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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 you have a triangle with side lengths a, b, and c, then:

a + b > c

a + c > b

b + c > a

This is represented best in the diagram given below in the link: Check out the image.

These inequalities must hold true for any triangle. If any of these conditions are not met, then it’s impossible to form a triangle with those side lengths.

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For example, if you have a triangle with side lengths of 3, 4, and 7, you would check:

3 + 4 > 7 (True)

3 + 7 > 4 (True)

4 + 7 > 3 (True)

So, a triangle can be formed with these side lengths.

If you have side lengths of 1, 2, and 6, then

1 + 2 > 6 (False)

1 + 6 > 2 (True)

2 + 6 > 1 (True)

So, a triangle cannot be formed with these side lengths.

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## Example Questions

1. Could a triangle have side lengths of 6 m, 7 m, and 5 m?

Solution

6 + 7 > 5 (True)

6 + 5 > 7 (True)

7 + 5 > 6 (True)

So, a triangle can be formed with these side lengths.

2. If the two sides of a triangle are 3 and 8. Find all the possible lengths of the third side.

Solution

To find the possible values of the third side of the triangle we can use the formula:

A difference of two sides< Unknown side < Sum of the two sides

8 – 3 < x < 8 + 3

5 < x < 11

There could be any value for the third side between 5 and 11

3. If 4 inches, 8 inches and 2 inches are the measures of three lines segment. Can it be used to draw a triangle?

Solution

4 + 8 > 2 (True)

8 + 2 > 4 (True)

4 + 2 > 8 (False)

So, a triangle cannot be formed with these side lengths.

## Triangle Inequality Theorem FAQS

#### What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

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

The theorem is fundamental in geometry and helps in determining whether a given set of side lengths can form a valid triangle.

#### Can any three lengths form a triangle?

No, not every combination of three lengths can form a triangle. The Triangle Inequality Theorem provides a condition for the validity of a triangle based on its side lengths.

#### What happens if the sum of the lengths of two sides equals the length of the third side?

If the sum of the lengths of two sides equals the length of the third side, the triangle formed is said to be a degenerate triangle, which appears as a straight line.

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

No, the Triangle Inequality Theorem specifically applies to triangles. It does not extend to other polygons.

#### How does the Triangle Inequality Theorem relate to the perimeter of a triangle?

The Triangle Inequality Theorem ensures that the sum of the lengths of any two sides of a triangle is greater than the third side. This condition ensures that the triangle can be closed, forming a polygon with finite perimeter.

#### Does the Triangle Inequality Theorem apply to both the interior and exterior angles of a triangle?

No, the Triangle Inequality Theorem deals only with the lengths of the sides of a triangle, not with its angles.

#### What are some real-life applications of the Triangle Inequality Theorem?

Real-life applications include designing bridges, determining whether a set of three given measurements (such as the sides of a box) can form a closed figure, and in navigation and surveying.

#### Does the Triangle Inequality Theorem hold true for all types of triangles?

Yes, the Triangle Inequality Theorem applies to all types of triangles, including equilateral, isosceles, and scalene triangles.

#### How can the Triangle Inequality Theorem be used to classify triangles?

The Triangle Inequality Theorem can be used to classify triangles based on the relationships between their side lengths, helping to identify whether a triangle is acute, obtuse, or right-angled.

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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