Calculus/Centroid:

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Expert: - 10/2/2009


Question

triangle
from triangle shown
we have to find the ratio between AF,BF using vectors only
the line segments drawn from vertices are not perpendicular.
thanks anyway


Answer
AF = BF , so the ratio is 1

To simplify the calculations , we may assume that C is the origin , C = (0,0).

Then D = B/2 , E = A/2

The vectors G - B and A/2 - B are parallel , so G - B = r(A/2 - B)
for some number r

The vectors A - G and A - B/2 are parallel , so A - G = s(A - B/2)
for some number s

Add the left sides and right sides in these equations

G - B = r(A/2 - B)
A - G = s(A - B/2)

to get

A - B = rA/2 - rB + sA - sB/2

and then

(1-r/2-s)A = (1-r-s/2)B

if either coefficient is non zero , then A and B are parallel vectors , which is impossible
in a triangle , so

(1-r/2-s) = 0
(1-r-s/2) = 0


then

r+2s=2
2r+s=2

solve for r and s and get

r = 2/3 , s = 2/3

Now substitute r=2/3 in

G - B = r(A/2 - B)

to get

G - B = (2/3)(A/2 - B)

G = A/3 + B/3


G and F are parallel vectors , so
F = tG = t(A/3 + B/3)
for some number t


A-B and F-B are parallel vectors , so
F-B = w(A-B)
for some number w


Substitute F = t(A/3 + B/3) in

F-B = w(A-B) and get


t(A/3+B/3)-B = w(A-B)

then

(t/3-w)A = (1-t/3-w)B

If either coefficient is not zero , then A and B are parallel vectors , which is impossible in a triangle, so

t/3-w = 0
1-t/3-w = 0

and then

t-3w = 0
t+3w = 3

Solving for t and w

t = 3/2 , w = 1/2


Now substitute  w = 1/2 in

F-B = w(A-B)

to get

F-B = (1/2)(A-B)

then

F = A/2 + B/2


But this means that F is the midpoint of the side AB

so AF/BF = 1
    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    Commentthanx so much for your help and iam terribly sorry that it took me long to say so i hope you will keep the good work & we be fourtane with luck


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I can answer questions from the standard four semester Calculus sequence. I am not prepared for questions on Tensor Calculus. Everything else is welcome. Derivatives, partial derivatives, ordinary differential equations, single and multiple integrals, change of variable, vector integration (Green`s Theorem, Stokes, and Gauss) and applications.

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Ph.D. in Mathematics and many years teaching Calculus at state universities.

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