# Triangle Inequality Theorem

The sum any two sides in a triangle is always greater than the third side of triangle.

If one side is longer than the sum of other two sides as shown below, then it is not a triangle.

See the example below:

In the above case, *AD* + *BC* = 18 + 9 = 27 units, which is less than side *AB* of length 30 units. Here since one side *AB* is longer, so the other two sides, *AD* and *BC* don’t meet.

In the above triangle, following holds true:

*b* + *c* > *a*

*a* + *b* > *c*

*a* + *c* > *b*

** Example:** What is the possible range for

*x*?

According to Triangle Inequality Theorem, the sum any two sides in a triangle is always greater than the third side of triangle.

So, the three inequalities for the above triangle are:

- (7 –
*x*) + 7 > 3*x*=> 14 –*x*> 3*x*=> 14 > 4*x*=>*x*=>*x*< - 3
*x*+ (7 –*x*) > 7 => 2*x*+ 7 > 7 => 2*x*> 0 =>*x*> 0 - 3
*x*+ 7 > 7 –*x =>*4*x*> 0 =>*x*> 0

Combining the three inequalities, we get the possible range as 0 < *x *<

#### Check Point

- Can the numbers 4, 5, 10 be the sides of a triangle?
- If 13 and 8 are the two sides of a triangle, then what is the possible range for the third side?
- Do the side lengths, 8, 11, and 16 form a triangle?
- If 3 + 2
*x*, 7 –*x*and 2*x*are the sides of a triangle, then what is the possible range for*x*? - If 10 and 15 are the two sides of a triangle, then what is the possible range for the third side?

##### Answer key

- No
- 5 <
*x*< 21 - Yes
- <
*x*< 10 - 5 <
*x*< 25

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