Triangle Inequality Theorem: Determining the Length of the Third Side

When faced with the task of determining the length of the third side of a triangle when two sides are known—specifically, a 7 cm side and a 10 cm side, the Triangle Inequality Theorem provides a clear and concise answer. This theorem, a fundamental principle in geometry, ensures that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. Let's explore how this theorem applies to find the possible range for the third side.

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem is a critical concept in Euclidean geometry. It states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem not only ensures the existence of a triangle but also sets boundaries for the possible lengths of its sides.

Applying the Triangle Inequality Theorem

Given two sides of a triangle measuring 7 cm and 10 cm, we can apply the Triangle Inequality Theorem to determine the range of possible lengths for the third side. Specifically, the theorem dictates the following:

The third side, denoted as c, must be greater than the difference between the two given sides: c 10 cm - 7 cm 3 cm. The third side must also be less than the sum of the two given sides: c 10 cm 7 cm 17 cm.

Therefore, the possible lengths of the third side are any value between 3 cm and 17 cm, inclusive. For example, the third side could be 5 cm, 12 cm, or 16 cm, all of which satisfy the triangle inequality theorem.

Why These Bounds?

The bounds of 3 cm and 17 cm are derived from the following conditions:

c 3 cm: If the third side were 3 cm or less, it would not be possible for the two 7 cm and 10 cm sides to meet and form a triangle, as they could not reach each other. c 17 cm: If the third side were 17 cm or more, the two known sides (7 cm and 10 cm) would not be able to reach the end of the third side, thus violating the triangle inequality.

These conditions ensure that the sides can form a triangle and that the figure remains a valid polygon with three sides.

Connecting the Theorem to Practical Examples

Consider a more complex problem where one side is 5 cm and another is 9 cm. Applying the triangle inequality theorem again:

The third side, denoted as c, must be greater than the difference between the two given sides: c 9 cm - 5 cm 4 cm. It must also be less than the sum of the two given sides: c 9 cm 5 cm 14 cm.

So, the length of the third side must be greater than 4 cm and less than 14 cm. This range ensures that the sides 5 cm, 9 cm, and the third side can indeed form a triangle.

Summary

In conclusion, the length of the third side of a triangle with two given sides of 7 cm and 10 cm is bounded between 3 cm and 17 cm. Similarly, for sides of 5 cm and 9 cm, the third side must be between 4 cm and 14 cm. These examples highlight how the triangle inequality theorem is a powerful tool in geometry for determining the possible lengths of the sides of a triangle.