PUMaC 2019 · 团队赛 · 第 12 题

PUMaC 2019 — Team Round — Problem 12

In quadrilateral ABCD , angles A, B, C, D form an increasing arithmetic sequence. Also, ◦ ∠ ACB = 90 . If CD = 14 and the length of the altitude from C to AB is 9, compute the area of ABCD .

解析

In quadrilateral ABCD , angles A, B, C, D form an increasing arithmetic sequence. Also, ◦ ∠ ACB = 90 . If CD = 14 and the length of the altitude from C to AB is 9, compute the area of ABCD . Proposed by: Eric Neyman Answer: 198 Let angles A, B, C, D have measures 90 − 3 x, 90 − x, 90 + x, 90 + 3 x . Observe that angles B and C add up to 180 degrees, so ABCD is a trapezoid with legs AB and CD . Let T and U be the feet of the altitudes to AB from C and D , respectively. Let BT = y and AU = z . Then y BT CT 9 4 BT C ∼ 4 CT A , so = . We can write this as = , i.e. y (14 + z ) = 81. We also CT AT 9 14+ z 3 3 tan x − tan x have y = 9 tan x and z = 9 tan 3 x . Writing tan 3 x = (which can be verified easily 2 1 − 3 tan x 4 3 2 using the tangent of sum formula), we find that 9 tan x +42 tan x − 54 tan x − 14 tan x +9 = 0. 1 This seems daunting, but the rational root theorem lets us find the solution tan = . We know 3 ◦ 1 √ that 0 < x < 30 , so 0 < tan x < . There are no other solutions to the quartic in this range. 3 3 2 (Factoring out 3 tan x − 1, we get 3 tan x + 15 tan x − 13 tan x − 9 = 0. We find that this 4 polynomial is positive at − 1, negative at 0 and positive at 1, which means its remaining roots are less than − 1; between − 1 and 0; and greater than 1.) 1 Now that we have tan x = , we immediately get y = 3 and z = 13, so the area of the trapezoid 3 14+30 is 9 · = 198 . 2