# Ex.7.5 Q4 Triangles Solution - NCERT Maths Class 9

## Question

Complete the hexagonal and star shaped rangolies (see the given figures) by filling them with as many equilateral triangles of side \(1\,\rm cm\) as you can. Count the number of triangles in each case. Which has more triangles?

## Text Solution

**Steps:**

It can be observed that hexagonal-shaped rangoli has 6 equilateral triangles in it.

Area of an equilateral Triangle=\( \frac { \sqrt { 3 } } { 4 } ( \text { side } ) ^ { 2 }\)

\[\begin{align}\text{Area of } \Delta OAB &= \frac { \sqrt { 3 } } { 4 } ( \text {side} )^{2}\\&= \frac { \sqrt { 3 } } { 4 } ( 25 ) \\ &= \frac { 25 \sqrt { 3 } } { 4 }\; cm ^ { 2 } \end{align}\]

Area of hexagonal-shaped Rangoli

\[\begin{align} &= 6 \times \frac { 25 \sqrt { 3 } } { 4 } \\ &= \frac { 75 \sqrt { 3 } } { 2 }\;cm ^ { 2 } \end{align}\]

Area of equilateral triangle having its side as \(1\,\rm cm\)

\[\begin{align} & = \frac { \sqrt { 3 } } { 4 } ( 1 ) ^ { 2 } \\ & = \frac { \sqrt { 3 } } { 4 } \; cm ^ { 2 } \end{align}\]

Number of equilateral triangles of 1 cm side that can be filled in this hexagonal-shaped Rangoli

\[\begin{align} & = \left( \frac { \frac { 75 \sqrt { 3 } } { 2 } } { \frac { \sqrt { 3 } } { 4 } } \right)\\ & =150 \end{align}\]

Star-shaped rangoli has \(12\) equilateral triangles of side \(5\,\rm cm\) in it.

Area of star-shaped rangoli

\[\begin{align}& = 12 \times \frac { \sqrt { 3 } } { 4 } \times ( 5 ) ^ { 2 } \\ & = 75 \sqrt { 3 } \end{align}\]

Number of equilateral triangles of 1 cm side that can be filled in this star-shaped rangoli

\[\begin{align} &= \left( \frac { 75 \sqrt { 3 } } { \frac { \sqrt { 3 } } { 4 } } \right) \\ &= 300 \end{align}\]

Therefore, star-shaped rangoli has more equilateral triangles in it.