PUMaC 2008 · 团队赛 · 第 5 题

PUMaC 2008 — Team Round — Problem 5

(4 points) Quadrilateral ABCD has both an inscribed and a circumscribed circle and sidelengths BC = 4, CD = 5, DA = 6. Find the area of ABCD .

解析

Quadrilateral ABCD has both an inscribed and a circumscribed circle and sidelengths BC = 4, CD = 5, DA = 6. Find the area of ABCD . √ ( ANS: 10 6 Because the quadrilateral has an inscribed circle, pair of opposite sides must add up to the same thing, so AB = 5. Half the perimeter is s = 10. This also has a circumscribed circle, so using Brahmagupta’s formula for the area of a quadrilateral inscribed in a circle, the area is √ √ √ ( s − AB )( s − BC )( s − CD )( s − DA ) = 5 · 6 · 5 · 4 = 10 6. Alternately, the quadrilateral is √ an isosceles trapezoid, whose height is easily calculable as 24 by the Pythagorean Theorem and the average of whose bases is 5. CB: AL6, IAF, ACH)