Results 1 to 6 of 6

Thread: Show that length of arc is between that of line segements

  1. #1
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,784
    Thanks
    581

    Show that length of arc is between that of line segements

    This is probably a really stupid question, but how (without resorting to calculus in any way) would you show that the length of the arc AC is between the lengths of the line segments AB and AD? ( \sin x < x < \tan x is not sufficient).

    Show that length of arc is between that of line segements-sinlimit.png
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,250
    Thanks
    2838

    Re: Show that length of arc is between that of line segements

    Start by taking angle x to be measured in radians. Then the length of arc AC is exactly x. What you want to show is that sin(x)\le x\le tan(x). Notice that you have two right triangles- one, triangle OAB, with hypotenuse of length 1, the other, triangle OAD, with a leg of length 1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,784
    Thanks
    581

    Re: Show that length of arc is between that of line segements

    Yes, I know all of that - I drew the diagram. I'm trying to find an (hopefully intuitive) geometrical argument that the length of the arc is between the lengths of the two line segments.

    x \ge \sin x because \sin x is the shortest distance between the line OD and the point A. But I can't think of a good reason why the arc must be shorter than AD. There are certainly curves within the triangle \triangle ABD that are longer than AD, so how can I argue (without reference to x and \tan x) that the arc AC is shorter than the line segment AD?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    124
    Thanks
    52

    Re: Show that length of arc is between that of line segements

    Quote Originally Posted by Archie View Post
    But I can't think of a good reason why the arc must be shorter than AD. There are certainly curves within the triangle \triangle ABD that are
    longer than AD, so how can I argue (without reference to x and \tan x) that the arc AC is shorter than the line segment AD?
    The largest triangle given is half of an equilateral triangle. Flip it over upwards to get the full equilateral triangle. Line segment OA is an altitude of that triangle.
    Extend the green arc around to make a circle. There are five more equilateral triangles of the same size that are adjacent to each other, all having point O as a
    vertex, and beginning with a triangle that shares a side with line segment OD directly below.

    The six equilateral triangles overlayed on the green circle divide the circle into six congruent sectors. The altitudes of each of these equilateral triangles divide
    each sector in to two smaller congruent sector pieces. The green arc in the problem is associated with one of these smaller sector pieces. There are 12 of them.

    The length of each green arc for each smaller sector piece, radius equals 1, is (2*pi*1)/12 = pi/6 ~ 0.5236, and so that is what arc AC's length is approximately.

    The outside figure is a regular hexagon. Look at one of its sides. But line segment AD equals half the length of one of its sides. Triangle OAD is a right triangle,
    and it's an equal half of an equilateral triangle. Then it's a 30-60-90 triangle. Because the altitude equals 1, then the shortest side equals 1/sqrt(3). That shortest
    side is line segment AD.

    The shortest side of triangle OAD is line segment AD, and its length equals \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3} \approx 0.5774.

    Therefore, line segment AD is longer than arc AC.
    Last edited by greg1313; Jan 17th 2017 at 01:55 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,784
    Thanks
    581

    Re: Show that length of arc is between that of line segements

    Quote Originally Posted by greg1313 View Post
    The largest triangle given is half of an equilateral triangle.
    I'm pretty sure it isn't and I'm absolutely sure it's not in general since x is going to head towards zero.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2013
    From
    Colombia
    Posts
    1,784
    Thanks
    581

    Re: Show that length of arc is between that of line segements

    I realised that I was working with the wrong diagram. This is the one (which make life much easier).

    Show that length of arc is between that of line segements-sinlimit.png
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: Oct 17th 2012, 08:56 PM
  2. Replies: 5
    Last Post: Sep 1st 2012, 04:15 PM
  3. Show that the parameter S is the arc length
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Aug 10th 2010, 01:26 PM
  4. length of line
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 20th 2010, 03:54 AM
  5. length of a line
    Posted in the Calculus Forum
    Replies: 12
    Last Post: Apr 10th 2010, 08:44 PM

/mathhelpforum @mathhelpforum