Question

The lengths of three sides of a triangle are m units, n units, and p units, respectively. Which inequality must be true?

1) m > n + p
2) n < m + p
3) p > m + n
4) p < m - n

Answer 1

The correct inequality for the lengths of the sides m, n, and p of a triangle is n < m + p, which is in accordance with the triangle inequality theorem stating that each side must be less than the sum of the other two sides.

Step-by-step explanation:

In the context of the triangle inequality theorem, which states that the length of any side of a triangle must be less than the sum of the other two sides, only one of the supplied inequalities is correct. Specifically, option 2) n < m + p must be true. This inequality along with two similar inequalities, m < n + p and p < m + n, ensures that a triangle can actually be formed with sides of lengths m, n, and p. If any one of these inequalities did not hold, you would not be able to form a triangle. The triangle inequality theorem is essential to understand the basic properties of geometrical figures and is fundamental in various applications within mathematics.