Question

Two sides of a triangle have lengths 5 and 18. Which inequalities represent the possible lengths for the third side x?

A) 13 < x < 18
B) 5 < x < 18
C) 13 < x < 5
D) 13 < x < 23

Answer 1

The length of the third side of a triangle with two sides measuring 5 and 18 must follow the Triangle Inequality Theorem, resulting in a length x that must satisfy the inequality 13 < x < 23.

Step-by-step explanation:

The third side of a triangle is bound by the Triangle Inequality Theorem, which dictates that the sum of the lengths of any two sides must be greater than the length of the third side. Given two sides of lengths 5 and 18, the possible range of the third side x must satisfy two inequalities: x + 5 > 18 and x + 18 > 5. After simplifying these, we get x > 13 and x > -13, respectively. However, since x must be a positive length, we ignore the second inequality. Additionally, by flipping the first inequality, we enhance it with the fact that the length of the third side must also be less than the sum of the other two: x < 23. Combining these facts, we end up with the inequality: 13 < x < 23.