Vector Applications: Ratios of Areas in Geometric Figures

Vector Applications: Ratios of Areas in Geometric Figures

Secondary 4

In the diagram, \(AQ = 3QC\), \(QB = 5QX\), and \(P\) is the mid-point of \(AB\). \(\overrightarrow{AQ} = 3\mathbf{a}\) and \(\overrightarrow{AP} = \mathbf{b}\).

Triangle ABC with points P, Q, and intersection X
Triangle ABC with points P, Q, and intersection X
  1. Express, as simply as possible, in terms of \(\mathbf{a}\) and or \(\mathbf{b}\),
    1. \(\overrightarrow{BQ}\),
    2. \(\overrightarrow{QX}\),
    3. \(\overrightarrow{CX}\),
    4. \(\overrightarrow{CP}\).
  2. Calculate the value of
    1. \(\frac{CX}{CP}\),
    2. \(\frac{\text{area of}\hspace{0.5em} \Delta BXC}{\text{area of}\hspace{0.5em} \Delta BPX}\).

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Answer:(a)(i) \(3\mathbf a-2\mathbf b\) (ii) \(\dfrac25\mathbf b-\dfrac35\mathbf a\) (iii) \(\dfrac25\mathbf b-\dfrac85\mathbf a\) (iv) \(-4\mathbf a+\mathbf b\) (b)(i) \(\dfrac25\) (ii) \(\dfrac23\)

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