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.
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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $12$