Triangle Inequality and Side Relations

The triangle inequality is a fundamental rule about the lengths of the sides of any triangle. It tells us which three line segments can actually form a triangle.

Triangle Inequality Theorem:

In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Using sides \(a\), \(b\), and \(c\):

\( a + b > c \)
\( b + c > a \)
\( c + a > b \)

    A
   / \
  /   \
 B-----C
(a, b, c must satisfy all three inequalities)

This rule ensures that the three sides can actually meet to form a closed shape. If the sum of two sides is equal to or smaller than the third side, the sides will lie flat and cannot form a triangle.

For example:

• Segments 3 cm, 4 cm, 5 cm can form a triangle.
• Segments 2 cm, 3 cm, 5 cm cannot (because 2 + 3 = 5, not > 5).

Imagine walking along the sides of a triangle. To return to the starting point, the two shorter sides must be able to “reach” the end of the longest side. This is only possible if their combined length is greater than the longest side.

This is why the inequality uses the symbol “greater than” (\(>\)), not “greater than or equal to”.

A triangle also follows a simple but powerful rule about sides and angles:

The longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

Symbolically, in a triangle:

• If \( a > b \), then \( \angle A > \angle B \)
• If \( b > c \), then \( \angle B > \angle C \)

      A
     / \
    /   \
   B-----C
(opposite side reflects angle size)

To verify whether three given side lengths can form a triangle, check all three inequalities:

• \( a + b > c \)
• \( b + c > a \)
• \( c + a > b \)

If all are true, a triangle is possible. If any one fails, a triangle cannot be formed.

The triangle inequality appears naturally whenever you compare distances:

• Finding the shortest path between two places.
• Designing ramps, roofs and frame structures.
• Ensuring parts of a mechanical structure can actually connect.

The rule ensures that physical triangular shapes are possible and stable.