Determining Triangle Side Lengths Using The Triangle Inequality Theorem

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One of the fundamental concepts in geometry is the Triangle Inequality Theorem. This theorem dictates the relationship between the sides of any triangle, stating that the sum of the lengths of any two sides must be greater than the length of the third side. This principle is crucial for determining the validity of a triangle given three side lengths. In this article, we'll explore the Triangle Inequality Theorem in detail and apply it to a specific problem: Hang knows one side of a triangle is 13 cm. Which set of two sides is possible for the lengths of the other two sides of this triangle? We'll analyze the given options and determine which set of side lengths satisfies the theorem.

Exploring the Triangle Inequality Theorem

The Triangle Inequality Theorem is a cornerstone of Euclidean geometry. It essentially states that for any triangle with side lengths a, b, and c, the following three inequalities must hold true:

  1. a + b > c
  2. a + c > b
  3. b + c > a

In simpler terms, this means that if you take any two sides of a triangle and add their lengths, the result must be greater than the length of the remaining side. If this condition is not met for all three combinations of sides, then a triangle cannot be formed with those side lengths. To truly grasp the essence of this theorem, let's delve into the reasons why it holds true and how it impacts our understanding of triangles.

Imagine trying to construct a triangle with sides of length 3 cm, 4 cm, and 8 cm. If you lay down the 8 cm side as the base, you'll find that the 3 cm and 4 cm sides cannot possibly meet to form a closed triangle. This is because their combined length (7 cm) is less than the length of the base (8 cm). The theorem ensures that the two shorter sides are always long enough to "reach" each other and close the triangle. The theorem's validity is rooted in the fundamental properties of straight lines and the shortest distance between two points. A straight line represents the shortest path, and in a triangle, any two sides combined must offer a path longer than the remaining side, which forms a direct line. This principle extends to all types of triangles, whether they are acute, obtuse, or right-angled. Understanding the theorem is vital for various geometric applications, including determining triangle existence, solving for unknown side lengths, and proving geometric relationships.

Applying the Theorem to Hang's Problem

Now, let's apply the Triangle Inequality Theorem to the problem presented. Hang knows that one side of a triangle is 13 cm, and we need to determine which of the given sets of side lengths is possible for the other two sides. We'll analyze each option by checking if the three inequalities of the theorem hold true.

Option A: 5 cm and 8 cm

Let's consider the side lengths 5 cm, 8 cm, and 13 cm. We need to check if the following inequalities are satisfied:

  1. 5 + 8 > 13 => 13 > 13 (False)
  2. 5 + 13 > 8 => 18 > 8 (True)
  3. 8 + 13 > 5 => 21 > 5 (True)

Since the first inequality is false (13 is not greater than 13), this set of side lengths cannot form a triangle.

Option B: 6 cm and 7 cm

Now, let's examine the side lengths 6 cm, 7 cm, and 13 cm:

  1. 6 + 7 > 13 => 13 > 13 (False)
  2. 6 + 13 > 7 => 19 > 7 (True)
  3. 7 + 13 > 6 => 20 > 6 (True)

Again, the first inequality is false, so this set of side lengths cannot form a triangle.

Option C: 7 cm and 2 cm

Let's analyze the side lengths 7 cm, 2 cm, and 13 cm:

  1. 7 + 2 > 13 => 9 > 13 (False)
  2. 7 + 13 > 2 => 20 > 2 (True)
  3. 2 + 13 > 7 => 15 > 7 (True)

As before, the first inequality is false, indicating that this set of side lengths cannot form a triangle.

Option D: 8 cm and 9 cm

Finally, let's consider the side lengths 8 cm, 9 cm, and 13 cm:

  1. 8 + 9 > 13 => 17 > 13 (True)
  2. 8 + 13 > 9 => 21 > 9 (True)
  3. 9 + 13 > 8 => 22 > 8 (True)

In this case, all three inequalities are true. Therefore, the side lengths 8 cm, 9 cm, and 13 cm can form a triangle.

Conclusion: Finding the Possible Side Lengths

Based on our analysis using the Triangle Inequality Theorem, we can conclude that the only set of side lengths that is possible for the other two sides of the triangle is Option D: 8 cm and 9 cm. The other options failed to satisfy the crucial requirement that the sum of any two sides must be greater than the third side. This problem highlights the importance of the Triangle Inequality Theorem in determining the validity of triangles and understanding the relationships between their sides. By mastering this theorem, you gain a fundamental tool for solving geometric problems and exploring the fascinating world of triangles.

Geometry, a branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids, often presents us with intriguing puzzles. One such puzzle involves triangles and the lengths of their sides. The Triangle Inequality Theorem serves as a cornerstone principle in resolving these puzzles. This theorem dictates a fundamental rule: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This article explores the application of this theorem to a specific problem: Hang knows one side of a triangle is 13 cm. Which set of two sides is possible for the lengths of the other two sides of this triangle? We will dissect the problem, understand the theorem, and systematically evaluate the potential solutions.

Understanding the Core of the Triangle Inequality Theorem

At its heart, the Triangle Inequality Theorem provides a simple yet powerful criterion for determining whether a triangle can exist given three side lengths. It asserts that if we denote the side lengths of a triangle as a, b, and c, the following three inequalities must simultaneously hold true:

  1. a + b > c
  2. a + c > b
  3. b + c > a

These inequalities are not merely abstract mathematical statements; they embody the very essence of triangular geometry. Imagine attempting to construct a triangle with sides measuring 2 cm, 3 cm, and 10 cm. If you were to lay the 10 cm side as the base, you would quickly realize that the 2 cm and 3 cm sides are too short to meet and form a closed figure. This is because their combined length (5 cm) is less than the length of the base (10 cm). This simple illustration encapsulates the essence of the theorem: the two shorter sides must possess sufficient combined length to