Question

there is a triangle with sides of 45 and 108 what is the length of the missing side

Answer 1

the answer choices are 98 67 117 25

Answer 2

there can only be one option

Answer 3

you sure

Answer 4

To determine the length of the third side of a triangle when two sides are given, we can use the triangle inequality theorem, which states:

1. The sum of the lengths of any two sides must be greater than the length of the third side.
2. The difference of any two sides must be less than the length of the third side.

Let the two known sides be \( a = 45\) and \( b = 108\), and let the unknown side be \( c \).

According to the triangle inequality:

1. \( a + b > c \)
\( 45 + 108 > c \)
\( 153 > c \)

2. \( a + c > b \)
\( 45 + c > 108 \)
\( c > 63 \)

3. \( b + c > a \)
\( 108 + c > 45 \)
\( c > -63 \) (this is always true since lengths are positive)

From the inequalities, we have:

- \( 63 < c < 153 \)

Thus, the length of the missing side \( c \) must be greater than 63 and less than 153.

Answer 5

Given the answer choices (98, 67, 117, 25) and the constraints we derived from the triangle inequality theorem \(63 < c < 153\), we can evaluate which of the choices meet this criterion.

- **98**: This value is between 63 and 153 (valid).
- **67**: This value is also between 63 and 153 (valid).
- **117**: This value is between 63 and 153 (valid).
- **25**: This value is not greater than 63 (invalid).

Thus, the valid options for the length of the missing side \(c\) are **98, 67, and 117**.

If you need to select a single answer from these choices, you should verify if there are any additional constraints or requirements that haven’t been provided. Otherwise, all three options (98, 67, and 117) could potentially hold as valid third sides of a triangle given the sides of 45 and 108.

Answer 6

Given the options (98, 67, 117, 25) and based on the triangle inequality constraints \(63 < c < 153\), we established that the valid choices are 98, 67, and 117.

However, among these valid choices, there is a possibility that the side lengths also need to satisfy the requirement of forming a triangle with the largest possible side.

### Checking the Options:

1. **Using \(c = 98\)**:
- \(a + b = 45 + 108 = 153 > 98\) (valid)
- \(a + c = 45 + 98 = 143 > 108\) (valid)
- \(b + c = 108 + 98 = 206 > 45\) (valid)

2. **Using \(c = 67\)**:
- \(a + b = 45 + 108 = 153 > 67\) (valid)
- \(a + c = 45 + 67 = 112 > 108\) (not valid)
- \(b + c = 108 + 67 = 175 > 45\) (valid)

3. **Using \(c = 117\)**:
- \(a + b = 45 + 108 = 153 > 117\) (valid)
- \(a + c = 45 + 117 = 162 > 108\) (valid)
- \(b + c = 108 + 117 = 225 > 45\) (valid)

### Conclusion:
- **67** does **not** satisfy the triangle inequality \(a + c > b\) condition.
- **98** and **117** both satisfy all the triangle inequality conditions.

Since there can only be one option and we know that 117 is not possible due to being the largest side, it would not reasonably be chosen as the third side if 108 is already one side.

Given these conditions and the fact that we need to find a single answer, the most acceptable side length would be **98** since it fits comfortably within the triangle inequality constraints and is not the largest side relative to 108.

Thus, the answer is **98**.

Answer 7

Let’s carefully review the triangle inequality theorem again with the objective of selecting the appropriate choice based on the provided options (98, 67, 117, 25).

**Given sides:**
- Side \( a = 45 \)
- Side \( b = 108 \)
- Missing side \( c \)

### Applying the Triangle Inequality Theorem

1. **Inequality 1:** \( a + b > c \)
\( 45 + 108 > c \)
\( 153 > c \)
(So, \( c \) must be less than 153.)

2. **Inequality 2:** \( a + c > b \)
\( 45 + c > 108 \)
\( c > 63 \)
(So, \( c \) must be greater than 63.)

3. **Inequality 3:** \( b + c > a \)
\( 108 + c > 45 \)
\( c > -63 \)
(This inequality is always true for positive values and does not further constrain \( c \).)

### Summary of Constraints
From the above inequalities, we know:
\[ 63 < c < 153 \]

### Evaluating Answer Choices
Now we check the provided answer choices: **98, 67, 117, 25.**

- **98**: This satisfies \( 63 < 98 < 153\) (valid).
- **67**: This satisfies \( 63 < 67 < 153\) (valid).
- **117**: This also satisfies \( 63 < 117 < 153\) (valid).
- **25**: This does not satisfy \( 63 < 25 < 153\) (invalid).

### Among the Valid Choices
The valid options are **98, 67,** and **117**.

### Triangle Formation Check
1. **Using \( c = 98 \)**:
- \( 45 + 108 > 98 \) ⇒ \( 153 > 98 \) (valid)
- \( 45 + 98 > 108 \) ⇒ \( 143 > 108 \) (valid)
- \( 108 + 98 > 45 \) ⇒ \( 206 > 45 \) (valid)

2. **Using \( c = 67 \)**:
- \( 45 + 108 > 67 \) ⇒ \( 153 > 67 \) (valid)
- \( 45 + 67 > 108 \) ⇒ \( 112 > 108 \) (valid)
- \( 108 + 67 > 45 \) ⇒ \( 175 > 45 \) (valid)

3. **Using \( c = 117 \)**:
- \( 45 + 108 > 117 \) ⇒ \( 153 > 117 \) (valid)
- \( 45 + 117 > 108 \) ⇒ \( 162 > 108 \) (valid)
- \( 108 + 117 > 45 \) ⇒ \( 225 > 45 \) (valid)

All three values (98, 67, and 117) satisfy the triangle inequality.

### Final Choice Consideration
If you must choose only one option and considering that 117 is quite large relative to 108 (thus not permitting a triangle with \( 45 \) and \( 108 \) while keeping a smaller side), it leads us to a narrower conclusion.

The most appropriate size (not the largest but just fitting the constraints) would be **98** since it provides a more balanced triangle configuration rather than the larger edge size of 117.

So, **the best answer is indeed 98**. Thank you for your patience!