Referred to an origin ${O}$, the position vectors of four non-collinear points ${A}$, ${B}$, ${C}$ and ${D}$ are ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{c}}$ and ${\mathbf{d}}$ respectively. 
Given that&nbsp;$\mathbf{a}-\mathbf{b}=\mathbf{d}-\mathbf{c}$, show that&nbsp;${ABCD}$ is a parallelogram.
Given also that&nbsp;$\left| \mathbf{a}-\mathbf{c} \right|=\left| \mathbf{b}-\mathbf{d} \right|$, identify the shape of the parallelogram ${ABCD}$, justifying your answer.[6]

Referred to an origin ${O}$, the position vectors of four non-collinear points ${A}$, ${B}$, ${C}$ and ${D}$ are ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathbf{c}}$ and ${\mathbf{d}}$ respectively. 

Given that $\mathbf{a}-\mathbf{b}=\mathbf{d}-\mathbf{c}$, show that ${ABCD}$ is a parallelogram.

Given also that $\left| \mathbf{a}-\mathbf{c} \right|=\left| \mathbf{b}-\mathbf{d} \right|$, identify the shape of the parallelogram ${ABCD}$, justifying your answer.

N2004 P1 Q3

N2004 P1 Q3

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