Tim Gan Math
Sign up free

Applying Vectors to Geometric Proofs and Ratio Theory

Applying Vectors to Geometric Proofs and Ratio Theory

TGM Original

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 \(\overrightarrow{WZ}=4\mathbf{q}\).

Trapezium WXYZ with vectors p and q
Trapezium WXYZ with vectors p and q
  1. Find in terms of \(\mathbf{p}\) and/or \(\mathbf{q}\),
    1. \(\overrightarrow{ZX}\),
    2. \(\overrightarrow{WP}\).
  2. Show that the line \(XY\) is parallel to the line \(WP\).
  3. Find the value of
    1. \(\frac{\text{area of}\hspace{0.5em} \Delta WPX}{\text{area of}\hspace{0.5em} \Delta XYZ}\),
    2. \(\frac{\text{area of}\hspace{0.5em} \Delta WZP}{\text{area of}\hspace{0.5em} \Delta XYZ}\).

Video Solution:

Video Solution

Video solution locked

A free account saves your progress; this video needs tuition enrolment.

Create free accountSign inMath tuition

Solution:

Solution locked

680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs tuition enrolment.

Create free accountSign inMath tuition

Create a free account to practise similar questions.

Create free accountSign inMath tuition
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) \(\dfrac3{16}\)

Need help? Join our JC Math tuition classes.

Learn more