Chapter 2.1, Problem 40E

Solution

To find: Area of the triangle with all three sides equal to $\frac{x}{3}$ .

The required area of triangle is $A=\frac{\sqrt{3}{x}^2}{16}$ square units.

Given: All three equal sides of triangle is $\frac{x}{3}$ .

Concept Used:

Formula of area of triangle is $A=\frac{\sqrt{3}}{4}{s}^2,\text{Here }s\text{ is the side length of triangle}\text{.}$

Properties of exponent: ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

Calculation:

  $\begin{array}{l}A=\frac{\sqrt{3}}{4}{\left(\frac{x}{2}\right)}^2\\ A=\frac{\sqrt{3}}{4}\times \frac{x^2}{2^2}\\ A=\frac{\sqrt{3}}{4}\times \frac{x^2}{4}\\ A=\frac{\sqrt{3}{x}^2}{16}\end{array}$

Conclusion:

The required area of triangle is $A=\frac{\sqrt{3}{x}^2}{16}$ square units.