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

Question

Philoid · Accepted Answer

Let l is a line segment and B is a point lying to it. We draw a line AB perpendicular to l. Let C be any other point on l

To prove: AB < AC

Proof: In ,

∠B = 90^o

Now,

∠A + ∠B + ∠C = 180^o

∠A + ∠C = 90^o

Therefore,

∠C must be acute angle or ∠C < ∠B

AB < AC (Sides opposite to the larger angle is larger)