[中學] 矩形內一點問題

看板Math作者 (腦海裡依然記得妳)時間11年前 (2014/06/11 03:38), 編輯推噓1(1推 0噓 28→)
留言29則, 5人參與, 6年前最新討論串1/1
ABCD為一個矩形 P為ABCD內一點 已知 PA = 1 PB = 2 PC = 3 AD = 2*AB 求矩形面積 = ??? -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 111.252.196.167 ※ 文章網址: http://www.ptt.cc/bbs/Math/M.1402429128.A.E60.html
推
06/11 07:11, , 1F
坐標法:令B(0,0),A(k,0),C(0,2k),P(2cosθ,2sinθ)
06/11 07:11, 1F
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06/11 07:12, , 2F
用k來湊出三角恆等式,答案有點醜...
06/11 07:12, 2F
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06/11 08:15, , 3F
這是國中資優題 請麻煩用國中生可以理解的解法 謝
06/11 08:15, 3F
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06/11 10:53, , 4F
國中一樣可以座標化啊...
06/11 10:53, 4F
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06/11 10:53, , 5F
設A(0,0) B(k,0) C(k,2k) D(0,2k) P(x,y) , x,y,k>0
06/11 10:53, 5F
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06/11 10:53, , 6F
(1)：PA = 1 => x^2 + y^2 = 1
06/11 10:53, 6F
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06/11 10:54, , 7F
(2)：PB = 2 => (x-k)^2 + y^2 = 4
06/11 10:54, 7F
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06/11 10:54, , 8F
(3)：PC = 3 => (x-k)^2 + (y-2k)^2 = 9
06/11 10:54, 8F
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06/11 10:54, , 9F
(2)-(1)： -2kx + k^2 = 3 => x = (k^2-3)/2k
06/11 10:54, 9F
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06/11 10:54, , 10F
(3)-(2)： -4ky + 4k^2 = 8 => y = (k^2-2)/k
06/11 10:54, 10F
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06/11 10:54, , 11F
兩式代回(1)可解k,再式回來可解x,y
06/11 10:54, 11F
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06/11 10:56, , 12F
ps.在解k時可以把k^2叫做t，會變成2次式，
06/11 10:56, 12F
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06/11 10:59, , 13F
不過數字真的有點醜......
06/11 10:59, 13F
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06/11 12:47, , 14F
另外，若不座標化的做法是：
06/11 12:47, 14F
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06/11 12:47, , 15F
作P在AB線段的垂足M、P在BC線段的垂足N
06/11 12:47, 15F
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06/11 12:47, , 16F
設：AB=k,AD=2k,PM=m,PN=n
06/11 12:47, 16F
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06/11 12:47, , 17F
(1)：PA = 1 => m^2 + (k-n)^2 = 1
06/11 12:47, 17F
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06/11 12:48, , 18F
(2)：PB = 2 => m^2 + n^2 = 4
06/11 12:48, 18F
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06/11 12:48, , 19F
(3)：PC = 3 => (2k-m)^2 + n^2 = 9
06/11 12:48, 19F
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06/11 12:48, , 20F
式子和座標化一樣，解法也一樣
06/11 12:48, 20F
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06/11 12:48, , 21F
應該說式子很像，解法一樣
06/11 12:48, 21F
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06/11 13:20, , 22F
將ΔAPB對B點旋轉-90度，再對B點放大2倍得ΔCEB
06/11 13:20, 22F
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06/11 13:21, , 23F
ΔAPB~ΔCEB，且∠PBE=90度，然後座標化
06/11 13:21, 23F
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06/11 13:23, , 24F
取B(0,0)、P(0,2)、E(4,0),設C(x,y)
06/11 13:23, 24F
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06/11 13:25, , 25F
PC=3得x^2+(y-2)^2=9
06/11 13:25, 25F
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06/11 13:26, , 26F
CE=2得(x-4)^2+y^2=4
06/11 13:26, 26F
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06/11 13:27, , 27F
長邊BC=2a得x^2+y^2=4a^2，所求為2a^2
06/11 13:27, 27F
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06/11 13:30, , 28F
這組方程組比較好解面積為4+(√95)/5
06/11 13:30, 28F
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07/07 12:14, 6年前 , 29F
作P在AB線段的垂足M http://yofuk.com
07/07 12:14, 29F
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