Consider the triangle with vertices \(A(4,5,1), B(8,1,2)\) and the origin \(O(0,0,0)\). The triangle has three medians.
The median through \(B\) has vector equation
\(\lambda \, \overrightarrow{O M}+(1-\lambda) \overrightarrow{O B}=\left(\begin{array}{l}8 \\ 1 \\ 2\end{array}\right)+\lambda\left(\begin{array}{c}-6 \\ 1.5 \\ -1.5\end{array}\right)\) (Do NOT prove this.)
for a parameter \(\lambda \in[0,1]\) and where \(M\) is the midpoint of \(O A\).
- Write down the median through \(A\) as a vector equation. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- The three medians of any triangle meet at a point known as the centroid.
- Find the value of \(\lambda\) corresponding to the centroid. (2 marks)
--- 3 WORK AREA LINES (style=lined) ---
- Into what ratio does the centroid divide the median \(B M\)? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
a. \(\text{Let}\ N = \text{midpoint of}\ OB:\)
\(N \equiv \left(\dfrac{8+0}{2}, \dfrac{1+0}{2}, \dfrac{2+0}{2}\right) \equiv (4,0.5,1).\)
\(\text{Direction vector of the median through} \ A \ \text {is}\left(\begin{array}{c}4-4 \\0.5-5 \\1-1\end{array}\right) = \left(\begin{array}{c}0 \\-4.5 \\0\end{array}\right).\)
\(\text{Equation of median from}\ A:\)
\(\left(\begin{array}{c}4 \\5 \\1\end{array}\right) + \mu \left(\begin{array}{c}0 \\-4.5 \\0\end{array}\right)\ \ \text{for}\ \ \mu \in[0,1]\)
b. \(\text{Equation of median from}\ B:\)
\(\left(\begin{array}{c}8 \\1 \\2\end{array}\right) + \lambda \left(\begin{array}{c}-6 \\1.6 \\-1.5\end{array}\right)\ \ \text{for}\ \ \lambda \in[0,1]\)
\(\text{Medians intersect at centroid}\)
\(x\text{-coordinate of median through}\ B = 8-6\lambda\)
\(x\text{-coordinate of median through}\ A = 4\)
\(\text{Equating}\ x\text{-coordinates:}\)
\(8-6 \lambda=4\ \ \Rightarrow\ \ \lambda=\dfrac{2}{3}\)
\(\therefore\ \text{Point of intersection occurs at } \lambda=\dfrac{2}{3}.\)
c. \(\text{The centroid divides the median in the ratio of} \ 2:1.\)