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Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

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Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

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First, let “l” be a line segment and “B” be a point lying on it. A line AB perpendicular to l is now drawn. Also, let C be any other point on l.

The diagram will be as follows:

To prove:

AB < AC

Proof:

In ΔABC, B = 90°

Now, we know that

A+B+C = 180°

∴ A +C = 90°

Hence, C must be an acute angle which implies C < B

AB < AC (As the side opposite to the larger angle is always larger)

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