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Offline Centres

All line segments drawn from a given point to the line, not on it, the perpendicular line segment is the ____.

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Detailed Solution

Assume

l

is a straight line. Draw a perpendicular from the point A (which is not on the line

l

) to

l .

We take a point C (other than B) on

l

and join A and B.
Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest. - Maths Q&A

Here, we have to prove that

A B < A C .

We have a right-angled triangle

Δ A B C

with

∠ B = 90 ° .

We know that the sum of three angles of a triangle

180 ° .

∠ A + ∠ B + ∠ C = 180 °

⇒ ∠ A + 90 ° + ∠ C = 180 ° ⇒ ∠ A + ∠ C = 90 ° ⇒ ∠ A + ∠ C = ∠ B ∵ ∠ B = 90 °

∠ B

is the sum of

∠ A

and

∠ C

, therefore we have,

∠ A < ∠ B and ∠ C < ∠ B

Since

A B and A C

are the opposite sides of

∠ C

and

∠ B

respectively, therefore,

A B < A C

This implies AB is the shortest distance from A to

l .

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

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