RMO ( #1)

Geometry Level 4

Let ABCD be a unit square. Draw a quadrant of circle with A as centre and B,D as end points of the arc. Similarly , draw a quadrant of a circle with B as centre A and C as end points of the arc. Inscribe a circle S touching arcs AC and BD both externally and also touching the side CD. Find the radius of the circle S

0.125 0.625 6.25E-2 0.25

Ratul Pan by Ratul Pan , 21, India

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2 solutions

Ratul Pan
Sep 16, 2015

we can easily prove that AOE congruent to BOE(R.H.S.)
therefore, AE=EB=1/2
By pythagoras theorem A O 2 AO^2 = A E 2 AE^2 + E O 2 EO^2
or, ( R + r ) 2 (R+r)^2 = ( R − r ) 2 (R-r)^2 +1/4
or, ( 1 + r ) 2 (1+r)^2 = ( 1 − r ) 2 (1-r)^2 +1/4 [as R=1]
or, r = 1/16
or, r=0.0625

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Good solution. I always lack in Geom :3

Nice one. :)

Mehul Arora - 5 years, 8 months ago

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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