The perimeter of a triangle is$97$ feet. One side of the triangle is eleven feet more than the smallest side. The third side is six feet more than twice the smallest side. Find the lengths of all sides.

Given that the perimeter of a triangle is $97$feet.

One side is eleven feet more than the smallest side.

And the third side is six feet more than twice the smallest side.

The objective is to find the lengths of all sides.

Let the length of the smallest side be $x$feet.

One side of the triangle is eleven feet more than the smallest side, so it is represented as $x+11$feet.

The third side is six feet more than twice the smallest side, so it is represented as$2x+6$feet.

The perimeter of a triangle is the sum of lengths of all three sides.

For the given triangle the perimeter is $97$feet and the lengths are given as $x,x+11,2x+6$. So an equation can be written as

$\begin{array}{rcl}x+x+11+2x+6& =& 97\\ 4x+17& =& 97\\ 4x+17-17& =& 97-17\\ 4x& =& 80\end{array}$

Divide both sides by $4$.

$\begin{array}{rcl}\frac{4x}{4}& =& \frac{80}{4}\\ x& =& 20\end{array}$

The length of the smallest side is $x=20$feet.

The length of the second side is

$\begin{array}{rcl}x+11& =& 20+11\\ & =& 31feet\end{array}$

And the length of the third side is

$\begin{array}{rcl}2x+6& =& 2\times 20+6\\ & =& 40+6\\ & =& 46feet\end{array}$