Question

prove that through a given point we can draw only one perpendicular to a given line.
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Answer 1

Answer:

Construction Draw two intersecting lines passing through the point P and which is perpendicular to l

To prove Only one perpendicular line can be drawn through a given point i.e., to prove `/_P = 0^(@)`.

proof In `DeltaAPB, " " /_A +/_P + /_B = 180^(@) " "` [by angle sum property of a triangle is `180^(@)`]

`rArr " "90^(@) + /_ + 90^(@) = 180^(@)`

`rArr " "/_P = 180^(@)- 180^(@)`

`:." "/_P = 0^(@)`

So, lines m and m coincide.

Hence, only one perpendicular line can be drawn through a given point.

Answer 2

Answer:

From the point P , a perpendicular PM is drawn to the given line AB.

∴ ∠ PMB = 90

∘

Let if possible , we can draw another perpendicular PN to the line AB. Then ,

∠ PMB = 90

∘

∴ ∠ PMB = ∠ PNB , which is possible only when PM and PN coincide with each other.

Hence , through a given point , we can draw only one perpendicular to a given line.