## What is Triangle Inequality Theorem ?

**If I want to give you a perfect definition for Triangle Inequality then I can say **: –

*The sum of the lengths of any two sides of a triangle is always greater than the length of the third side of that triangle.*

**It follows from the fact that a straight line is the shortest path between two points. The inequality is strict if the triangle is non-degenerate (meaning it has a non-zero area). **

**So in other words we can say that :** **It is not possible to construct a triangle from three line segments if any of them is longer than the sum of the other tw***o.* This is known as The Converse of the Triangle Inequality theorem .

*So suppose we have three sides lengths as 6 m, 4 m and 3 m then can we draw a triangle with this side ? The answer will be YES we can.*

**Suppose side a = 3 m**

**length of side b = 4 m**

**Length of side c = 6 m **

**if side a + side b > side c then only we can draw the triangle or **

**side b + side c > side a or**

**side a + side c > side b **

**So from the above example we can find that 4 m + 3 m > 6 m**

**But look if we try to take 4 m + 6 m \(\geq \) 3 m .**

*This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a metric spaces and analysis.*

## Problem using Triangle Inequality :

*In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? *

- 43
- 44
- 45
- 46

**Key Concepts**

**Triangle Inequality**

**Inequality**

**Geometry**

## Check the Answer

But try the problem first…

**The answer is 43 m**

**AMC – 2006 – 10 B – Problem 10**

**Secrets in Inequalities.**

## Try with Hints

First Hint……..

**This can be a very good example to show Triangle Inequality**

**Let ‘ x ‘ be the length of the first side of the given triangle. So the length of the second side will be 3 x and that of the third side be 15 . Now apply triangle inequality and try to find the possible values of the sides. **

If you really need another hint try this out …………….

**If we apply Triangle Inequality here then the expression will be like **

**\(3 x < x + 15 \)**

**\( 2 x < 15 \)**

**\( x < \frac {15}{2}\)**

**x < 7.5 **

**Now do the rest of the problem ………..**

Final Step

*I am sure you have already got the answer but let me show the rest of the steps for this sum*

**If x < 7.5 then **

**The largest integer satisfying this inequality is 7.**

**So the largest perimeter is 7 + 3.7 +15 = 7 + 21 + 15 = 43.**