# The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. The lengths of two sides of a triangle are 15 ft and 27 ft. Find the possible lengths of the third side.

The third side must have a length greater than
? ft and less than
? ft.

## According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, in this case, the third side must have a length greater than the difference between the lengths of the other two sides, and less than the sum of the lengths of the other two sides.

Given that the lengths of the two sides are 15 ft and 27 ft, the possible lengths of the third side can be found as follows:

Third side > 27 ft - 15 ft = 12 ft,

and

Third side < 27 ft + 15 ft = 42 ft.

Therefore, the possible lengths of the third side are greater than 12 ft and less than 42 ft.

## To find the possible lengths of the third side of a triangle, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given that the lengths of two sides of the triangle are 15 ft and 27 ft, we can find the range of possible lengths for the third side.

To apply the triangle inequality theorem, we can perform the following calculations:

1. Determine the minimum possible length of the third side:
- Subtract the length of the shorter side from the length of the longer side.
- In this case, 27 ft - 15 ft = 12 ft.

2. Determine the maximum possible length of the third side:
- Add the lengths of the two known sides and subtract 1 from the sum.
- In this case, 15 ft + 27 ft - 1 ft = 41 ft.

Therefore, the third side must have a length greater than 12 ft and less than 41 ft.

## let the third side be x , where x > 0

then x+15 > 27 ---> x > 12
and
x+27> 15 ---> x > -12
so far we have x > 12
and
12+27 > x
x < 39

so :
12 < x < 39