Saturday, April 25, 2009

Problem 284: Circular Sector 90 degrees, Semicircles, Tangent, Radius

Proposed Problem

 Problem 284: Circular Sector 90 degrees, Semicircles, Tangent, Radius.

See also: Complete Problem 284, Collection of Geometry Problems

Level: High School, SAT Prep, College geometry

2 comments:

  1. The two inner circles are given by:
    (x-R/2)^2+y^2=R^2/4
    x^2+(y-R+r)^2=r^2
    Solve these simultaneously and set the discriminant to zero for the circles to touch each other. This gives us:-R^6+4rR^5-3r^2R^4=0 or -R^2+4rR -3r^2 =0 which can be solved to obtain r = R/3 or r=R of which only the first solution is admissible.
    Ajit: ajitathle@gmail.com

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  2. We can use Pythagorean Theorem to solve it.
    Since D, E, C are collinear,OD=R-r, OR=R/2 and DC=r+R/2,
    So solving (R-r)^2+(R/2)^2=(r+R/2)^2 will give r=R/3.

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