Question #160084

 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.


1
Expert's answer
2021-02-03T03:39:49-0500

Solution.

By the rule of adding vectors we get

OA+AB=OB,\vec{OA}+\vec{AB}=\vec{OB},

from here AB=OBOA=ba.\vec{AB}=\vec{OB}-\vec{OA}=b-a.

By the rule of adding vectors we get

OB+BC=OC,\vec{OB}+\vec{BC}=\vec{OC},

from here BC=OCOB=cb.\vec{BC}=\vec{OC}-\vec{OB}=c-b.

DD is the midpoint of BCBC, then BD=cb2.\vec{BD}=\frac{c-b}{2}.

By the rule of adding vectors we get

OD=OB+BD=b+cb2=b+c2.\vec{OD}=\vec{OB}+\vec{BD}=b+\frac{c-b}{2}=\frac{b+c}{2}.

By the rule of adding vectors we get

AD=AB+BD=ba+cb2=b+c2a2.\vec{AD}=\vec{AB}+\vec{BD}=b-a+\frac{c-b}{2}=\frac{b+c-2a}{2}.

EE is the point on ADAD such that AE:ED=2:1,AE:ED=2:1, then AE=23AD=23b+c2a2=b+c2a3.\vec{AE}=\frac{2}{3}\vec{AD}=\frac{2}{3}\frac{b+c-2a}{2}=\frac{b+c-2a}{3}.

By the rule of adding vectors we get

OE=OA+AE=a+b+c2a3=a+b+c3.\vec{OE}=\vec{OA}+\vec{AE}=a+\frac{b+c-2a}{3}=\frac{a+b+c}{3}.

Answer.     

OD=b+c2\vec{OD}=\frac{b+c}{2}

 OE=a+b+c3\vec{OE}=\frac{a+b+c}{3}

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