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

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## Question

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

Video Solution
Triangles
Ex 7.4 | Question 6

## 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.

Video Solution
Triangles
Ex 7.4 | Question 6

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