Monday, June 16, 2008

Elearn Geometry Problem 123



See complete Problem 123
Area of triangle, Similarity. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

3 comments:

  1. see P122

    SAGEC = SAGFEC + SFGE

    using the fact h is the same for AGF and GFE,
    and AF = 6/10 AA", FE = 3/20 AA" ( 4/10 - 1/4 )

    SFGE = 1/80
    =>
    SAGEC = 1/20 + 1/5 + 1/80 + 1/20

    SAGEC = 5/16

    for three sides of tr
    3∙5/16 = 15/16

    SGME = 16/16 - 15/16

    SGME = 1/16
    ----------------------------------------

    ReplyDelete
  2. Apply Menelaus’s theorem to ABA’ and secant CMC”
    CA’/CB x C”B/C”A x MA/MA’= 1
    Replace value of CA’/CB and C”B/C”A in above we get MA/MA’= 3 so AM/AA’=3/4
    Similarly apply Menelaus’s theorem to triangle CAA’ and secant BEB’
    We get EA/EA”=3 => AE/AA’=>3/4
    ME//BC and triangle MGE similar to triangle BAC with the factor ME/BC= ¼
    So S1/S= (1/4)^2= 1/16

    ReplyDelete
  3. Solution to problem 123.
    Let K be the center of gravity of triangle ABC. Let S be the area of ABC. As proved in problem 121, KE/KC = 1/4, and also KG/KA = 1/4. So triangles KGE and KAC are similar with
    S(KGE) = (1/16).S(KAC). Similarly,
    S(KEM) = (1/16).S(KCB) and S(KMG) = (1/16).S(KBA).
    Adding these three equalities we get S1 = S/16.

    ReplyDelete