Circles 2

Geometry Level 4

In the above figure, given that A B C D ABCD is square with side-length 6 cm and that circle (c) has radius r r cm, enter r r as your answer.


The answer is 1.

Refaat M. Sayed by Refaat M. Sayed , 24, Egypt

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2 solutions

Draw perpendiculars from center of small circle F to BD to meet it at E, and to CD to meet it at G. So EF=GD=r.
In triangles FCG and EFD, using Pythagoras, F C 2 − C G 2 = F G 2 = F D 2 − G D 2 ⟹ ( 6 + r ) 2 − ( 6 − r ) 2 = ( 6 − r ) 2 − r 2 . ⟹ 36 + 12 r − 36 + 12 r = 36 − 12 r . ∴ r = 1 FC^2 - CG^2=FG^2=FD^2 - GD^2\\ \implies ~(6+r)^2 -(6 - r)^2=(6 - r)^2 - r^2. ~~\implies ~36+12r - 36 + 12r=36 - 12r.\\ \therefore ~\Large \color{#D61F06}{\boxed{~~r=1~~}}

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Ahmad Saad
Feb 17, 2016

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Well done :)

Refaat M. Sayed - 5 years, 3 months ago

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@Refaat M. Sayed This might come as a clue to people, but you haven't entered "Decimals OK" in the answer box. So people can fluke the answer.

You should've entered the answer as 1.00 to make sure no one can fluke the answer.

Mehul Arora - 5 years, 3 months ago

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