In the triangle $OAB$, $M$ is the midpoint of $OA$. $N$ is a point on $OB$ such that

Question

In the triangle $OAB$, $M$ is the midpoint of $OA$. $N$ is a point on $OB$ such that $ON : NB = 2 : 1$. $MN$ is produced to $P$ so that $MN : NP = 1 : 2$. Given $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Express, in terms of $\mathbf{a}$ and/or $\mathbf{b}$, $\overrightarrow{NB}$, $\overrightarrow{MN}$, $\overrightarrow{NP}$. Express $\overrightarrow{AB}$ and $\overrightarrow{BP}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Write down two facts about $\overrightarrow{AB}$ and $\overrightarrow{BP}$. Find $\frac{\text{area of triangle}\hspace{0.5em} PMB}{\text{area of triangle}\hspace{0.5em} PMA}$.

Accepted Answer

(a)(i) NB = (1/3)b; (ii) MN = -(1/2)a + (2/3)b; (iii) NP = -a + (4/3)b. (b) AB = BP = -a + b. (c) AB and BP have equal magnitude and the same direction; A, B and P are collinear with B the midpoint of AP. (d) [PMB]/[PMA] = 1/2.

Properties of Parallelograms and Trapezia in Vector Form

Properties of Parallelograms and Trapezia in Vector Form

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