Chapter 3.6, Problem 10PSB

To determine

To Find: In the given figure whether the given $\Delta RST$ is equilateral.

Answer to Problem 10PSB

The given triangle is an equilateral triangle

Explanation of Solution

Given information: In the given figure $\Delta RST$ is an isosceles triangle. Also $RS\text{ = }x\text{ + 7, }ST\text{ = 9 - }x\text{ , }TR\text{ = 3}x\text{ + 5}$

Concept used: In an isosceles triangle measurement of two sides is equal.

Calculation: As per the given question we have $RS\text{ = }x\text{ + 7, }ST\text{ = 9 - }x\text{ , }TR\text{ = 3}x\text{ + 5}$ .Also given triangle is an isosceles triangle which means two of its sides are equal in measurement. So we have three cases here:

Case $1:$ When $RS=ST$

Case $2:$ When $RS=TR$

Case $3:$ When $ST=TR$

Considering Case $1:$ When $RS=ST$

  $\begin{array}{l}\Rightarrow x\text{ + 7 = 9 - }x\\ \\ \Rightarrow x\text{ + }x\text{ = 9 - 7}\\ \\ \Rightarrow \text{ 2}x\text{ = 2}\\ \\ \Rightarrow x\text{ = 1}\mathrm{.........................................................}\text{(i)}\end{array}$

Case $2:$ When $RS=TR$

  $\begin{array}{l}\Rightarrow x\text{ + 7 = 3}x\text{ + 5}\\ \\ \Rightarrow x\text{ - 3}x\text{ = 5 - 7}\\ \\ \Rightarrow -2x\text{ = -2}\\ \\ \Rightarrow x\text{ = 1}\mathrm{.....................................................................}\text{(ii)}\end{array}$

Case $3:$ When $ST=TR$

  $\begin{array}{l}\Rightarrow \text{ 9 - }x\text{ = 3}x\text{ + 5}\\ \\ \Rightarrow -x-3x\text{ = 5 - 9}\\ \\ \Rightarrow \text{ - 4}x\text{ = - 4}\\ \\ \Rightarrow x\text{ = 1}\mathrm{..................................................................}\text{(iii)}\end{array}$

  $$ Now in all the three cases we have got value of $x=1$ ,putting this value to find the measurement of the sides of triangle we will get:

  $\begin{array}{l}RS\text{ = }x\text{ + 7 = 1 + 7 = 8 }\mathrm{....................................}\text{(iv)}\\ \\ ST\text{ = 9 - }x\text{ = 9 - 1 = 8 }\mathrm{......................................}\text{(v)}\\ \\ TR\text{ = 3}x\text{ + 5 = (3}\times \text{1) + 5 = 3 + 5 = 8}\mathrm{................................}\text{(vi)}\\ \end{array}$

From (iv),(v) and (vi) $RS\text{ = }ST\text{ = }TR\text{ = 8 }$

As the measurement of all the sides is equal so the given triangle is an equilateral triangle

Conclusion:

The given triangle is an equilateral triangle