Hello there.
To answer this question, we need to remember some properties of the isosceles triangle.
First (and most importantly), an isosceles triangle is composed by three sides, in which only two of the sides have equal length.
The perimeter of a triangle is given by the sum of the lengths of the sides, such that: [tex]2p={\ell}_1+{\ell}_2+{\ell}_3[/tex], in which [tex]p[/tex] is the semiperimeter.
Given that the perimeter of such triangle is equal to [tex]155~\text{in}[/tex] and the longest side is [tex]14~\text{in}[/tex] longer than both the other sides, we have that:
[tex]\begin{cases}{\ell}_2={\ell}_3\\ {\ell}_1+{\ell}_2+{\ell}_3=155\\ {\ell}_1={\ell}_2+14\\\end{cases}[/tex]
Using these informations, in terms of [tex]{\ell}_2[/tex], we get:
[tex]{\ell}_2+14+{\ell}_2+{\ell}_2=155[/tex]
Sum the equal terms on the left hand side of the equation
[tex]3\,{\ell}_2+14=155[/tex]
Subtract [tex]14[/tex] on both sides of the equation
[tex]3\,{\ell}_2=141[/tex]
Divide both sides of the equation by a factor of [tex]3[/tex]
[tex]{\ell}_2=47~\text{in}[/tex]
Substituting this value on the first and second equations, we get:
[tex]{\ell}_3=47~\text{in}\\\\\\ {\ell}_1=47+14=61~\text{in}[/tex]
Thus, the length of the sides of this triangle are, respectively from the largest to the smallest: [tex]{\ell}_1=61~\text{in},~{\ell}_2=47~\text{in},~{\ell}_3=47~\text{in}~~\checkmark[/tex]
These are the values we were searching for.
