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# Triangle Inequality Theorem

## Sum of the lengths of any two sides of a triangle is greater than the third side.

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Triangle Inequality Theorem

What if you had to determine whether the three lengths 5, 7 and 10 make a triangle?

### Triangle Inequality Theorem

Can any three lengths make a triangle? The answer is no. There are limits on what the lengths can be. For example, the lengths 1, 2, 3 cannot make a triangle because \begin{align*}1 + 2 = 3\end{align*}, so they would all lie on the same line. The lengths 4, 5, 10 also cannot make a triangle because \begin{align*}4 + 5 = 9\end{align*}.

The arc marks show that the two sides would never meet to form a triangle. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third.

#### Determining if Three Lengths make a Triangle

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. \begin{align*}4+8=12\end{align*} and \begin{align*}12>11\end{align*} so yes these lengths make a triangle

#### Solving for an Unknown Length

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, so the third side, \begin{align*}s\end{align*}, can either be the shortest side or the longest side. For example \begin{align*}s\end{align*} could be 5 because \begin{align*}6 + 5 > 10\end{align*}. It could also be 15 because \begin{align*}6 +10 > 15\end{align*}. Therefore, the range of values for \begin{align*}s\end{align*} is \begin{align*}4 < s < 16\end{align*}.

Notice the range is no less than 4, and not equal to 4. The third side could be 4.1 because \begin{align*}4.1 + 6 > 10\end{align*}. For the same reason, \begin{align*}s\end{align*} cannot be greater than 16, but it could 15.9, \begin{align*}10 + 6 > 15.9\end{align*}.

#### Making Conclusions about the Length of Legs

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 \begin{align*}x\end{align*} is the length of a leg:

\begin{align*}x+x&>24\\ 2x &>24\\ x&>12\end{align*}

Each leg must have a length greater than 12.

#### Earlier Problem Revisited

The three lengths 5, 7, and 10 do make a triangle. The sum of the lengths of any two sides is greater than the length of the third.

### Examples

Do the lengths below make a triangle?

Use the Triangle Inequality Theorem. Test to see if the smaller two numbers add up to be greater than the largest number.

#### Example 1

4.1, 3.5, 7.5

\begin{align*}4.1 + 3.5 > 7.5\end{align*}. Yes, this is a triangle because \begin{align*}7.6 > 7.5\end{align*}.

#### Example 2

4, 4, 8

\begin{align*}4 + 4 = 8\end{align*}. No this is not a triangle because two lengths cannot equal the third

#### Example 3

6, 7, 8

\begin{align*}6 + 7 > 8\end{align*}. Yes this is a triangle because \begin{align*}13 > 8\end{align*}.

### Review

Determine if the sets of lengths below can make a triangle. If not, state why.

1. 6, 6, 13
2. 1, 2, 3
3. 7, 8, 10
4. 5, 4, 3
5. 23, 56, 85
6. 30, 40, 50
7. 7, 8, 14
8. 7, 8, 15
9. 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.

1. 8 and 9
2. 4 and 15
3. 20 and 32
4. 2 and 5
5. 10 and 8
6. \begin{align*}x\end{align*} and \begin{align*}2x\end{align*}
7. The legs of an isosceles triangle have a length of 12 each. What can you say about the length of the base?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 5.7.

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### Vocabulary Language: English

Triangle Inequality Theorem

The Triangle Inequality Theorem states that in order to make a triangle, two sides must add up to be greater than the third side.

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