Finding the Length of a triangle side

Question

one side of a triangle is 6 inches longer than the shortest side and 6 inches shorter than the longest side. If the perimeter of the triangle is 39 inches. What is the length of the shortest side of the triangle?

Answer 1

Another way to think about this is if you consider everything in relation to the medium side.

Let's call the length of the medium side, x.

The medium side is 6 inches longer than the shortest side, so the shortest side has a length of x-6.

The medium side is 6 inches shorter than the longest side, so the longest side has a length of x+6.

Since the perimeter is the sum of all sides, x + (x-6) + (x+6) = 39

Simplifying this further by grouping by like terms:

3x = 39

x = 13

The shortest side has a length of x-6, so the answer is 13-6, or 7 inches.

Answer 2

We could solve the system of 3 linear equations in 3 unknowns:

L + M + S = 39

M - S = 6

L - M = 6

OR,

In this case, we can notice that the length of the "medium" side is halfway between the short and long side, so the medium side must be the average of 3 values that add to 39. That is, the medium side must be 39/3 = 13 inches.

Add 6 to get the longest side, 19 inches, and subtract 6 to get the length of the shortest side, 7 inches.