Question
x is the length in inches of the third side of a triangle. The range of all possible values of x is shown on the number line. Which of the following options has possible lengths of the other two sides of the triangle? (1 point) Responses 42 inches and 50 inches 42 inches and 50 inches - no response given 28 inches and 64 inches 28 inches and 64 inches - no response given 36 inches and 92 inches 36 inches and 92 inches - incorrect 48 inches and 76 inches
Answers
Answer
so which one
Answered by
GPT-4o mini
To determine the possible lengths of the other two sides of the triangle given a third side \( x \), we need to apply the triangle inequality theorem, which states:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.
Since we are considering \( c = x \) and the lengths of the other two sides are \( a \) and \( b \), we can express the inequalities as:
1. \( a + b > x \)
2. \( a + x > b \)
3. \( b + x > a \)
From the inequalities above, we can derive a range for \( x \) from the lengths \( a \) and \( b \).
Let's examine each given option:
1. **42 inches and 50 inches**:
- \( x < a + b = 42 + 50 = 92 \)
- \( x > |a - b| = |42 - 50| = 8 \)
- So, \( x \) can range from 8 to 92.
2. **28 inches and 64 inches**:
- \( x < a + b = 28 + 64 = 92 \)
- \( x > |28 - 64| = 36 \)
- So, \( x \) can range from 36 to 92.
3. **36 inches and 92 inches**:
- \( x < a + b = 36 + 92 = 128 \)
- \( x > |36 - 92| = 56 \)
- So, \( x \) can range from 56 to 128.
4. **48 inches and 76 inches**:
- \( x < a + b = 48 + 76 = 124 \)
- \( x > |48 - 76| = 28 \)
- So, \( x \) can range from 28 to 124.
To summarize, for each option:
- 42 and 50: Valid range for \( x \) is 8 to 92 (possible).
- 28 and 64: Valid range for \( x \) is 36 to 92 (possible).
- 36 and 92: Valid range for \( x \) is 56 to 128 (not appropriate).
- 48 and 76: Valid range for \( x \) is 28 to 124 (possible).
From the given options, the possible lengths of the other two sides of the triangle could be either **42 inches and 50 inches** or **28 inches and 64 inches** or **48 inches and 76 inches**. The only incorrect option, based on exclusion, is **36 inches and 92 inches**.
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.
Since we are considering \( c = x \) and the lengths of the other two sides are \( a \) and \( b \), we can express the inequalities as:
1. \( a + b > x \)
2. \( a + x > b \)
3. \( b + x > a \)
From the inequalities above, we can derive a range for \( x \) from the lengths \( a \) and \( b \).
Let's examine each given option:
1. **42 inches and 50 inches**:
- \( x < a + b = 42 + 50 = 92 \)
- \( x > |a - b| = |42 - 50| = 8 \)
- So, \( x \) can range from 8 to 92.
2. **28 inches and 64 inches**:
- \( x < a + b = 28 + 64 = 92 \)
- \( x > |28 - 64| = 36 \)
- So, \( x \) can range from 36 to 92.
3. **36 inches and 92 inches**:
- \( x < a + b = 36 + 92 = 128 \)
- \( x > |36 - 92| = 56 \)
- So, \( x \) can range from 56 to 128.
4. **48 inches and 76 inches**:
- \( x < a + b = 48 + 76 = 124 \)
- \( x > |48 - 76| = 28 \)
- So, \( x \) can range from 28 to 124.
To summarize, for each option:
- 42 and 50: Valid range for \( x \) is 8 to 92 (possible).
- 28 and 64: Valid range for \( x \) is 36 to 92 (possible).
- 36 and 92: Valid range for \( x \) is 56 to 128 (not appropriate).
- 48 and 76: Valid range for \( x \) is 28 to 124 (possible).
From the given options, the possible lengths of the other two sides of the triangle could be either **42 inches and 50 inches** or **28 inches and 64 inches** or **48 inches and 76 inches**. The only incorrect option, based on exclusion, is **36 inches and 92 inches**.
Answered by
GPT-4o mini
To clarify the possible lengths of the other two sides of the triangle based on the triangle inequality, we look at the provided options and their compatibility with \( x \):
1. **42 inches and 50 inches**: Possible.
2. **28 inches and 64 inches**: Possible.
3. **36 inches and 92 inches**: Not possible (incorrect).
4. **48 inches and 76 inches**: Possible.
Given that the question asks for options that have possible lengths of the other two sides of the triangle, any of the valid options (1, 2, or 4) is appropriate.
However, since you've mentioned that "36 inches and 92 inches" is marked as incorrect, the definitive answer would be **42 inches and 50 inches**, **28 inches and 64 inches**, or **48 inches and 76 inches** as the correct pairs.
If you are looking for the best single answer based on the conventions of quiz-type questions, I recommend going with **42 inches and 50 inches** as it is confirmed as a valid option fulfilling the triangle inequality conditions.
1. **42 inches and 50 inches**: Possible.
2. **28 inches and 64 inches**: Possible.
3. **36 inches and 92 inches**: Not possible (incorrect).
4. **48 inches and 76 inches**: Possible.
Given that the question asks for options that have possible lengths of the other two sides of the triangle, any of the valid options (1, 2, or 4) is appropriate.
However, since you've mentioned that "36 inches and 92 inches" is marked as incorrect, the definitive answer would be **42 inches and 50 inches**, **28 inches and 64 inches**, or **48 inches and 76 inches** as the correct pairs.
If you are looking for the best single answer based on the conventions of quiz-type questions, I recommend going with **42 inches and 50 inches** as it is confirmed as a valid option fulfilling the triangle inequality conditions.
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