Describe the possible lengths of the third side of the triangle given that the lengths of the other two sides are 12 feet and 18 feet.

Answer 1

For any triangle, the length of any two sides must be greater than the third side. Likewise, the difference between any two sides must be less than the third side.

For the given triangle, let us call it ABC, with side lengths:

A=12ft

B=18ft

and

C= unknown

Adding sides A and B must give us a sum that is greater than side C:

A + B =12ft +18ft = 30ft > C

Therefore; side C must be less than 30ft long.

At the same time, the difference between sides A and B must be less than side C:

B - A = 18ft - 12ft = 6ft < C

Therefore; side C must be greater than 6ft long.

Combining the two inequalities will describe all possible values of C such that a triangle can be formed with side lengths 12ft (side A) and 18ft (side B):

6ft < C < 30ft

In other words, C must be greater than 6ft and less than 30ft.

Answer 2

The sum of the lengths of any two sides of a triangle must be greater then the 3rd side (otherwise a triangle cannot be formed).

Let x = the length of the 3rd side, This allows us to create 3 different inequalities.

I. 12+18 > x

II. x+12 > 18

III. x + 18 > 12

Solving I. gives x < 30.

Solving II. gives x > 6.

Solving III. gives x > -6.

What are the numbers that satisfy all three inequalities?

Well x has to be greater than 6 (which means x also greater than -6) and x must be less than 30.

So:

6 < x < 30.

Any number in that interval can be the length of the 3rd side.