NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions

Question 80

Method (i): Using Set Squares

1. Place one set square so that one of its long edges lies exactly on line l.
2. Hold the set square fixed with one hand so it doesn’t move.
3. Slide the second set square along the first until one short edge passes through point P.
4. That short edge is at a right angle to l.

   ( ext{Right angle} = 90^circ )

5. Draw the line along this short edge through P. This is the perpendicular to l at P.

Method (ii): Using a Protractor

1. Place the protractor so that its center mark is exactly on point P.
2. Align the base line of the protractor with line l.
3. Find the mark for

   ( 90^circ )

   on the protractor (from either side).
4. Put a small dot at this

   ( 90^circ )

   position from P.
5. Remove the protractor and draw a straight line through P and the dot. This line is perpendicular to l.

Method (iii): Using Ruler & Compasses (Construction)

1. Keep the compass point on P. Open it to a small, convenient width.
2. Draw two small arcs on line l, one on each side of P. Call the points where the arcs meet line l as A and B.
3. Without changing the compass width, place the compass on A and draw an arc above the line.
4. Place the compass on B and draw another arc to cross the previous arc. Call the intersection point C.
5. Draw a straight line through P and C. This PC is perpendicular to l.

   ( angle APC = 90^circ )

Why is there only one perpendicular through P?

At a point on a line, there is exactly one direction that makes a right angle with the line.

If there were two different lines through P, both making

( 90^circ )

with l, they would have to overlap—so they are actually the same line.

Therefore, the perpendicular through P is unique.