In a quadrilateral OABC, D is the midpoint of BC and E is the point on AD such that AE : ED = 2 : 1. Given that OA = A, OB = B, and OC = c express OD and OE in terms of a,b and c.
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Expert's answer
2021-02-01T12:00:23-0500
The question does not contain complete information that is needed to resolve it. Depending on angles ∠BOC and ∠AOB , the answer may be different. If we change these angles and and all conditions are done, OD and OE change. It is shown on the picture
Solution:
Introduce vectors: OA=a , OB=b , OC=c .
Then OD=2b+c (used BD = DC).
DA=a−OD=a−2b+c=22a−(b+c) .
AD and AM are medians. E is the point of the intersection of AD and AM. In this point each median divides as 2:1 (AE:AD = 2:1). That's why
DE=31DA=62a−(b+c)
OE=OD+DE=2b+c−22a−(b+c)=3a+b+c
OD=∣OD∣=∣2b+c∣=21(b+c)2=21b2+c2+2bccos∠BOC
OE=∣OE∣=∣3a+b+c∣=31(a+b+c)2=
=31a2+b2+c2+2(abcos∠AOB+bccos∠BOC+accos∠AOC)
An answer depends on angles ∠BOC , ∠AOB and ∠AOC=∠BOC+∠AOB.
We are free to change them without breaking any condition (as shown on the picture). So these angles cannot be expressed in terms of a, b and c.
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