KutaSoftware: Geometry- Inequalities In One Triangle Part 1
KutaSoftware: Geometry- Inequalities In One Triangle Part 1

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Triangle Inequality Theorem

The triangle inequality theorem states that, in a triangle, the sum of lengths of any two sides is greater than the length of the third side. Suppose a, b and c are the lengths of the sides of a triangle, then, the sum of lengths of a and b is greater than the length c. Similarly, b + c > a, and a+ c > b. If, in any case, the given side lengths are not able to satisfy these conditions, it means it is not possible to draw a triangle with those measurements.

 1 What Is Triangle Inequality Theorem? 2 Triangle Inequality Theorem Formula 3 FAQs on Triangle Inequality Theorem

## What is Triangle Inequality Theorem?

The triangle inequality theorem states, “The sum of any two sides of a triangle is greater than its third side.” This theorem helps us to identify whether it is possible to draw a triangle with the given measurements or not without actually doing the construction. Let’s understand this with the help of an example. Triangle ABC has side lengths of 6 units, 8 units, and 12 units.

Here, AB = 6 units, BC = 8 units and CA = 12 units.

• The sum of sides AB and BC is 6 + 8 = 14 units and 14 is greater than side CA (12 units).
• The sum of sides BC and CA is 8 + 12 = 20 units and 20 is greater than side AB (6 units).
• The sum of sides CA and AB that is 12 + 6 = 18 units and 18 is greater than side BC (8 units).

Thus, lengths of all the sides satisfy the triangle inequality theorem. In this, not only one, but all 3 cases should satisfy the triangle inequality theorem.

Let’s take another example. Let’s check whether a triangle with sides lengths 5 units, 3 units, and 10 units satisfy the triangle inequality theorem or not.

Here,

• 5 + 3 = 8 which is less than 10
• 3 + 10 = 13 which is greater than 5
• 10 + 5 = 15 which is greater than 3

We can see that two cases are satisfying the triangle inequality theorem but one case is not satisfying. This means the triangle with these side lengths does not exist. All three sides should satisfy the triangle inequality theorem.

## Triangle Inequality Theorem Formula

Before understanding the formula, first, we need to understand the proof of the triangle inequality theorem. Consider a triangle ABC as shown below.

Let us extend side AB to the point D such that AC = AD and △BDC will form a right angled triangle at angle C.

Applying angle sum property in △BDC, we get,

∠BDC + ∠CBD + ∠BCD = 180°

∠BDC + ∠CBD + 90° = 180°

∠BDC + ∠CBD = 90°

This implies, ∠BCD > ∠BDC.

As the side opposite to the greater angle is longer, we have BD > BC.

This implies:

BD > BC

AB + AC > BC

Hence proved.

Similarly, we can prove that AC + BC > AB and AB + BC > AC.

So, the triangle inequality theorem formula is,

• AB + AC > BC
• AC + BC > AB
• AB + BC > AC

## Triangle Inequality Theorem Examples

1. Example 1: Suzie has three sticks of lengths 4 units, 8 units, and 2 units. Using the triangle inequality theorem, find out whether Suzie can form a triangle using these sticks or not?

Solution: The triangle formed by the given sticks must satisfy the triangle inequality theorem.

Let’s check if the sum of the two sides is greater than the third side.

• 4 + 8 > 2 ⟹ 12 > 2 ⟹ TRUE
• 2 + 8 > 4 ⟹ 10 > 4 ⟹ TRUE
• 4 + 2 > 8 ⟹ 6 > 8 ⟹ FALSE

So, the lengths of the sticks do not satisfy the triangle inequality theorem. Thus, Suzie can’t form a triangle using the sticks of the given lengths.

2. Example 2: Ron wants to decorate his triangular flag with a ribbon. The two sides of the flag are 8 units and 2 units. Using the triangle inequality theorem, find out how much ribbon is required for the third side?

Solution: By using the triangle inequality theorem, we can say that the length of the third side must be less than the sum of the other two sides.

So, the third side is less than 8 units + 2 units = 10 units.

Also, the third side cannot be less than the difference between the other two sides.

So, the third side is greater than 8 units – 2 units= 6 units.

Thus, the length of the ribbon can be 7, 8, or 9 units.

## FAQs on Triangle Inequality Theorem

### What is Meant by Triangle Inequality Theorem?

The triangle inequality theorem states that the sum of any two sides of a triangle is greater than the third side, and if the sum of any two sides of a triangle is not greater than the third side it means the triangle does not exist.

### Why is Triangle Inequality Theorem Important?

The triangle inequality theorem is important to find out whether the triangle with the given three measurements exists or not. As the theorem states that sum of any two sides should be greater than the measurement of the third side. For example, the triangle with sides 3 units, 4 units, and 9 units does not exist as it does not satisfy the triangle inequality theorem.

• 3 + 4 > 9 ⟹ 7 > 9 ⟹ False
• 4 + 9 > 3 ⟹ 13 > 3 ⟹ True
• 9 + 3 > 4 ⟹ 12 > 4 ⟹ True

Thus, by using the triangle inequality theorem we can say that the given measurements do not form a triangle.

### How is the Triangle Inequality Theorem used in Real Life?

One example of the application of the triangle inequality theorem in real life is by Engineers. Civil engineers use the triangle inequality theorem in real life. Since their work is related to surveying, transportation, and urban planning. With the help of the triangle inequality theorem, they calculate the unknown lengths and estimate the remaining dimension.

### Does the Triangle Inequality Theorem Apply to all Triangles?

Yes, the triangle inequality theorem applies to all triangles. Any side of a triangle must be shorter than the sum of the other two sides. If a side is greater than or equal to the sum of the other two sides, then it is not a triangle.

### What is an example of the Triangle Inequality Theorem?

Following is the example of the triangle inequality theorem. Triangle with side lengths 5, 7, and 9 units exists, as lengths of all sides satisfy the theorem.

• 5 + 7 > 9 ⟹ 12 > 9 ⟹ TRUE
• 7 + 9 > 5 ⟹ 16 > 5 ⟹ TRUE
• 9 + 5 > 7 ⟹ 14 > 7 ⟹ TRUE

### How do you write a Triangle Inequality Theorem?

Suppose ABC is a triangle. We will write the triangle inequality theorem in this form:

• AB + AC > BC
• AC + BC > AB
• AB + BC > AC

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