Answer to Question #160084 in Geometry for Kelvin Osei

Question

 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.

Answer 1

Solution.

By the rule of adding vectors we get

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

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

By the rule of adding vectors we get

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

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

$D$ is the midpoint of $BC$, then $\vec{BD}=\frac{c-b}{2}.$

By the rule of adding vectors we get

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

By the rule of adding vectors we get

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

$E$ is the point on $AD$ such that $AE:ED=2:1,$ then $\vec{AE}=\frac{2}{3}\vec{AD}=\frac{2}{3}\frac{b+c-2a}{2}=\frac{b+c-2a}{3}.$

By the rule of adding vectors we get

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

Answer.     

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

 $\vec{OE}=\frac{a+b+c}{3}$