In the diagram, $WXYZ$ is a trapezium and $WX$ is parallel to $ZY$. The point $P$ on $XZ$

Question

In the diagram, $WXYZ$ is a trapezium and $WX$ is parallel to $ZY$. The point $P$ on $XZ$ is such that $ZP:ZX=1:4$ and $4WX=3ZY$. $\overrightarrow{WX}=12\mathbf{p}$ and $\overrightarrow{WZ}=4\mathbf{q}$. Find in terms of $\mathbf{p}$ and/or $\mathbf{q}$, $\overrightarrow{ZX}$, $\overrightarrow{WP}$. Show that the line $XY$ is parallel to the line $WP$. Find the value of $\frac{\text{area of}\hspace{0.5em} \Delta WPX}{\text{area of}\hspace{0.5em} \Delta XYZ}$, $\frac{\text{area of}\hspace{0.5em} \Delta WZP}{\text{area of}\hspace{0.5em} \Delta XYZ}$.

Accepted Answer

(a)(i) $12\mathbf p-4\mathbf q$ (ii) $3\mathbf p+3\mathbf q$ (b) $\overrightarrow{XY}=\dfrac43\overrightarrow{WP}$, hence $XY\parallel WP$ (c)(i) $\dfrac9{16}$ (ii) $\dfrac3{16}$

Applying Vectors to Geometric Proofs and Ratio Theory

Applying Vectors to Geometric Proofs and Ratio Theory

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