2017 CJC P1 Q11

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

For the triangle shown below, \(O\), \(A\) and \(B\) are vertices, where \(O\) is the origin, \(\overrightarrow{OA}=\mathbf{a}\) and \(\overrightarrow{OB}=\mathbf{b}\).

The midpoints of \(OB\), \(OA\) and \(AB\) are \(M\), \(N\) and \(T\) respectively.

It is given that \(X\) is the point of intersection between the medians of triangle \(OAB\) from vertices \(A\) and \(B\).

1. Show that \(\overrightarrow{OX}=\frac{1}{3}\left( \mathbf{a}+\mathbf{b} \right)\).[4]
2. Prove that \(X\) also lies on \(OT\), the median of triangle \(OAB\) from vertex \(O\).[2]

The centroid of triangle \(OAB\) is the common point of intersection \(X\) between all three medians of the triangle.

Ray tracing is a technique in computer graphics rendering used to realistically capture the lighting effect in a scene being modelled. Starting from a chosen viewpoint, different rays are being traced backwards towards different parts of an object in the scene and reflected off the object. For each ray, if it reflects off the object and intersects a light source, then the part of the object at which the ray is reflected off would be made to appear brighter.

In a particular scene depicting a dolphin jumping out of the ocean, a ray is being traced back from a chosen viewpoint at \(V\) to the centroid \(X\) of a particular triangular facet defined by the vertices comprising the origin \(O\), \(A\left( 5,4,6 \right)\) and \(B\left( -2,2,3 \right)\), and then reflected off the facet at \(X\), as shown in Figure 1.

3. Show that the plane \(p\) which contains the triangular facet \(OAB\) can be represented by the cartesian equation \(-3y+2z=0\).[2]
4. Given \(V\left( 1,-68,-37 \right)\), determine the coordinates of the foot of perpendicular \(F\) from \(V\) to plane \(p\).[4]

The reflected ray travels along a line \(m\) such that:

• both line \(VX\) (denoted by \(l\) ) and line \(m\) lie in a plane that is perpendicular to plane \(p\), and

• the angle between line \(l\) and plane \(p\) equals the angle between line \(m\) and plane \(p\).

5. By first finding two suitable points lying on line \(m\), or otherwise, find a cartesian equation for line \(m\).[5]