like this or just some?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      Except that you may be given a set of numbers that does not match this pattern. The reason that it doubles is the fact that the two lines formed from the outside point to the two tangents have to be equal. So if you label the point between AB as X, between BC as Y, and AC as Z, this theorem says that AX = AZ, XB=BY, and YC = ZC. Thus, two segments are 16 (XB and BY), and we also know that AZ + CZ =14 which substituting means that AX + CY =14 (even though we do not know the measures of any of these 4). Thus, adding what is known and doubling it gives the answer.
      (2 votes)
  • blobby green style avatar for user Arbaaz Ibrahim
    Why and how is it possible to label the unknown sides "x", and add all of the lengths to get the perimeter?
    Can someone please explain?

    Why can't we put the above expressions (for both questions) as equations and then solve for the unknown?
    (1 vote)
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