Concept:The triangle inequality theorem states that the length of any side of a triangle is less than the sum of the lengths of the other two sides.
Explanation:For any triangle
ABC, the three side lengths satisfy:
AB+BC>AC,
BC+AC>AB, and
AC+AB>BC.
Each given inequality can be obtained by rearranging one of these fundamental statements.
Consider statement 1:
AC−AB<BC. Adding
AB to both sides gives
AC<AB+BC, which is exactly the triangle inequality for side
AC. This is always true.
Statement 2:
BC−AC<AB is equivalent to
BC<AC+AB after adding
AC to both sides. This is true for any triangle.
Statement 3:
AB−BC<AC becomes
AB<BC+AC after adding
BC, which holds in every triangle.
Therefore, all three inequalities are simply different forms of the triangle inequality and are always correct.
Answer:Option D: 1, 2 and 3