Chapter 3.6, Problem 11PSB

To determine

To Find: In the given figure base side of $\Delta VSY$ .

Answer to Problem 11PSB

The base side of $\Delta VSY=VY$

Explanation of Solution

Given information: In the given figure $\Delta VSY$ is an isosceles triangle with the perimeter less than $\text{ 45}$ .Also $VS\text{ = 10, }SY\text{ = 2}x-\text{ 8 , }VY\text{ = }x\text{ + 7}$

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

Calculation: As per the question we have $VS\text{ = 10, }SY\text{ = 2}x-\text{ 8 , }VY\text{ = }x\text{ + 7}$ .The perimeter of the given triangle is less than $\text{ 45}$ . Also $\Delta VSY$ is an isosceles triangle which means it has two of its sides equal. So, we get three cases here:

Case $1:$ When $VY=VS$

Case $2:$ When $VY=SY$

Case $3:$ When $SY=VS$

Considering Case $1:$

When $VY=VS$

  $\begin{array}{l}\Rightarrow x\text{ +7 = 10 }\\ \\ \Rightarrow x\text{ = 3}\end{array}$

Substituting this value of $x$ in the given values of $VS,VY,SY$ we will get:

  $VS\text{ = 10 , }SY\text{ = 2}x\text{ - 8 = ( 2}\times \text{3 ) - 8 = 6 - 8 = -2 and }VY\text{= 3 +7 = 10 }$

Now here $VS=VY=10$ but $SY=-2$ ,which is not possible as measurement of side of triangle can’t be negative. So we can’t consider value of $x=3$ ………………………………….(i)

Considering Case $2:$ When $VY=SY$

  $\begin{array}{l}\Rightarrow x\text{ +7 = 2}x\text{ - 8}\\ \\ \Rightarrow x\text{ - 2}x\text{ = - 8 - 7}\\ \\ \Rightarrow \text{ - }x\text{ = - 15}\\ \\ \Rightarrow x\text{ = 15}\end{array}$

Substituting this value of $x$ in the given values of $VS,VY,SY$ we will get:

  $VS\text{ = 10 , }SY\text{ = 2}x\text{ - 8 = ( 2}\times 15\text{ ) - 8 = 30 - 8 = 22 and }VY\text{= 15+7 = 22 }$

Now here $SY=VY=22$ and $VS=10$ .Thus the perimeter = $22+22+10=52$ which his more than $45$ . So we can’t consider value of $x=15$ ………………………………………….(ii)

Considering Case $3:$ When $SY=VS$

  $\begin{array}{l}\Rightarrow \text{ 2}x\text{ - 8 = 10 }\\ \\ \Rightarrow \text{ 2}x\text{ = 10 + 8 }\\ \\ \Rightarrow \text{ 2}x\text{ = 18}\\ \\ \Rightarrow x\text{ = 9 }\end{array}$

Substituting this value of $x$ in the given values of $VS,VY,SY$ we will get:

  $VS\text{ = 10 , }SY\text{ = 2}x\text{ - 8 = ( 2}\times 9\text{ ) - 8 = 18 - 8 = 10 and }VY\text{= 9 +7 = 16 }$

Now here $VS=SY=10$ and $VY=16$ . Thus, the perimeter = $\text{10 + 10 + 16 = 36 }$ .

As perimeter is less than $45$ and measurement of two of the sides of triangle are equal so we will consider value of $x=9$ ………………………………………….(iii)

Considering results of (i) ,(ii) and (iii) we can say case $3$ satisfies all the given conditions thus according to that the non-equal side will be the base. $$

Conclusion:

The base side of $\Delta VSY=VY$