Median Formula/Examples/(1, -2), (-3,4), (2,2)

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Example of Use of Median Formula

Consider the triangle $\triangle ABC$ whose vertices are:

$A = \tuple {1, -2}, B = \tuple {-3, 4}, C = \tuple {2, 2}$

The length of the median of $\triangle ABC$ which which bisects $AB$ is $\sqrt {10}$.


Proof

Let $\triangle ABC$ be embedded in a complex plane.

Let the position vectors of $A$, $B$ and $C$ be $z_1 = 1 - 2 i$, $z_2 = -3 + 4 i$, $z_3 = 2 + 2 i$ respectively.


Length-of-Triangle-Median-Complex-(1,-2),(-3,4),(2,2).png


Then:

\(\ds AC\) \(=\) \(\ds z_3 - z_1\)
\(\ds \) \(=\) \(\ds \paren {2 + 2 i} - \paren {1 - 2 i}\)
\(\ds \) \(=\) \(\ds 1 + 4 i\)
\(\ds BC\) \(=\) \(\ds z_3 - z_2\)
\(\ds \) \(=\) \(\ds \paren {2 + 2 i} - \paren {-3 + 4 i}\)
\(\ds \) \(=\) \(\ds 5 - 2 i\)
\(\ds AB\) \(=\) \(\ds z_2 - z_1\)
\(\ds \) \(=\) \(\ds \paren {-3 + 4 i} - \paren {1 - 2 i}\)
\(\ds \) \(=\) \(\ds -4 + 6 i\)
\(\ds AD\) \(=\) \(\ds \dfrac {AB} 2\)
\(\ds \) \(=\) \(\ds \paren {\dfrac {-4 - 6 i} 2}\)
\(\ds \) \(=\) \(\ds -2 + 3 i\)

Then:

\(\ds AC + CD\) \(=\) \(\ds AD\)
\(\ds \leadsto \ \ \) \(\ds CD\) \(=\) \(\ds AD - AC\)
\(\ds \) \(=\) \(\ds \paren {-2 + 3 i} - \paren {1 + 4 i}\)
\(\ds \) \(=\) \(\ds -3 - i\)


Hence:

\(\ds \size {CD}\) \(=\) \(\ds \cmod {-3 - i}\)
\(\ds \) \(=\) \(\ds \sqrt {3^2 + 1^2}\)
\(\ds \) \(=\) \(\ds \sqrt {10}\)

$\blacksquare$


Sources