In this section, we are going to see "The Triangle Inequality Theorem" which can be used to know whether a triangle actually exists.

**Triangle Inequality Theorem : **

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

**Example : **

A triangle with the side lengths 5 cm, 6 cm and 4 cm actually exists.

Because, sum of the lengths of any two sides is greater than the third side.

That is, 5 cm + 6 cm > 4 cm.

6 cm + 4 cm > 5 cm

5 cm + 4 cm > 6 cm

The diagram given below illustrates the theorem :

**Problem 1 : **

State if the three numbers given below can be the measures of the sides of a triangle.

8, 12 and 9

**Solution :**

According to the theorem explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply the theorem for the given numbers.

8 + 12 > 9

12 + 9 > 8

8 + 9 > 12

Because the given numbers meet the condition said in the theorem, the numbers 8, 12 and 9 can be the measures of the sides of a triangle.

**Problem 2 : **

State if the three numbers given below can be the measures of the sides of a triangle.

10, 7 and 13

**Solution :**

According to the theorem explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply the theorem for the given numbers.

10 + 7 > 13

7 + 13 > 10

10 + 13 > 7

Because the given numbers meet the condition said in the theorem, the numbers 10, 7 and 13 can be the measures of the sides of a triangle.

**Problem 3 : **

State if the three numbers given below can be the measures of the sides of a triangle.

6, 12 and 3

**Solution :**

According to the theorem explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply the theorem for the given numbers.

6 + 12 > 3

12 + 3 > 6

6 + 3 < 12 (Does not satisfy the theorem)

Because the given numbers do not meet the condition said in the theorem, the numbers 6, 12 and 3 can not be the measures of the sides of a triangle.

**Problem 4 : **

Two sides of a triangle have the measures 6 and 7. Find the range of possible measures for the third side.

**Solution :**

Let "x" be the length of the third side of the triangle.

Sum of the lengths of the given two sides :

6 + 7 = 13

Because the sum of the lengths of the two sides 6 and 7 is 13, the maximum length of the third side must be less than 13.

That is

x < 13 -----(1)

Let us find the minimum value of "x".

According to the theorem, we must have

x + 6 > 7

x + 7 > 6

To satisfy both the inequalities above, the value of "x" must be greater than 1.

That is

x > 1 (or) 1 < x -----(2)

From (1) and (2), the range of "x" is

1 < x < 13

**Problem 5 : **

Find the range of possible measures of x in the following given sides of a triangle :

10, 7, x

**Solution :**

Sum of the lengths of the given two sides :

10 + 7 = 13

Because the sum of the lengths of the two sides 10 and 7 is 17, the value of "x" must be less than 17.

That is

x < 17 -----(2)

Let us find the minimum value of "x".

According to the theorem, we must have

x + 10 > 7

x + 7 > 10

To satisfy both the inequalities above, the value of "x" must be greater than 3.

That is

x > 3 (or) 3 < x -----(2)

From (1) and (2), the range of "x" is

3 < x < 17

**Shortcut : **

For better understanding, problem 4 and 5 have been explained in detail.

But there is a shortcut to find the range of possible measures for the third side.

**Problem 4 : **

Lengths of the given two sides are 6 and 7.

Difference of the lengths = 7 - 6 = 1

Sum of the lengths = 7 + 6 = 13

Hence, the range of possible measures for the third side is

1 < x < 13

**Problem 5 : **

Lengths of the given two sides are 10 and 7.

Difference of the lengths = 10 - 7 = 3

Sum of the lengths = 10 + 7 = 17

Hence, the range of possible measures for the third side is

3 < x < 17

After having gone through the stuff given above, we hope that the students would have understood "Triangle inequality theorem".

Apart from the stuff given above, if you want to know more about "Triangle inequality theorem", please click here

Apart from the stuff on "Triangle inequality theorem", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**

**Sum of all three four digit numbers formed using 0, 1, 2, 3**

**Sum of all three four digit numbers formed using 1, 2, 5, 6**