Thursday, December 25, 2014

Geometry Problem 1070: Circle, Chord, Equilateral Triangle, Square, Rectangle, Area, Diameter, Perpendicular

Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the diagram below to enlarge it.

Online Math: Geometry Problem 1070: Circle, Chord, Equilateral Triangle, Square, Rectangle, Area, Diameter, Perpendicular.

2 comments:

  1. It is equivalent to show that the srea of the three squares is equal to the blue rectangle.
    Algebraically, to show that
    EA² + EA×EB + EB² = 3×OD²

    Now consider
    OD² = OA²
    = OO₁² + [(EA + EB)/2]²
    = (EO₁ tan30°)² + [(EA + EB)/2]²
    = 1/3×[(EA − EB)/2]² + [(EA + EB)/2]²
    = 1/3×[EA² + EA×EB + EB²]

    Hence, EA² + EA×EB + EB² = 3×OD².

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  2. Let AD=AE= l , EB=d and OD= R
    We need to that l^2+d^2+l.d=3.R^2
    In triangle ABD we have BD/sin(60)= 2.R => BD= R.SQRT(3)
    Apply cosine formula in triangle ABD we have
    BD^2=AD^2+AB^2-2.AB.AD.cos(60) => l^2+(l+d)^2-l(l+d)
    So 3.R^2=l^2+d^2+l.d or Blue area= yellow areas

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