<p>Prove that the medians of a triangle intersect one another at two-thirds the distance from the vertex to the mid-point of the opposite side.</p>

Prove that the medians of a triangle intersect one another at two-thirds the distance from the vertex to the mid-point of the opposite side.

Vectors - General

Vectors

H2 Math

International Baccalaureate Higher Level Mathematics

IB HL Math

Proving the Properties of intersection of medians of triangle using Vectors

Let \(O\), \(A\) and \(B\) be the vertices of a triangle.

\(\therefore \overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\)

Let points \(M\) and \(N\) be the midpoints of \(OB\) and \(AB\) respectively.

\(\overrightarrow{OM} = \frac{1}{2}\overrightarrow{OB} = \frac{1}{2}\mathbf{b}\), \(\overrightarrow{ON} = \dfrac{\overrightarrow{OA} + \overrightarrow{OB}}{2} = \dfrac{\mathbf{a} + \mathbf{b}}{2}\)

Let point \(E\) be the intersection point of medians \(AM\) and \(ON\).

\(l_{AM}: \mathbf{r} = \overrightarrow{OA} + \lambda \overrightarrow{AM}\)

\(\mathbf{r} = \mathbf{a} + \lambda \left( \frac{1}{2}\mathbf{b} - \mathbf{a} \right)\)

\(\mathbf{r} = (1-\lambda)\mathbf{a} + \frac{1}{2}\lambda \mathbf{b}\)

\(l_{ON}: \mathbf{r} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} + \mu \overrightarrow{ON}\); \(\mu \in \mathbb{R}\)

\(\mathbf{r} = \mu \left( \dfrac{\mathbf{a} + \mathbf{b}}{2} \right)\)

\(\mathbf{r} = \frac{1}{2}\mu \mathbf{a} + \frac{1}{2}\mu \mathbf{b}\)

\((1-\lambda)\mathbf{a} + \frac{1}{2}\lambda \mathbf{b} = \frac{1}{2}\mu \mathbf{a} + \frac{1}{2}\mu \mathbf{b}\)

Compare the coefficients of \(\mathbf{a}\) and \(\mathbf{b}\):

\(1 - \lambda = \frac{1}{2}\mu\), \(\frac{1}{2}\lambda = \frac{1}{2}\mu\)

\(1 - \mu = \frac{1}{2}\mu\), \(\lambda = \mu\)

\(\overrightarrow{ON} = \dfrac{\mathbf{a} + \mathbf{b}}{2} = \frac{1}{2}(\mathbf{a} + \mathbf{b})\) --- (3)

\(\overrightarrow{OE} = \frac{2}{3} \cdot \dfrac{\mathbf{a} + \mathbf{b}}{2} = \frac{1}{3}(\mathbf{a} + \mathbf{b})\) --- (4)

From (3) and (4): \(\dfrac{ON}{OE} = \dfrac{\frac{1}{2}}{\frac{1}{3}} = \frac{3}{2}\)

\(\overrightarrow{AE} = \frac{1}{3}(\mathbf{a} + \mathbf{b}) - \mathbf{a} = -\frac{2}{3}\mathbf{a} + \frac{1}{3}\mathbf{b} = \frac{2}{3}\left( -\mathbf{a} + \frac{1}{2}\mathbf{b} \right)\) --- (1)

\(\overrightarrow{AM} = \frac{1}{2}\mathbf{b} - \mathbf{a}\) --- (2)

From (1) and (2): \(\overrightarrow{AE} = \frac{2}{3}\overrightarrow{AM} \Rightarrow \dfrac{AE}{AM} = \frac{2}{3}\)

\(\therefore\) Medians of a triangle intersect at \(\frac{2}{3}\) the distance from vertex to midpoint of the opposite side.

Proving the Properties of intersection of medians of triangle using Vectors

Proving the Properties of intersection of medians of triangle using Vectors

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