NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Fill in the Blanks

Question 37

37. The number of lines of symmetry in a regular polygon of ( n ) sides is ____.

Answer: ( n ).

Beginner-friendly explanation (step by step)

1. What is a regular polygon?

   All sides are equal.

   All angles are equal.

2. What is a line of symmetry?

   A line that splits a shape into two mirror-image halves.

3. How do we find symmetry lines in a regular polygon?

   • Through vertices: Draw a line from a vertex, through the centre.

     This line reflects the polygon onto itself.

   • Through side midpoints: Draw a line from the midpoint of a side, through the centre.

     This can also reflect the polygon onto itself (for even-sided polygons).

4. Count them carefully.

   • Odd ( n ):

     Each line goes through a vertex and the midpoint of the opposite side.

     Total lines (=) number of vertices.

     So, lines of symmetry (= n).

   • Even ( n ):

     There are two kinds of lines:

     ( frac{n}{2} ) lines through opposite vertices,

     ( frac{n}{2} ) lines through midpoints of opposite sides.

     Total (= frac{n}{2} + frac{n}{2} = n).

5. Check with small examples.

   Equilateral triangle: ( n=3 Rightarrow ) lines (=3).

   Square: ( n=4 Rightarrow ) lines (=4).

   Regular pentagon: ( n=5 Rightarrow ) lines (=5).

6. Therefore:

   The number of lines of symmetry in a regular ( n )-gon is

   ( n ).