Question

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

[Hint: Use proof by contradiction].

Answer 1

Given Consider a line l and a point P.

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 ΔAPB, ∠A + ∠P + ∠B = 180°  ...[By angle sum property of a triangle is 180°]

⇒ 90 + ∠P + 90° = 180°

⇒ ∠P = 180° – 180°

∴ ∠P = 0°

So, lines n and m coincide.

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