TRIANGLE INEQUALITY THEOREM - ppt download - Free Printable

The image you've uploaded explains the Triangle Inequality Theorem, which is a fundamental concept in geometry. Let's break down the problem and solution step by step.

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🔷 What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states:

> *The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.*

This must be true for all three combinations of sides in a triangle.

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🔷 Given Triangle Sides: 5, 12, and 15

We are given a triangle with side lengths:
- Side A = 5
- Side B = 12
- Side C = 15

We need to check whether these side lengths can form a valid triangle using the Triangle Inequality Theorem.

We check all three possible sums:

#### ✔ Check 1: 5 + 12 > 15
→ 17 > 15 → True

#### ✔ Check 2: 5 + 15 > 12
→ 20 > 12 → True

#### ✔ Check 3: 12 + 15 > 5
→ 27 > 5 → True

Since all three conditions are satisfied, the triangle with side lengths 5, 12, and 15 can exist.

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🔷 Explanation of the Diagram

The image shows three versions of the same triangle (with sides 5, 12, 15) and checks each pair of sides:

1. 5 + 12 > 15 → 17 > 15 ✔
2. 5 + 15 > 12 → 20 > 12 ✔
3. 12 + 15 > 5 → 27 > 5 ✔

All are true, so this confirms that a triangle with these side lengths is possible.

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🔷 Conclusion

✔ Yes, a triangle with side lengths 5, 12, and 15 satisfies the Triangle Inequality Theorem, so it is a valid triangle.

This theorem helps us determine whether three given lengths can form a triangle. If even one of the inequalities fails, a triangle cannot be formed.

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📌 Summary

| Sum of Two Sides | Compare to Third Side | Result |
|------------------|------------------------|--------|
| 5 + 12 = 17 | > 15 | ✔ True |
| 5 + 15 = 20 | > 12 | ✔ True |
| 12 + 15 = 27 | > 5 | ✔ True |

✔ Valid Triangle

Let me know if you'd like to explore what happens if one inequality fails!