Determine the Third Side of a Triangle with Given Sides

Introduction: When solving geometric problems involving triangles, understanding the triangle inequality theorem is essential. This theorem provides a simple yet powerful way to determine if a triangle can exist given the lengths of its sides. In this content, we will explore how to find the possible lengths of the third side of a triangle when two sides are given, using a specific example.

The Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This principle ensures that the three sides can form a triangle and not a straight line or degenerate shape.

Given Sides: 7 cm and 9 cm

To determine the possible lengths of the third side of a triangle whose two sides are 7 cm and 9 cm, let's denote the sides as:

Side A (a) 7 cm Side B (b) 9 cm Side C (c) Third side (unknown)

According to the triangle inequality theorem, the following inequalities must hold:

a b > c a c > b b c > a

Applying the Inequalities

Substituting the values of a and b, we get:

7 9 > c 7 c > 9 9 c > 7

Now, let's solve each inequality step by step:

Inequality 1: 7 9 c

16 > c

Result: c 16

Inequality 2: 7 c 9

c 2

Result: c 2

Inequality 3: 9 c 7

c -2

Result: This condition is always satisfied for positive c.

Combining the Inequalities

Combining the results of the three inequalities, we get:

c 16 c 2

Therefore, the possible lengths for the third side (c) can be any value between 2 cm and 16 cm, not inclusive.

Real-World Application

Understanding these principles can be particularly useful in fields such as architecture, engineering, and design. For example, if you are designing a triangular structure, ensuring that the lengths of the sides meet the triangle inequality theorem is crucial to prevent the structure from collapsing or deforming.

Conclusion

The triangle inequality theorem is a fundamental concept in geometry that ensures the physical realizability of a triangle. By applying this theorem, we can determine the possible lengths of the third side when two sides of a triangle are known. In our specific example, the third side (c) must lie between 2 cm and 16 cm, not inclusive. This understanding is key to solving geometric and truss design problems.