The longest side of the triangular section is 7ft. Shorter than twice the shortest side. The third side is 6ft. Longer than the shortest side. how long is each.

Question

first read the problem carefull, determine what info you will need to solve the problem, and choose a variable to represent one unknown quantity. Look for key words and translate the words into algebraic symbols. Use a relationship given in the problem or an appropriate formula on order to write an equation. Finally solve the equation. Recall that the perimeter of a triangle is the sum of the lengths of its sides.

Answer 1

You have a problem statement and a pretty good explanation of the method for solving it.  Let's follow the instructions.

Read and re-read the problem until you can understand it and put it into your own words.  How's this: "Find the length of three sides of a triangle if the longest side is 7 ft shorter than twice the shortest side.  The third side is 6 ft longer than the shortest side."?

Assign variables.  We will need to report all three lengths.  The length of the shortest side is mentioned twice, so:
    Let S = length of shortest side
         L = length of longest side
         T = length of third side

Now, translate:
  "longest side of the triangular section is 7ft. Shorter than twice the shortest side"    means
            L                                           =     -7              +    2  *                S

  "third side is 6ft. Longer than the shortest side"    means
       T         =   6     +                           S

This gives us three lengths:   S,   (S+6),    (2S-7)    for the three sides.
    The perimeter, P = S + (S+6) + (2S-7)  =  4S-1

Now, if we knew the perimeter, we could find S and then find the lengths of the other sides.
 
Without a picture or other information, the only thing we know is that the sum of the lengths of any two sides of a triangle must be more than the third side (or else there can't be a triangle).  So --
      (S) + (S+6) > 2S-7
      (S+6) + (2S-7) > S
      (2S-7) + (S) > (S+6)
 
The first inequality:
      (S) + (S+6) > 2S-7
      2S + 6 > 2S - 7
             6 > -7
The second inequality:
     (S+6) + (2S-7) > S
      3S -1 > S
       2S > 1
The third inequality:
     (2S-7) + (S) > (S+6)
     3S - 7 > S + 6
      2S > 13
       S > 13/2       (Ah!  There's something important)
     
  P= 4S-1, so P>(4)(13/2)-1  or P>25
With what we know (you may know more from the figure or problem statement, so use it):
   The "longest side" is 2S-7, so
     2S-7 > S+6
        S > 13
 
So, pick S=14, then T=20, L=21.   Then P=55.  
[Note: this continues to be true for larger values of S, so see the diagram for the information you need.]
       

Answer 2

Latifah,

Either I am missing something, or you have left out something. To solve any system with three variables, we must have three equations. The final hint about perimeter makes me think that that is what was intended for inclusion. I'll just assume that you have that piece and I'll show you the setup.

 

If L = the length of the longest side   and    S = the length of the shortest side,

Then we'll assume that the "third side" is the one of mid-length = M.

 

Given:  L = 2S-7

           M = S+6

           Perimeter = L+M+S

3 equations : 3 variables

 

Using substitution:  Perimeter = (2S-7) + (S-6) + S = 4S-13

                            Perimeter + 13 = 4S

                            (Perimeter + 13)/4 = S

 

So if you know the perimeter, you can find S.  Once you have S, plug it into the other two equations to find L and M.