1) To draw a perpendicular to a given line from a point within it.
Method 1:-
(A) When a point P is near the middle of the line.
(i) Let us consider AB be the given line and P the point on it.
(ii) with P as centre and any convenient radius R1 draw arcs cutting AB at point C and D.
(iii) with any radius R2 greater than R1 and centres C and D, draw arcs intersecting each other at O.
(iv) by using scale draw a line connecting points P and O. Then this is the required perpendicular to the given line.
Method 2:-
(B) When a point P is near the end of the line.
(i) Let us consider AB be the given line and P the point on it.
(ii) Mark a point O, with O as centre OP as radius, draw an arc greater than the semi-circle cutting AB at C.
(iii) Draw a line joining C and O, and extend it up to arc to cut it at Q,
(iv) Draw the line joining P and Q. Then PQ is the required perpendicular to the given line.
Method 3:-
(C) When a point P is near the end of the line.
(i) Let us consider AB be the given line and P the point on it.
(ii) with P as centre and any convenient radius, draw an arc greater than the semi-circle cutting AB at C.
(iii) With the same radius cut the arc into two equal divisions CD and DE.
(iv) With the same radius drawn an arc from D and cut this arc by drawing same radius arc from E.
(v) Mark this intersecting point as Q.
(vi) Draw a line joining P and Q, This is the perpendicular to the line AB.