# Triangle Inequality Theorem

Let's say there are three random spots \(A\), \(B\), and \(C\) on a floor and join the paths \(AB\), \(BC\), and \(AC\).

We then ask Saumya to start from \(A\) and reach \(C\), walking along these three paths.

She can first walk along \(\overline{AB}\) and then walk along \(\overline{BC}\) to reach \(C\) or she can directly walk along \(\overline{AC}\).

What do you think? What path will she prefer?

She will naturally prefer the direct path \(\overline{AC}\) because \(\overline{AB}+\overline{BC}>\overline{AC}\).

Certainly, most of us will follow that path only.

\(\overline{AC}\) because \(\overline{AB}+\overline{BC}>\overline{AC}\), gives us the crux of the Triangle Inequality Theorem, i.e, “the sum of any two sides of a triangle is greater than its third side."

In this lesson, we will explore triangle inequality theorem, triangle inequality theorem-proof, triangle inequality theorem problems and try our hands-on triangle inequality theorem calculator and interactive problems to have a better understanding of it.

Let's begin!

**Lesson Plan**

**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."

Let's do an activity to implement this theorem, and later we will solve some triangle inequality theorem problems.

Take a few small strips of different lengths, say, 2 cm, 3 cm, 4 cm, 5 cm, ...,10 cm.

Form a triangle taking any three strips.

Suppose you first choose two strips of length 6 cm and 12 cm.

What do you think about the length of the third strip?

Your third strip has to be of length more than 12 – 6 = 6 cm, and less than 12 + 6 = 18 cm.

Repeat this activity by choosing different combinations of the three strips.

Do you know what we observe when we do this?

We observe that the sum of the lengths of any two sides of a triangle is greater than the third side.

- The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
- The difference between the lengths of any two lengths is less than the length of the third side.

**What is the General Formula of Triangle Inequality Theorem?**

**Triangle Inequality Theorem Proof**

Consider a triangle \(ABC\) as shown below.

We will do construction in this triangle.

Let's produce the side \(BA\) to a point \(D\) such that \(AC=AD\).

Applying Angle Sum Property in \(\triangle BDC\), we get:

\(\begin{align}\angle BDC+\angle CBD+\angle BCD&=180^{\circ}\\\angle BDC+\angle CBD+90^{\circ}&=180^{\circ}\\\angle BDC+\angle CBD&=90^{\circ}\end{align}\)

So, \(\angle BCD > \angle BDC\).

Since the side opposite to the greater angle is longer, we have \(BD>BC\).

This implies:

\(\begin{align}BD&>BC\\AB+AD&>BC\\AB+AC&>BC\end{align}\)

Hence proved.

A triangle inequality theorem calculator is designed as well to discover the multiple possibilities of the triangle formation.

- Triangle Inequality Theorem: The sum of lengths of any two sides of a triangle is greater than the length of the third side.
- The converse of Triangle Inequality Theorem: If there are 3 real numbers such that the sum of two is greater than the third number, then there exists a triangle with side lengths equal to the real numbers.

**Solved Examples**

Example 1 |

Rita is happy doing some craftwork.

She has three sticks of lengths 4 cm, 8 cm, and 2 cm.

Can she form a triangle using these sticks?

**Solution**

The triangle formed by these sticks must satisfy the triangle inequality theorem.

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

\(\begin{align}4+8&>2 \implies 12>2 \implies TRUE\\2+8&>4 \implies 10>4 \implies TRUE\\4+2&>8 \implies 6>8 \implies FALSE\end{align}\)

So, the lengths of the sticks do not satisfy the triangle inequality theorem.

\(\therefore\) Rita can't form a triangle using the sticks of the given lengths. |

Example 2 |

Soham wants to decorate his triangular flag with a ribbon.

The two sides of the flag are 7 metres and 2 metres.

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 \(7\;\text{metres}+2\;\text{metres}=9\;\text{metres}\).

Also, the third side cannot be less than the sum of the other two sides.

So, the third side is more than \(7\;\text{metres}-2\;\text{metres}=5\;\text{metres}\).

\(\therefore\) The length of the ribbon is between 5 metres and 9 metres. |

Keep in mind two useful properties while drawing a triangle when the lengths of the three sides are known.

**Interactive Questions**

**Here are a few activities for you to practice. Select/type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about Triangle Inequality Theorem with the simulations and practice questions. Now you will be able to easily solve problems on triangle inequality theorem proof, triangle inequality theorem problems, and triangle inequality theorem calculator.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

## 1. How to find the third side of a triangle?

The third side of a triangle is always less than the sum of the other two sides and always greater than the difference between the other two sides.

## 2. How do you solve the triangle inequality theorem?

Let's consider a triangle \(ABC\) and produce the sides \(BA\) to a point \(D\) such that \(AC=AD\).

Using the angle sum property of a triangle, prove that \(\angle BCD>\angle BDC\).

Since the side opposite to a greater angle is longer, you'll arrive at \(BA+AC>BC\)

## 3. Why is the triangle inequality theorem true?

The Triangle inequality theorem is true because of the shortest distance property that states that the shortest distance between a point \(A\) and a line \(L\) is the perpendicular line to \(L\) drawn from the point \(A\).

Let's consider a triangle \(RST\).

We'll draw a perpendicular line \(RV \perp ST\) as shown below.

Also, \(RV \perp SV\)

So, the shortest distance from \(S\) to \(RV\) is \(SV\).

This implies that \(SV<RS\).