2025 TGM P2 Q5

The diagram above shows the triangle \(ABC\). The points \(P\), \(Q\) and \(R\) are the midpoints of the line segments \(AB\), \(BC\) and \(AC\) respectively.
Let \(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\).

1. Find \(\overrightarrow{BR}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\).

   [2]

Point \(G\) is the point where \(BR\) and \(AQ\) intersect.

2. By finding the vector line equations of lines \(BR\) and \(AQ\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) and their respective parameters. Show that \(\overrightarrow{OG} = \frac{1}{3}(\mathbf{a} + \mathbf{b} + \mathbf{c})\).

   [5]

A point \(X\) is such that \(GX\) is perpendicular to the plane \(ABC\). The coordinates of \(A\), \(B\) and \(C\) are \((1, 3, 1)\), \((3, 7, -5)\) and \((2, 2, 1)\) respectively.

3. Given that the tetrahedron \(ABCX\) has volume of \(12\text{ units}^3\), find the possible coordinates of \(X\).[4]