# Ex.7.4 Q6 Triangles Solution - NCERT Maths Class 9

## Question

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

## Text Solution

**What is Known?**

Line segments drawn from a given point is not on line.

**To prove:**

The perpendicular line segment is the shortest.

**Reasoning:**

We know that in a triangle if one angle is \(90\) degree then other have to be acute.

**Steps:**

Let us take a line \(l\) and from point \(P\) (i.e., not online \(l\)), draw two line segments \(PN\) and \(PM\). Let \(PN\) be perpendicular to line \(l\) and \(PM\) is drawn at some other angle.

In \(\Delta PNM\),

\[\begin{align} & \angle N = 90 ^ { \circ } \\ & \angle P + \angle N + \angle M = 180 ^ { \circ }\\& \left( \begin{array} & \text{Angle sum property } \\ \text{of a triangle} \\ \end{array} \right)\\ \\ & \angle P + \angle M = 90 ^ { \circ } \end{align}\]

Clearly, \(\angle M\) is an acute angle.

\[\begin{align} & \angle M \lt \angle N \\ & PN \lt PM \\&\left(\begin{array}{} \text {side opposite to the} \\\text{smaller angle is smaller}\end{array}\right) \end{align}\]

Similarly, by drawing different line segments from \(P\) to \(l\), it can be proved that \(PN\) is smaller in comparison to them. Therefore, it can be observed that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.