Two side of a triangle have lengths 13 and 19. Write an inequality to represent the possible lengths for the third side, x.
In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This rule is known as the triangle inequality theorem. Given two sides of a triangle with lengths 13 and 19, you can use this theorem to write inequalities to represent the possible lengths for the third side, x.
The three inequalities you would write based on the possible pairings of sides are:
1. The sum of the lengths of the third side x and the side with length 13 must be greater than the length of the side with length 19.
x + 13 > 19
x > 6 (after subtracting 13 from both sides)
2. The sum of the lengths of the third side x and the side with length 19 must be greater than the length of the side with length 13.
x + 19 > 13
x > -6 (after subtracting 19 from both sides, but since the length of a side can't be negative, this inequality doesn't provide a useful bound)
3. The sum of the lengths of the side with length 13 and the side with length 19 must be greater than the length of the third side x.
13 + 19 > x
32 > x (after adding 13 and 19 together)
So, combining inequalities 1 and 3, we have:
6 < x < 32
The third side length x must therefore be greater than 6 but less than 32.