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In each of the following, draw perpendicular through point P to the line segment AB :

(i)

(ii)

(iii)

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#### Solution

**(i) Steps of Construction :**

- With P as a centre, draw an arc of a suitable radius which cuts AB at points C and D.
- With C and D as centres, draw arcs of equal radii and let these arcs intersect each other at point Q.

[The radius of these arcs must be more than half of CD and both the arcs must be drawn on the other side] - Join P and Q
- Let PQ cut AB at the point O.

Thus, OP is the required perpendicular clearly, ∠AOP = ∠BOP = 90°

**(ii) Steps of Construction :**

- With P as a centre, draw an arc of any suitable radius which cuts AB at points C and D.
- With C and D as centres, draw arcs of equal radii. Which intersect each other at point A.

[This radius must be more than half of CD and let these arc intersect each other at the point 0] - Join P and O. Then OP is the required perpendicular.

∠OPA = ∠OPB = 90°

**(iii) Steps of Construction :**

- With P as a centre, draw an arc of any suitable radius which cuts AB at points C and D.
- With C and D as a centre, draw arcs of equal radii

[The radius of these arcs must be more than half of CD and both the arcs must be drawn on the other side.]

and let these arcs intersect each other at the point Q. - Join Q and P. Let QP cut AB at the point O. Then OP is the required perpendicular.

Clearly, ∠AOP = ∠BOP = 90°

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