Problem 22 · 2016 Math Kangaroo
Stretch
Geometry & Measurement
area
A quadrilateral has an inner circle (i.e. all four sides of the quadrilateral are tangents to the circle). The ratio of the perimeter of the quadrilateral to the circumference of the circle is 4:3. The ratio of the area of the quadrilateral to that of the circle is therefore
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Answer: E — 4:3
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Hint 1 of 2
A tangential polygon's area is the inradius times its semiperimeter.
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Hint 2 of 2
Combine that with the given perimeter-to-circumference ratio.
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Approach: area = r * semiperimeter for a tangential quadrilateral
- For a quadrilateral with an inscribed circle of radius r, area = r * (perimeter/2).
- Given perimeter:circumference = 4:3, perimeter = (4/3)(2*pi*r) = 8*pi*r/3.
- Area = r * (4*pi*r/3) = 4*pi*r^2/3; circle area = pi*r^2; ratio = 4:3 (E).
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