Square and circles

Square with sides 68 km is circumscribed and inscribed with circles. Determine the radiuses of both circles.

Correct answer:

r =  34 km
R =  48.08 km

Step-by-step explanation:

a=68 km  r=a/2=68/2=34 kma = 68 \ \text{km} \ \\ \ \\ r = a/2 = 68/2 = 34 \ \text{km}
a2+a2 = (2R)2 2a2 = 4R2 a2 = 2R2 a=2⋅ R  R=r⋅ 2=34⋅ 2=34 2=48.08 kma^2+a^2\ = \ (2R)^2 \ \\ 2a^2\ = \ 4R^2 \ \\ a^2\ = \ 2R^2 \ \\ a = \sqrt{ 2 } \cdot \ R \ \\ \ \\ R = r \cdot \ \sqrt{ 2 } = 34 \cdot \ \sqrt{ 2 } = 34 \ \sqrt{ 2 } = 48.08 \ \text{km}

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