What if you were given three lengths, like 5, 7 and 10? How could you determine if sides with these lengths form a triangle? After completing this Concept, you'll be able to use the Triangle Inequality Theorem to determine if any three side lengths make a triangle.
Can any three lengths make a triangle? The answer is no. For example, the lengths 1, 2, 3 cannot make a triangle because
The arcs show that the two sides would never meet to form a triangle.
To make a triangle, two sides must add up to be greater than the third side. This is called the Triangle Inequality Theorem. This means that if you know two sides of a triangle, there are only certain lengths that the third side could be. If two sides have lengths
Do the lengths 4, 11, 8 make a triangle?
To solve this problem, check to make sure that the smaller two numbers add up to be greater than the biggest number.
Find the length of the third side of a triangle if the other two sides are 10 and 6.
The Triangle Inequality Theorem can also help you find the range of the third side. The two given sides are 6 and 10. The third side,
Notice the range is no less than 4, and not equal to 4. The third side could be 4.1 because
The base of an isosceles triangle has length 24. What can you say about the length of each leg?
To solve this problem, remember that an isosceles triangle has two congruent sides (the legs). We have to make sure that the sum of the lengths of the legs is greater than 24. In other words, if
Each leg must have a length greater than 12.
Do the lengths below make a triangle?
1. 4.1, 3.5, 7.5
2. 4, 4, 8
3. 6, 7, 8
Use the Triangle Inequality Theorem. Test to see if the smaller two numbers add up to be greater than the largest number.
Determine if the sets of lengths below can make a triangle. If not, state why.
- 6, 6, 13
- 1, 2, 3
- 7, 8, 10
- 5, 4, 3
- 23, 56, 85
- 30, 40, 50
- 7, 8, 14
- 7, 8, 15
- 7, 8, 14.99
If two lengths of the sides of a triangle are given, determine the range of the length of the third side.
- 8 and 9
- 4 and 15
- 20 and 32
- 2 and 5
- 10 and 8
- The legs of an isosceles triangle have a length of 12 each. What can you say about the length of the base?
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 5.7.