NCERT Class 9 Solutions: Triangles (Chapter 7) Exercise 7.5 – Part 4
Equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
Q4 Complete the hexagonal rangoli and the star rangolies (See figure) but filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?
Solution:

In hexagonal, Number of equilateral triangles each of side 5 cm are 6.

Area of equilateral triangle =

Area of hexagonal = Area of an equilateral triangle rangoli
……………………..equation(1)

Now area of equilateral triangle of side ………..equation(2)

Number of equilateral triangles each of side 1cm in hexagonal rangoli
………………equation(3)

Now in star

Number of equilateral triangles each of side 5 cm=12

Therefore , total area of star area of an equilateral triangle of side 5cm
……….equation(4)

Number of equilateral triangles each of side 1 cm in star
……….equation (4)

From eq. (3) and (4), we observe that star has more equilateral triangles each of side 1 cm.