Drawing a perpendicular to a line from a point outside the line

Perpendicular to a Line Through a Point Not on It:

A. By folding the paper:

• Draw a line MN on a paper. Take a point P anywhere outside the line.

• Keeping the line MN in view, fold the paper along the line MN.

• Now fold the paper through point P in such a way that the part of line MN on one side of the fold falls on the part of line MN on the other side of the fold.
                 
• Unfold the paper. Name the point of intersection of the two folds Q. Draw the line PQ. This line falls on a fold in the paper.
               
  Using a protractor, measure every angle formed at the point Q.
  Line PQ is perpendicular to line MN.
  Line PQ ⊥ line MN.

A. Method using ruler and compasses:

Step 1: Given a line l and a point P not on it.

Step 2: With P as a centre, draw an arc that intersects line l at two points A and B.

Step 3: Using the same radius and with A and B as centers, construct two arcs that intersect at a point, say Q, on the other side.

Step 4: Join PQ. Thus, `bar"PQ"` is perpendicular to l.

B. Method using a ruler and a set-square (An optional activity):

Step 1: Let l be the given line and P be a point outside l.

Step 2: Place a set-square on l such that one arm of its right angle aligns along l.

Step 3: Place a ruler along the edge opposite to the right angle of the set-square.

Step 4: Hold the ruler fixed. Slide the set-square along the ruler till the point P touches the other arm of the set-square.

Step 5: Join PM along the edge through P, meeting l at M.
Now, `bar"PM"` ⊥ l.