Perpendicular is Shortest Straight Line from Point to Straight Line
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Theorem
Let $AB$ be a straight line.
Let $C$ be a point which is not on $AB$.
Let $D$ be a point on $AB$ such that $CD$ is perpendicular to $AB$.
Then the length of $CD$ is less than the length of all other line segments that can be drawn from $C$ to $AB$.
Proof
Let $E$ on $AB$ such that $E$ is different from $D$.
Then $CDE$ forms a right triangle where $CE$ is the hypotenuse.
- $CD^2 + DE^2 = CE^2$
and so $CD < CE$.
$\blacksquare$
Sources
- 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.21 \ \text{(i)}$