A straight line is the shortest distance connection between the two points. This article is presented to show that the Perpendicular is the Shortest Theorem in inequalities in triangles. Our main aim is to prove that from all the straight lines that can be drawn to a line from a given point exterior to it, then the perpendicular is the shortest among them.

## Perpendicular is the Shortest Theorem

Question:

Prove that the shortest segment from a point to a line is the perpendicular segment?

AB is a straight line and P is a point outside it. PQ is perpendicular to AB and PQ is an oblique

To Prove That:

PQ < PR

**Proof:**

In ∆PQR, ∠PQR = 90°, [ Therefore PQ perpendicular AB]

∠PRQ is an acute angle [ in a triangle if one angle is a right angle, the other two must be acute]

∠PRQ < ∠PQR [ from equation 1 and equation 2]

PQ < PR [ in a triangle the greater angle has the greater side opposite to it]

Hence proved

Thus it is shown that all line segments drawn from a given point, not on it, the perpendicular is the shortest.

### FAQs on Perpendicular is the Shortest Theorem

**1. Is the perpendicular distance the shortest?**

The shortest distance from a point to a straight line is always the path that is perpendicular to the line, starting from the given point.

**2. Is perpendicular the shortest side of a triangle?**

In a right-angled triangle, the shortest side is the perpendicular side.

**3. What is the shortest distance Theorem?**

The Shortest Distance Theorem states that the shortest distance between a point P, and a line, l, is the perpendicular line from P to l. It is also called the perpendicular distance.