P(5,9),Q(4,1) and R(2,5) are three points on AB,BC and CA respectively such that AP:PB=BQ:QC=CR:RA=2:1.Find the coordinate of C. And A,B,C are the three vertices of triangle. There is also a triangle inside the triangle ABC. And P,Q,Rare three vertices of the triangle inside the triangle ABC. Give me the exaction answer and tell me how to do with a clear solution.Please!!!!!!
Thank you!!!!!
有關maths coordinate 既題目,急!!!!!!
2013-06-13 6:02 am
回答 (2)
2013-06-14 10:04 pm
✔ 最佳答案
Please refer to the following figure:圖片參考：http://imgcld.yimg.com/8/n/HA00622999/o/20130614140240.jpg
參考: coordinates geometry
2013-06-13 3:56 pm
Let A be (x,y).
Let B be (a, b)
Since AP : PB = 2 : 1 and P is (5,9)
so (2a + x)/3 = 5 and (2b + y)/3 = 9
so B is [ (15 - x)/2, (27 - y)/2]
Similar C is [ (6 - 2x), (15 - 2y)]
But BQ : QC = 2 : 1
Applying point to division to the x - coordinates we get
[(15 - x)/2 + 2(6 - 2x)]/3 = 4
(15 - x) + 4(6 - 2x) = 24
15 - x + 24 - 8x = 24
15 = 9x
x = 5/3
Similarly,
[ (27 - y)/2 + 2(15 - 2y)]/3 = 1
(27 - y) + 4(15 - 2y) = 6
27 - y + 60 - 8y = 6
y = 9
So A is ( 5/3, 9)
B is [ (15 - 5/3)/2, (27 - 9)/2] = (20/3, 9)
C is [ ( 6 - 10/3), (15 - 18)] = (8/3, - 3)
Let B be (a, b)
Since AP : PB = 2 : 1 and P is (5,9)
so (2a + x)/3 = 5 and (2b + y)/3 = 9
so B is [ (15 - x)/2, (27 - y)/2]
Similar C is [ (6 - 2x), (15 - 2y)]
But BQ : QC = 2 : 1
Applying point to division to the x - coordinates we get
[(15 - x)/2 + 2(6 - 2x)]/3 = 4
(15 - x) + 4(6 - 2x) = 24
15 - x + 24 - 8x = 24
15 = 9x
x = 5/3
Similarly,
[ (27 - y)/2 + 2(15 - 2y)]/3 = 1
(27 - y) + 4(15 - 2y) = 6
27 - y + 60 - 8y = 6
y = 9
So A is ( 5/3, 9)
B is [ (15 - 5/3)/2, (27 - 9)/2] = (20/3, 9)
C is [ ( 6 - 10/3), (15 - 18)] = (8/3, - 3)
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