Ridiculous radii

Geometry Level 3

In the image above, a quarter circle is enclosed by a square with side length R R . There is also a smaller circle with radius r r that touches the quarter circle and the top and left side of the square.

Given that r = R ⋅ c + a c + b \displaystyle r= R\cdot \frac{\sqrt{c}+a}{\sqrt{c}+b} where − 4 ≤ a < b < c ≤ 4 -4 \leq a<b<c \leq 4 , find a + b + c \displaystyle a+b+c .


The answer is 2.

James Watson by James Watson , 16, United Kingdom

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1 solution

Vijay Simha
Sep 11, 2020

The equation which illustrates the relationship between the two radii is

sqrt(2) * r + r + R = sqrt(2) * R

Grouping the terms, we arrive at the relation

r = R * {(sqrt(2)-1)/(sqrt(2)+1)}

The final expression in the question is wrong

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