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AravindG

what is the maximum number of tangents to a triangle?

  • one year ago
  • one year ago

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  1. AravindG
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    @TuringTest , @UnkleRhaukus , @phi , @saifoo.khan , @ash2326

    • one year ago
  2. TuringTest
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    |dw:1346864028370:dw|try drawing as many different tangents on this figure as you can

    • one year ago
  3. AravindG
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    is it 3 ?

    • one year ago
  4. henpen
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    What point could you draw to satisfy that?

    • one year ago
  5. AravindG
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    |dw:1346864106521:dw|

    • one year ago
  6. CliffSedge
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    I count a whole bunch!

    • one year ago
  7. AravindG
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    @UnkleRhaukus isnt it 3?

    • one year ago
  8. phi
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    Can't you draw an infinite number through each vertex?

    • one year ago
  9. AravindG
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    @phi y so?

    • one year ago
  10. UnkleRhaukus
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    i count three infinite lines

    • one year ago
  11. TuringTest
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    tangents drawn from the vertices would have no direction the only question I see is if you want to talk about the number of \(different\) tangents, which I do figure to be 3

    • one year ago
  12. AravindG
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    ya different tangents

    • one year ago
  13. TuringTest
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    you could draw infinite tangents on any line segment technically, but that seems arbitrary

    • one year ago
  14. AravindG
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    my doubt is why dont we count the tangent at the 3 vertices?

    • one year ago
  15. UnkleRhaukus
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    |dw:1346864223266:dw|

    • one year ago
  16. TuringTest
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    which way would the tangents at the vertices point?

    • one year ago
  17. CliffSedge
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    Do we have to consider that the slope of a tangent at a sharp point is undefined?

    • one year ago
  18. AravindG
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    i am not sure

    • one year ago
  19. UnkleRhaukus
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    a triangle is made with three tangents

    • one year ago
  20. phi
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    OK, by definition a tangent results from a limiting process.... so they do not exist at a vertex... (per wikipedia)

    • one year ago
  21. AravindG
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    so the answer is 3?

    • one year ago
  22. TuringTest
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    @CliffSedge that is basically how I figured it

    • one year ago
  23. CliffSedge
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    If it's tangent to the triangle, then they cannot coincide with the sides of the triangle.

    • one year ago
  24. AravindG
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    @CliffSedge y so?

    • one year ago
  25. CliffSedge
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    Doesn't 'tangent' mean touch at only a single point?

    • one year ago
  26. AravindG
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    @CliffSedge yes , but the tangent to a straight line is the straight line itself!!!

    • one year ago
  27. AravindG
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    tangent is nothing but a flat approximation of curve!!

    • one year ago
  28. CliffSedge
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    hmmm... looks like there can be several definitions depending on if you're using geometry, calculus, or an even more generalized usage.

    • one year ago
  29. AravindG
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    i see what i need to knw is this :|dw:1346864494173:dw| can we draw tangent at point A?

    • one year ago
  30. TuringTest
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    I was thinking of it as like a line graph|dw:1346864498741:dw|there is only one unique equation of the tangent line to this graph\[y=mx+b\implies y'=m\]so \(y'=m\) is the only tangent line in the figure of the triangle you get the same effect for each line segment, of which there are three

    • one year ago
  31. CliffSedge
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    Here's my reasoning: 1. tangent is defined as touching at only a single point. 2. in calculus, the slope of *the* tangent line at a sharp point is undefined because there is no single/unique tangent line that can be drawn there. Therefore: infinite tangent lines are possible.

    • one year ago
  32. CliffSedge
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    @Turing but that's not the tangent line, that is the slope of the tangent line which is the line itself. You can't draw a line tangent to another line because it would touch at more than one point.

    • one year ago
  33. TuringTest
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    More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f'(c) where f' is the derivative of f. -Wikipedia that does not require the line to only touch the graph at a single point, the slope just has to match that of the graph at that point

    • one year ago
  34. AravindG
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    i asked this qn in my lecture and answered it as 3 then one of the srtudents asked if its 6 as in fig: |dw:1346864689277:dw| i couldnt justify it was 3

    • one year ago
  35. AravindG
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    i told the student i will answer it tmrw dats y i asked

    • one year ago
  36. TuringTest
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    consider|dw:1346864772930:dw|the tangent at point P intersects point Q on the graph, but that does not mean the line is not tangent to the graph at P

    • one year ago
  37. AravindG
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    agree @TuringTest can u answer my query above?

    • one year ago
  38. CliffSedge
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    Yes, but the tangent doesn't cross at P. It's free to cross elsewhere. see: http://en.wiktionary.org/wiki/tangent def'n. 1.

    • one year ago
  39. siddhantsharan
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    Is the slope of the tangent the always the same thing as the derivative?

    • one year ago
  40. CliffSedge
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    I'm still not convinced that there are fewer than infinite tangent lines that can be drawn to that triangle.|dw:1346864875620:dw|

    • one year ago
  41. TuringTest
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    That definition seems to hold for the 3 tangents of the triangle; they lines do not cross the sides again there is no defined slope at the vertices, so you can't do that @CliffSedge

    • one year ago
  42. CliffSedge
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    I don't think the slope of the tangent matters here, since we're not looking for a single unique tangent with a defined slope, just "how many lines?"

    • one year ago
  43. AravindG
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    The person who gave me this qn for presenting in lecture was @UnkleRhaukus and he has vanished :O

    • one year ago
  44. TuringTest
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    there can only be one unique tangent at each point, the idea that you can draw multiple tangents from each vertex shows that there is no tangent

    • one year ago
  45. TuringTest
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    unkle said that a triangle was composed of three tangents and therefor, I think, meant to imply that there are three tangents

    • one year ago
  46. AravindG
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    thats a nic reasoning @TuringTest

    • one year ago
  47. CliffSedge
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    Thinking of this in terms of first derivatives is over-thinking it, in my opinion. Ask Euclid or Archimedes what they think of this question. It's fundamentally geometrical.

    • one year ago
  48. AravindG
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    so can i tell the student that slope is not defined at sharp points ?

    • one year ago
  49. AravindG
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    as a reason for no:of tangent=3 and not 6?

    • one year ago
  50. TuringTest
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    I would like to see some reference that contradicts that, otherwise I would argue along the calculus approach.

    • one year ago
  51. AravindG
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    @TuringTest answer me pls

    • one year ago
  52. TuringTest
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    I say "yes" there are 3, but there is some debate, and I'm not perfect...

    • one year ago
  53. siddhantsharan
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    http://mathdl.maa.org/images/upload_library/22/Polya/07468342.di020721.02p01112.pdf

    • one year ago
  54. AravindG
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    is see but i kthink u r right at this ,the calculus approach is best at this qn

    • one year ago
  55. siddhantsharan
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    A really good reference I would say that clarifies that tangents are not defined at edges.

    • one year ago
  56. AravindG
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    thanks a lot evryone for sharing their ideas :")

    • one year ago
  57. TuringTest
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    nice link @siddhantsharan :)

    • one year ago
  58. siddhantsharan
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    Thank Google :D

    • one year ago
  59. phi
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    Per wolfram's definition (see the last paragraph) http://www.wolframalpha.com/input/?i=tangent+definition&x=0&y=0 a tangent line touches a curve at a single point.... so one could argue that polygons do not have tangents.

    • one year ago
  60. siddhantsharan
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    Although they have not defined it precisely there. The precise definition is the calculus one.

    • one year ago
  61. CliffSedge
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    Can anyone provide a reason why the slopes of the tangents need to be defined? If the claim is that tangents are not defined (i.e. are not unique) at sharp points, then doesn't that mean that the number of lines is infinite?

    • one year ago
  62. AravindG
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    @siddhantsharan that was really useful thx

    • one year ago
  63. TuringTest
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    page 136 of the link that @siddhantsharan gives a nice non-calculus way of circumventing the definition, but it still leads to the same conclusions; one tangent line per point, and it's okay to draw a tangent to a line segment

    • one year ago
  64. AravindG
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    the idea is derivative does not exist at sharp points

    • one year ago
  65. CliffSedge
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    Why does the existence of the derivative matter? I can draw a triangle, and I can draw numerous tangents at each vertex. I'm looking right at it, so what's the problem?

    • one year ago
  66. phi
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    You can define tangent any way you like. The question is, what is the recognized mathematical definition? Does it exclude straight lines? or is it only defined for "curves"

    • one year ago
  67. CliffSedge
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    This is the definition I'm using: (geometry) A straight line touching a curve at a single point without crossing it there. We can all agree that straight lines count as curves, right?

    • one year ago
  68. CliffSedge
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    There exists a triangle, and I may, in principle at least, draw as many tangents to its vertices as I wish. QED. I work with a bunch of math tutors/teachers. I'll bring this up with them this afternoon.

    • one year ago
  69. UnkleRhaukus
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    The slopes of the tangents need to be defined because the tangent Is the slope at that Point

    • one year ago
  70. siddhantsharan
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    @CliffSedge The definition you are using is not the mathematical one.

    • one year ago
  71. CliffSedge
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    Geometry is not mathematics?

    • one year ago
  72. CliffSedge
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    Triangles are geometrical, lines are geometrical; how many hairs are we going to split here?

    • one year ago
  73. siddhantsharan
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    That is the definition that is implied from the calculus one and is observed to be helpful in many cases. However it is one DERIEVED from the calculus one.

    • one year ago
  74. TuringTest
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    @CliffSedge the definition you said you are working with would imply that this is not a tangent|dw:1346866463441:dw|of course the term "there" is not well-defined

    • one year ago
  75. CliffSedge
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    So why is it not useful here? I see a triangle. I can draw tangent lines to its vertices. Why do I need to have a well-defined slope for each one? That was not a condition set forth in the problem statement.

    • one year ago
  76. TuringTest
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    that definition makes no statement about sharp points, so it doesn't seem to help us here

    • one year ago
  77. CliffSedge
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    False. It touches at P, but does not cross at P.

    • one year ago
  78. CliffSedge
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    Why would the definition need to make a statement about sharp points? Why not generalize? Why not employ Occam's Razor and not introduce more assumptions than are necessary?

    • one year ago
  79. TuringTest
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    Okay, fair enough, so you want to say that a sharp curve can have multiple tangents is the discrepancy. Occam's Razor is all well and good, unless there is a more formal definition such that we don't have to employ it.

    • one year ago
  80. CliffSedge
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    I know mathematics is meant to be precise, but it is also about simplifying problems so they may be solved more easily, not more difficultly.

    • one year ago
  81. UnkleRhaukus
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    the vertex is only a single point it can only have one tangent

    • one year ago
  82. TuringTest
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    In your attempt to simplify this problem we get infinite tangent lines at a sharp point, or any geometric figure with sharp points. I hardly find that simple, but I see what you are saying.

    • one year ago
  83. UnkleRhaukus
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    hmm now im not sure if it is one or zero

    • one year ago
  84. CliffSedge
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    Does the formal definition state that if their is no single defined/unique slope for a tangent line then the tangent line can't exist? If that is true, then why is it so easy for me to draw one on paper? I saw above that there exists a definition that states that there may exist only a single tangent line at any point, but if that is true then the triangle has zero tangent lines because each vertex has two lines touching it.

    • one year ago
  85. CliffSedge
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    What I meant by simple is that if you gave this problem to a child and gave a definition of 'tangent line' that a child could understand, then you'd get many lines drawn with no fuss over formal definitions.

    • one year ago
  86. TuringTest
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    \[y=|x|\]has tangents \(y=-x,y=x\) though both cross the vertex, so I don't think it matters that the tangents intersect on the graph

    • one year ago
  87. CliffSedge
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    It's not that the tangents intersect each other, it's that there is more than one tangent at a single point. If y=|x| can have two tangents, why can't any point have more than one tangent? By one definition (preferred) I say infinite lines, but another, it is zero (and that's not so bad - less work the pencil, eh?

    • one year ago
  88. TuringTest
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    my point was that they intersect \(on\) the graph, at (0,0)

    • one year ago
  89. TuringTest
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    at (0,0) we have to accept that there are either infinite slopes or none, and mathematics when it ever discusses the matter seems to say none.

    • one year ago
  90. CliffSedge
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    I'll concede that for now. Anyway, I'm in a rush to get to work, so I'll have to leave this for later. Interesting stuff, gets the mind all geared up for logical thinking!

    • one year ago
  91. TuringTest
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    indeed, let me know if you find out something :)

    • one year ago
  92. UnkleRhaukus
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    its a good question

    • one year ago
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