Using the Triangle Inequality Theorem to find possible third side lengths

📐 Understanding 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 theorem helps determine if a triangle can actually be formed with given side lengths and to find the possible range of values for the third side when two sides are known.

📜 History and Background

The Triangle Inequality Theorem has been known since ancient times. Euclid discussed it in his book "Elements" over 2000 years ago. It's a fundamental concept in Euclidean geometry, providing a basic rule for the construction of triangles.

🔑 Key Principles

• ➕ The Sum Principle: ➕ The sum of any two sides must be greater than the third side. If the sides are $a$, $b$, and $c$, then: $a + b > c$, $a + c > b$, and $b + c > a$.
• ➖ Finding the Range: ➖ If you know two sides, $a$ and $b$, the third side, $c$, must be between the difference and the sum of the other two sides: $|a - b| < c < a + b$.
• 🚫 Invalid Triangles: 🚫 If any of the three inequalities ($a + b > c$, $a + c > b$, $b + c > a$) are not met, a triangle cannot be formed.

✍️ Applying the Theorem: Examples

Example 1: Determining if a Triangle is Possible

Can a triangle have sides of length 3, 4, and 5?

• ✅ Check: $3 + 4 > 5$, $3 + 5 > 4$, and $4 + 5 > 3$. All inequalities hold.
• ✔️ Conclusion: Yes, a triangle can be formed.

Example 2: Finding the Range of the Third Side

Two sides of a triangle are 7 and 12. What is the possible range of values for the third side?

• ➕ Sum: $7 + 12 = 19$
• ➖ Difference: $|7 - 12| = 5$
• ✔️ Conclusion: The third side, $c$, must be between 5 and 19, or $5 < c < 19$.

Example 3: Another Range Example

If two sides of a triangle are 5 and 8, find the range of possible lengths for the third side.

• ➕ Sum: $5 + 8 = 13$
• ➖ Difference: $|8 - 5| = 3$
• ✔️ Conclusion: Therefore, the third side must be between 3 and 13, so $3 < x < 13$.

➗ Practice Quiz

Let's test your knowledge with these example problems:

1. Two sides of a triangle are 6 and 10. What is a possible length for the third side?
2. The sides of a triangle are 4, 5, and $x$. What range of values can $x$ have?
3. Can a triangle have sides of length 2, 5, and 9?
4. If two sides of a triangle are 11 and 15, what is the range of possible values for the third side?
5. The sides of a triangle are 7, $x$, and 12. What is the range of possible values for $x$?
6. Determine if a triangle can exist with side lengths 1, 2, and 3.
7. Two sides of a triangle measure 9 cm and 16 cm. What is the range of possible lengths for the third side?

💡 Tips and Tricks

• ✍️ Always remember the three inequalities must hold true for a triangle to exist.
• 🧮 When finding the range, calculate both the sum and the absolute difference.
• 🧐 Check your work to make sure the third side falls within the calculated range.

🏁 Conclusion

The Triangle Inequality Theorem is a simple but powerful tool for understanding the relationships between the sides of a triangle. By applying its principles, we can determine if a triangle is valid and find the possible range of lengths for its third side, given the other two sides. This theorem is foundational in geometry and has practical applications in various fields.