The key to solving this math puzzle is to first re-configure the puzzle diagram as shown above to more easily apply the geometry and algebra. We see immediately, for example, the large radius R can be expressed either as:
R = 210 + y or:
R = 330 + z
And: z + 330 = y + 210
So: z = y - 120 (solving simultaneous eqn. in y, z)
Then: z +330 = x + z + 165
So: x = 165
Also: r = x + z
Further, from Pythagoras:
x2 + y 2 =
x2 + y 2 = x 2 + 2xz + z 2
è
y 2 = 2xz + z 2
y 2 = 330 (y - 120) + (y - 120) 2
y 2 = 330 y - 39,600 + y 2 - 240 y + 14,400
è
90 y = 25,200
y = 25,200/ 90 = 280
But: R = 210 + y = 210 + 280 = 490
If: r = x + z and x = 165
Then, from the geometry:
r = R - x
r = 490 - 165 = 325
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