Square and Two Circles Problem

Geometry Level 4

Two circles and a square are tangent to eachother at one point. The height of the whole thing is 400 units, the square has a side length of 279 units, and the small circle has a radius of 65 units. Find the radius of the larger circle to the nearest whole unit.


The answer is 153.

Daniel Araujo by Daniel Araujo , 20, Canada

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1 solution

Let the radius of the large circle be r r . From the figure we note that 65 + ( r + 65 ) 2 − ( r − a ) 2 + r = 400 65 + \sqrt{(r+65)^2-(r-a)^2} + r = 400 , where a = 6 5 2 − b 2 a = \sqrt{65^2 - b^2} . We note that b = 400 − 65 − 279 = 56 b= 400 - 65 - 279 = 56 . ⟹ a = 6 5 2 − 5 6 2 = 33 \implies a = \sqrt{65^2-56^2} = 33 and:

65 + ( r + 65 ) 2 − ( r − a ) 2 + r = 400 ( r + 65 ) 2 − ( r − 33 ) 2 = 335 − r Squaring both sides 196 r + 3136 = r 2 − 670 r + 112225 ⟹ r 2 − 866 r + 109089 = 0 ( r − 713 ) ( r − 153 ) = 0 Since r < 400 ⟹ r = 153 \begin{aligned} 65 + \sqrt{(r+65)^2-(r-a)^2} + r & = 400 \\ \sqrt{(r+65)^2-(r-33)^2} & = 335 - r & \small \color{#3D99F6} \text{Squaring both sides} \\ 196r+3136 & = r^2 -670r + 112225 \\ \implies r^2 -866r + 109089 & = 0 \\ (r-713)(r-153) & = 0 & \small \color{#3D99F6} \text{Since }r < 400 \\ \implies r & = \boxed{153} \end{aligned}

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