+4624 Let us have four distinct collinear points A,B,C and D on the Cartesian plane. The point C is such that ABCB=12 and the point D is such that DABA=3 and DBBA=2. If C=(0,4),D=(4,0), and A=(x,y), what is the value of 2x+y ?
+502 I think if I answer any more math questions my mind will b**w up cuz its mixed with chemistry and maths and that a very bad mixture
+130466 AB / CB = 1/2 ⇒ 2AB = BC
AD/AB = 3 ⇒ 3AB = AD
BD/ AB = 2 ⇒ 2AB = BD
So this implies that BD = BC
So....B must be the midpoint of C and D = (2,2)
2AB = BC ⇒ AB = BC/2
3AB = AD ⇒ AB = AD/3
So logic dictates that A must be the midpoint of BC = (1,3)
So....AB = sqrt [ (1^2 + 1^2) ] = sqrt (2)
And AD = sqrt (3^2 + 3^2) = sqrt (18) = 3sqrt (2) = 3AB
Proof
AB = sqrt (2)
CB = sqrt [ 2^2 + (4 -2)^2] = sqrt (4 + 4) = sqrt (8) = 2sqrt (2)
So AB /CB = sqrt (2)/ 2sqrt (2) = 1/2
And
AD = 3sqrt(2)
BA = sqrt(2)
So DA /BA = 3sqrt(2)/ sqrt (3) = 3
And
DB = sqrt [ ( 4 - 2)^2 + 2^2 ] = sqrt (2^2 + 2^2 ) = sqrt (8) = 2sqrt(2)
BA = sqrt(2)
So DB / BA = 2sqrt(2)/ sqrt (2) = 2
So A = (1 ,3)
And 2x + y = 2(1) + 3 = 5
