WORKSHEET ON TRIANGLE INEQUALITY THEOREM

Problem 1 : 

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

8, 12 and 9

Problem 2 : 

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

10, 7 and 13

Problem 3 : 

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

6, 12 and 3

Problem 4 : 

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

Problem 5 : 

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

10, 7, x 

Answers

1. Answer :

8, 12 and 9

According to Triangle Inequality Theorem, 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.

2. Answer :

10, 7 and 13

Use Triangle Inequality Theorem. 

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.

3. Answer :

6, 12 and 3

Use Triangle Inequality Theorem

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.

4. Answer :

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

5. Answer :

10, 7, x

Sum of the lengths of the given two sides : 

10 + 7  =  17

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

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