Triangle Inequality Theorem Worksheet: Practice & Solutions

7 min read 11-16-2024

Triangle Inequality Theorem Worksheet: Practice & Solutions

The Triangle Inequality Theorem is a fundamental concept in geometry that asserts that, for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem is not only essential for understanding triangles but also for solving various geometric problems. In this post, we'll provide you with a comprehensive worksheet for practicing the Triangle Inequality Theorem, along with detailed solutions to enhance your understanding.

What is the Triangle Inequality Theorem? 📐

The Triangle Inequality Theorem states that for any triangle with sides of lengths ( a ), ( b ), and ( c ):

( a + b > c )
( a + c > b )
( b + c > a )

These three conditions must be satisfied for three lengths to form a triangle. If any one of these inequalities is not true, then the three lengths cannot form a triangle.

Importance of the Triangle Inequality Theorem

Understanding this theorem is vital for several reasons:

Triangle Construction: It helps in determining whether three given lengths can form a triangle.
Problem Solving: Many geometrical problems and proofs rely on this theorem.
Real-world Applications: From architecture to engineering, the Triangle Inequality Theorem finds relevance in various fields.

Triangle Inequality Theorem Worksheet 📄

To help you practice, here’s a worksheet with several problems based on the Triangle Inequality Theorem.

Problem Set

Given sides ( a = 7 ), ( b = 10 ), and ( c = 5 ):
- Can these lengths form a triangle? Explain.
Check whether the following sets of sides can form a triangle:
- ( (3, 4, 8) )
- ( (5, 5, 10) )
- ( (6, 7, 9) )
For the sides ( a = 12 ) and ( b = 15 ), find the range of possible values for ( c ) such that a triangle can be formed.
If ( a = 9 ) and ( c = 6 ), what is the maximum length ( b ) can be for a triangle to exist?
A triangle has sides of lengths ( a = 14 ), ( b = x ), and ( c = 20 ). What values of ( x ) will make a triangle possible?

Table of Problem Sets

<table> <tr> <th>Problem</th> <th>Sides (a, b, c)</th> <th>Can form a triangle?</th> </tr> <tr> <td>1</td> <td>(7, 10, 5)</td> <td>Yes</td> </tr> <tr> <td>2.1</td> <td>(3, 4, 8)</td> <td>No</td> </tr> <tr> <td>2.2</td> <td>(5, 5, 10)</td> <td>No</td> </tr> <tr> <td>2.3</td> <td>(6, 7, 9)</td> <td>Yes</td> </tr> </table>

Important Note

“Make sure to check all three inequalities for each set of sides to confirm if they can form a triangle.”

Solutions to the Worksheet 📝

Now, let's dive into the solutions for the problems listed in the worksheet.

Solution to Problem 1

Given ( a = 7 ), ( b = 10 ), and ( c = 5 ):
- Check:
  - ( 7 + 10 > 5 ) (17 > 5) ✅
  - ( 7 + 5 > 10 ) (12 > 10) ✅
  - ( 10 + 5 > 7 ) (15 > 7) ✅

Conclusion: Yes, these lengths can form a triangle.

Solution to Problem 2

Set (3, 4, 8):
- ( 3 + 4 > 8 ) (7 > 8) ❌

Conclusion: No, these lengths cannot form a triangle.

Set (5, 5, 10):
- ( 5 + 5 > 10 ) (10 > 10) ❌

Conclusion: No, these lengths cannot form a triangle.

Set (6, 7, 9):
- ( 6 + 7 > 9 ) (13 > 9) ✅
- ( 6 + 9 > 7 ) (15 > 7) ✅
- ( 7 + 9 > 6 ) (16 > 6) ✅

Conclusion: Yes, these lengths can form a triangle.

Solution to Problem 3

For ( a = 12 ) and ( b = 15 ):
- The inequalities are:
  - ( 12 + 15 > c ) → ( c < 27 )
  - ( 12 + c > 15 ) → ( c > 3 )
  - ( 15 + c > 12 ) → ( c > -3 ) (this condition is always true)

Thus, the range of possible values for ( c ) is:

( 3 < c < 27 )

Solution to Problem 4

Given ( a = 9 ) and ( c = 6 ):
- The inequalities are:
  - ( 9 + 6 > b ) → ( b < 15 )
  - ( 9 + b > 6 ) (this condition is always true)
  - ( 6 + b > 9 ) → ( b > 3 )

Thus, the maximum length ( b ) can be is:

( 3 < b < 15 )

Solution to Problem 5

For ( a = 14 ), ( b = x ), and ( c = 20 ):
- The inequalities are:
  - ( 14 + x > 20 ) → ( x > 6 )
  - ( 14 + 20 > x ) → ( x < 34 )
  - ( 20 + x > 14 ) (this condition is always true)

Thus, the values of ( x ) that will make a triangle possible are:

( 6 < x < 34 )

Conclusion

The Triangle Inequality Theorem is a crucial element in geometry that ensures the validity of triangle formation with given side lengths. By practicing various problems, you can solidify your understanding of this theorem. Use this worksheet to test your skills, and remember that geometric concepts build upon each other, so mastering the fundamentals will aid in your mathematical journey. Happy studying! 📚✨