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# 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.

• Triangle Inequality Theorem
• Formula
• Proof & Derivation
• Applications
• FAQs

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## 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.

In other words, if we have a triangle with sides a, b, and c, then:

a + b > c

b + c > a

a + c > b

This theorem applies to all triangles, whether they are acute, obtuse, or right-angled. It is a fundamental property of triangles and is essential in geometry, as it helps us determine whether a set of given side lengths can form the sides of a triangle. For example, if we are given the side lengths 5, 7, and 12, we can use the Triangle Inequality Theorem to determine whether they can form the sides of a triangle:

5 + 7 > 12 (False)
7 + 12 > 5 (True)
5 + 12 > 7 (True)

Since all three inequalities are true, we can conclude that the side lengths 5, 7, and 12 can form the sides of a triangle.

## Formulae of Triangle Inequality Theorem

The formula for the Triangle Inequality Theorem is:

a + b > c

b + c > a

a + c > b

where a, b, and c are the lengths of the sides of a triangle.

The inequalities state that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If any of these inequalities are not satisfied, then the given side lengths cannot form the sides of a triangle.

For example, if we have a triangle with sides a = 3, b = 4, and c = 9, we can check if they satisfy the Triangle Inequality Theorem:

a + b > c

3 + 4 > 9 (False)

b + c > a

4 + 9 > 3 (True)

a + c > b

3 + 9 > 4 (True)

Since the first inequality is false, the given side lengths do not form the sides of a triangle.

## Proof & Derivation of Triangle Inequality Theorem

The proof of the Triangle Inequality Theorem is a fundamental result in geometry and is based on the Euclidean Distance Formula, which is used to calculate the distance between two points in a two-dimensional space.

Let’s consider a triangle with sides a, b, and c, and let’s assume that a is the longest side, i.e., a > b and a > c. We can construct a right-angled triangle with the two sides of length b and c, and the hypotenuse of length a, as shown below:b

By the Pythagorean Theorem, we know that:

a^2 = b^2 + c^2

We can rearrange this equation as:

a^2 – b^2 = c^2

We also know that a > b and a > c, so we can write:

a – b > 0 and a – c > 0

Multiplying these two inequalities, we get:

(a – b)(a – c) > 0

Expanding this expression, we get:

a^2 – (b + c)a + bc > 0

Since a^2 = b^2 + c^2, we can substitute this expression to get:

b^2 + c^2 – (b + c)a + bc > 0

Rearranging terms, we get:

b^2 – 2bc + c^2 + (b – a)(c – a) > 0

This expression can be simplified further as:

(b – c)^2 + (b – a)(c – a) > 0

Since the square of any real number is non-negative, the first term on the left-hand side is always non-negative. Therefore, the second term must also be positive, i.e.,

(b – a)(c – a) > 0

This inequality can be written as:

bc – (b + c)a + a^2 > 0

or

a + b > c

a + c > b

b + c > a

which is the Triangle Inequality Theorem.

Therefore, we have proved that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

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## Applications of the Triangle Inequality Theorem

The Triangle Inequality Theorem has several applications in various fields, including mathematics, physics, and computer science. Some of the applications are:

1. Determining the feasibility of a triangle: The Triangle Inequality Theorem is used to determine whether a given set of side lengths can form the sides of a triangle. If the theorem is not satisfied, then it is not possible to construct a triangle with the given side lengths.

2. Optimization Problems: The theorem is used to solve optimization problems where we need to minimize or maximize a certain quantity subject to the constraint that it should be possible to construct a triangle with given side lengths.

3. Error Correction: The Triangle Inequality Theorem is used in error-correcting codes in computer science, where the theorem is used to detect and correct errors in data transmission. The theorem ensures that the received data satisfies the conditions required to form a triangle, and if not, the data is considered to be corrupted and discarded.

4. Calculating Distances: The theorem is used to calculate distances between points in a two-dimensional or three-dimensional space. The distance between two points can be calculated as the length of the shortest path between them, which is a straight line. The theorem ensures that the sum of the lengths of any two sides of a triangle is greater than the length of the third side, which ensures that the path is the shortest possible.

5. Analyzing Data: The theorem is used in data analysis and statistics to analyze the relationship between variables. For example, in a survey, the theorem can be used to determine whether the responses to two questions are related to each other or not.

Overall, the Triangle Inequality Theorem is a fundamental result in geometry with several practical applications in various fields.

## Triangle Inequality Theorem FAQS

##### What is 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.

##### Why is the Triangle Inequality Theorem important?

The theorem is important because it helps us determine whether a given set of side lengths can form the sides of a triangle. It is also used in optimization problems, error correction, calculating distances, and analyzing data.

##### What happens if the Triangle Inequality Theorem is not satisfied?

If the theorem is not satisfied, then it is not possible to construct a triangle with the given side lengths.

##### How do you use the Triangle Inequality Theorem to calculate distances?

The theorem ensures that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. This property is used to calculate distances between points in a two-dimensional or three-dimensional space.

##### Can the Triangle Inequality Theorem be applied to any polygon?

No, the Triangle Inequality Theorem only applies to triangles. Other polygons have their own unique properties and theorems. Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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