Wednesday, January 29, 2014

Geometry Problem 971: Equilateral Triangle, Rectangle, Common Vertex, Sum of Right Triangles Areas

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 971.

Online Geometry Problem 971: Equilateral Triangle, Rectangle, Common Vertex, Sum of Right Triangles Areas

3 comments:

  1. http://imagizer.imageshack.us/v2/800x600q90/835/8nxu.png

    Draw rectangular FDNM as per sketch
    Note that area ( EBC) = area(ENC)
    And Area(EAF)= area(FME)
    This problem becomes problem 969.

    ReplyDelete
  2. Let z(P) be the complex number representing P.
    Let z(C)=0, z(B)=−a, z(E)=−a+bi.

    Then
    z(F) = (−a+bi)(1/2 + √3/2 i) = (−1/2 a − √3/2 b) + (−√3/2 a + 1/2 b)i
    z(A) = −a + (−√3/2 a + 1/2 b)i
    z(D) = (−√3/2 a + 1/2 b)i

    Thus
    BE = b, BC = a
    AF = −1/2 a + √3/2 b, AE = √3/2 a + 1/2 b
    DC = √3/2 a − 1/2 b, DF = 1/2 a + √3/2 b

    Area of ΔEBC = 1/2 ab
    Area of ΔFAE = 1/8 (−√3 a² + 2 ab + √3 b²)
    Area of ΔFDC = 1/8 (√3 a² + 2 ab − √3 b²)

    Hence,
    Area of ΔEBC = Area of ΔFAE + Area of ΔFDC

    ReplyDelete
    Replies
    1. Sir, can you give me the idea how to get Z(F) Z(B) and Z(D)
      thx for a lot
      here is my Gmail account : bukolight@yahoo.com

      Delete