Results 1 to 8 of 8

Math Help - Incircle of a Triangle

  1. #1
    Newbie
    Joined
    Feb 2011
    Posts
    17

    Incircle of a Triangle

    Let the incircle of triangle ABC have radius 2 and let it be tangent to BC at D. Suppose BD = 3 and DC = 4. What is the length of the longest side of ABC?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,115
    Thanks
    68
    Can't comment on YOUR work on this: can't see it!

    Anyway, start with a diagram; like E = tangent point on AB, F = tangent point on AC.
    You now have BD=BE, CD=CF and AE=AF.
    Use pythagorean theorem and Law of Sines.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2011
    Posts
    17
    Quote Originally Posted by Wilmer View Post
    Can't comment on YOUR work on this: can't see it!

    Anyway, start with a diagram; like E = tangent point on AB, F = tangent point on AC.
    You now have BD=BE, CD=CF and AE=AF.
    Use pythagorean theorem and Law of Sines.
    I used the Pythagorean Theorem to get the lengths of BE and CF, but I'm stuck on getting EA and AF. I know the Law of Sines, but I don't know where to apply it.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,115
    Thanks
    68
    Quote Originally Posted by UNLVRich View Post
    I used the Pythagorean Theorem to get the lengths of BE and CF, but I'm stuck on getting EA and AF. I know the Law of Sines, but I don't know where to apply it.
    You seem unaware of the basics; like, no need to use pythagorean theorem to get BE and CF;
    BE=3 and CF=4. And similarly, AE = AF.
    Were you aware that the 2 tangent lines from a point outside circle to the tangent points are equal?

    You would use Law of Sines to calculate angles ABC and ACB, leaving angle BAC = 180 - ABC - ACB.
    Let M = inner circle center. Now work with the 3 inner triangles AMB, AMC and BMC.

    Were you aware of this formula:
    radius-of-inner-circle = 2[(area-of-triangle) / (perimeter of triangle)] ?
    If we let x = AE = AF, then perimeter becomes 2x + 14 and area (using Heron's formula)
    becomes sqrt[12(x^2 + 7x)].
    This would be another way of solving, NOT needing the Law of Sines.

    Hope this helps...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Feb 2011
    Posts
    17
    Quote Originally Posted by Wilmer View Post
    Were you aware that the 2 tangent lines from a point outside circle to the tangent points are equal?
    I had a feeling this was true, but I didn't use it because I hadn't proved it yet. I just proved it so I'll use it. Thank you!!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Feb 2011
    Posts
    17
    I knew those formulas you mentioned, but the key was the fact that the two tangent lines were congruent. Thanks for your help!!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,115
    Thanks
    68
    Quote Originally Posted by UNLVRich View Post
    I had a feeling this was true, but I didn't use it because I hadn't proved it yet. I just proved it so I'll use it. Thank you!!
    Good stuff Rich! Btw, don't think you needed to prove it: it's a theorem, right?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Feb 2011
    Posts
    17
    Quote Originally Posted by Wilmer View Post
    Good stuff Rich! Btw, don't think you needed to prove it: it's a theorem, right?
    Yeah...it's a theorem, but I don't like to use things that I can't prove by myself.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: April 23rd 2011, 09:10 AM
  2. Replies: 3
    Last Post: April 30th 2009, 08:41 AM
  3. Replies: 1
    Last Post: October 28th 2008, 08:02 PM
  4. Replies: 7
    Last Post: July 19th 2008, 07:53 AM
  5. Replies: 27
    Last Post: April 27th 2008, 11:36 AM

/mathhelpforum @mathhelpforum