1998 Paper 2 Q8

Year: 1998
Paper: 2
Question Number: 8

Course: LFM Pure and Mechanics
Section: Vectors

Difficulty: 1600.0 Banger: 1487.4

vectors geometric proof triangle centroid midpoint concurrent lines plane intersection 3D geometry

Problem

Points \(\mathbf{A},\mathbf{B},\mathbf{C}\) in three dimensions have coordinate vectors \(\mathbf{a},\mathbf{b},\mathbf{c}\), respectively. Show that the lines joining the vertices of the triangle \(ABC\) to the mid-points of the opposite sides meet at a point \(R\). \(P\) is a point which is {\bf not} in the plane \(ABC\). Lines are drawn through the mid-points of \(BC\), \(CA\) and \(AB\) parallel to \(PA\), \(PB\) and \(PC\) respectively. Write down the vector equations of the lines and show by inspection that these lines meet at a common point \(Q\). Prove further that the line \(PQ\) meets the plane \(ABC\) at \(R\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1487.4

Banger Comparisons: 1

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Problem source

Points $\mathbf{A},\mathbf{B},\mathbf{C}$ in three dimensions have coordinate vectors
$\mathbf{a},\mathbf{b},\mathbf{c}$, respectively. Show that the lines joining the vertices of the
triangle $ABC$ to the mid-points of the opposite sides meet at a point $R$.
$P$ is a point which is {\bf not} in the plane $ABC$.
Lines are drawn through the mid-points of $BC$, $CA$ and $AB$ parallel to
$PA$, $PB$ and $PC$ respectively. Write down the vector equations of the
lines and show by inspection that these lines
meet at a common point $Q$.
Prove further that the line $PQ$ meets the plane $ABC$ at $R$.

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