Results 1 to 2 of 2

Math Help - arithmetic sequence in a triangle

  1. #1
    Member
    Joined
    Dec 2008
    Posts
    152

    arithmetic sequence in a triangle

    In what ratio are the sides of a triangle if one of the angles is 120 and the sides of the triangle make up a arethmetic sequence..

    so we have 3 sides:
    a = a
    b = a +d
    c = a +2d
    and one of the angles is 120 = angle between a and a+d (the shortest sides)..

    i tried a lot of things but i couldnt find a way to get the right result. =(
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,861
    Thanks
    742
    Hello, metlx!

    In what ratio are the sides of a triangle if one of the angles is 120
    and the sides of the triangle make up a arethmetic sequence?

    so we have 3 sides: a = a, b = a +d , c = a +2d . . . . Good!

    and one of the angles is 120 = angle between a and a+d (the shortest sides). . Right!
    We need the Law of Cosines: . c^2 \;=\;a^2 + b^2 - 2ab\cos C


    So we have: . (a+2d)^2 \;=\;a^2 + (a+d)^2 - 2(a)(a+d)\cos120^o

    Since \cos120^o = \text{-}\tfrac{1}{2}, we have: . a^2 + 4ad + 4d^2 \;=\;a^2 + a^2 + 2ad + d^2 + a^2+ad

    . . which simplifies to: . 2a^2 - ad - 3d^2\:=\:0 \quad\Rightarrow\quad (a + d)(2a-3d) \:=\:0

    And we get: . \begin{Bmatrix}a + d \:=\:0 & \Rightarrow & a \:=\:\text{-}d & \text{not possible} \\<br />
2a-3d \:=\:0 & \Rightarrow & d \:=\:\frac{2}{3}a \end{Bmatrix}


    The shortest side is: . a

    The next side is: . b \:=\:a+d \:=\:a+\tfrac{2}{3}a \:=\:\tfrac{5}{3}a

    The longest side is: . c \:=\:a + 2d \:=\:a + \tfrac{4}{3}a \:=\:\tfrac{7}{3}a


    The ratio of the sides is: . a : \tfrac{5}{3}a : \tfrac{7}{3}a

    . . Multiply by 3\!:\;\;3a:5a:7a


    Therefore, the ratio of the sides is: . {\color{blue}3:5:7}

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. arithmetic sequence
    Posted in the Algebra Forum
    Replies: 4
    Last Post: September 29th 2010, 12:34 AM
  2. Replies: 2
    Last Post: August 24th 2010, 03:10 AM
  3. Arithmetic Sequence
    Posted in the Algebra Forum
    Replies: 4
    Last Post: February 5th 2010, 12:08 AM
  4. Replies: 12
    Last Post: November 15th 2006, 01:51 PM
  5. Arithmetic Sequence
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 7th 2006, 07:16 AM

Search Tags


/mathhelpforum @mathhelpforum