Wednesday, July 8, 2009

Problem 316: Circular segments and Inscribed Squares

Proposed Problem
Click the figure below to see the complete problem 316.

 Problem 316: Circular segments and Inscribed Squares.
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Complete Problem 316
Level: High School, SAT Prep, College geometry


  1. Draw OQ parallel to AB meeting EF in Q. Thus, OQ=a/2 and QF=a-x. We've ON^2=(b/2)^2+(b+x)^2 & OF^2=(a/2)^2+(a-x)^2 but ON = OF being the radii of the circle. So,(b/2)^2+(b+x)^2=(a/2)^2+(a-x)^2 which, on simplification, directly yields x=5(a-b)/8

  2. OG^2 = (a-x)^2 + a^2/4 and OM^2 = (x+b)^2 + b^2/4

    Since the Radius = OG = OM, (a-x)^2 + a^2/4 = (x+b)^2 + b^2/4

    Hence 2x(a+b) = (a^2 - b^2)/4

    Dividing by (a+b), x = (5/8)(a+b)

    Sumith Peiris
    Sri Lanka