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The ratio of the area of a regular n-sided polygon circumscribed about a circle to the area of the regular n-sided polygon inscribed in the same circle is 4:3. Find the value of n.

1. 3
2. 6
3. 9
4. 10

Correct answer: 6

Solution

A_circ / A_ins = [n*r²*tan(pi/n)] / [n*r²*sin(pi/n)*cos(pi/n)] = tan(pi/n) / (sin(pi/n)*cos(pi/n)) = 1/cos²(pi/n). Setting 1/cos²(pi/n) = 4/3 gives cos²(pi/n) = 3/4, so cos(pi/n) = sqrt(3)/2, meaning pi/n = pi/6, so n = 6.

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