Where two lines in a triangle cross
Where two lines in a triangle cross
IP 4
8 marks
In triangle \(OAB\), \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). The point \(C\) lies on \(AB\) with \(AC:CB=1:3\), and \(D\) is the midpoint of \(OB\). The lines \(OC\) and \(AD\) meet at \(X\).
- Express \(\overrightarrow{OC}\) and \(\overrightarrow{AD}\) in terms of \(\mathbf a\) and \(\mathbf b\).[2]
- Given that \(\overrightarrow{OX}=s\overrightarrow{OC}\) and \(\overrightarrow{AX}=t\overrightarrow{AD}\), find \(s\) and \(t\). Hence state the ratios \(OX:XC\) and \(AX:XD\).[4]
- Find the ratio of the area of triangle \(OXA\) to the area of triangle \(OCA\).[2]
Solution:
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Answer:(a) \(\overrightarrow{OC}=\dfrac34\mathbf a+\dfrac14\mathbf b\), \(\overrightarrow{AD}=\dfrac12\mathbf b-\mathbf a\) (b) \(s=\dfrac45\), \(t=\dfrac25\); \(OX:XC=4:1\), \(AX:XD=2:3\) (c) \(4:5\)
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