Perpendiculars

Two lines (or rays or segments) are perpendicular if they intersect in such a way that the angles formed between them are right angles (90°).

In the given figure, the lines l and m are perpendicular to each other.

You can find perpendicular lines in many everyday objects. Here are five examples:

• The corners of a notebook or a piece of paper

• The edges of a table or a book

• A window frame

Perpendicular Line Through a Given Point

Activity: Using Paper Folding

We can draw a perpendicular line using a simple folding technique:

1. Draw a Line:

• Take a sheet of tracing paper and draw a straight line (l).

2. Mark a Point:

• Choose a point P on this line where you want to construct the perpendicular.

3. Fold the Paper:

• Fold the paper at P in such a way that both halves of the line l overlap exactly.

4. Unfold to Reveal the Perpendicular:

• When you unfold the paper, the crease formed by the fold will be perpendicular to line l at point P.

Steps of Construction

1. Draw the Line Segment:

• Start by drawing a straight line segment AB.

2. Set the Compass Radius:

• Adjust the compass to a radius slightly more than half the length of AB.

3. Draw Arcs from Point A:

• Place the compass pointer at A and draw two arcs—one above and one below the line segment.

4. Draw Arcs from Point B:

• Without changing the compass radius, place the pointer at B and draw two more arcs.

• These arcs should intersect the previous arcs at two points—M (above AB) and N (below AB).

5. Draw the Perpendicular Bisector:

• Use a ruler to join M and N with a straight line.

• This line perpendicularly bisects AB at point P, meaning it divides AB into two equal parts at a 90° angle.

Now, line l is the required perpendicular bisector of AB, and P is its midpoint!

Steps of Construction

1. Draw the Given Line and Point:

• Draw a straight line l.

• Mark a point A somewhere above or below the line (not on it).

2. Draw an Arc from Point A:

• Place the compass pointer at A and draw an arc that cuts the line l at two points.

• Label these points as M and N.

3. Construct Intersecting Arcs Below the Line:

• Keep the same compass radius and place the pointer at M.

• Draw an arc below the line.

• Without changing the radius, place the pointer at N and draw another arc that intersects the first arc.

• Label this intersection point as B.

4. Draw the Perpendicular Line:

• Use a ruler to join points A and B with a straight line.

• This line AB is the perpendicular to l from A.

Now, you've successfully drawn a perpendicular to l from a point not on the line!