Question
two sides of a triangle mesure 6in, and 8in. what are possible lengths of the third side?
1in
2in
4in
10in
12in
14in
15in
explain your reasoning.
the ? (sum, difference) of the two ? (shorter, longer) side lengths is ?(greater, less) than the length of the ?(shorter, longest)
1in
2in
4in
10in
12in
14in
15in
explain your reasoning.
the ? (sum, difference) of the two ? (shorter, longer) side lengths is ?(greater, less) than the length of the ?(shorter, longest)
Answers
GPT-4o mini
To find the possible lengths of the third side of a triangle with two sides measuring 6 inches and 8 inches, we can use the triangle inequality theorem. The triangle inequality theorem states:
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. The difference between the lengths of any two sides must be less than the length of the third side.
Let’s denote the two given sides as \( a = 6 \) inches and \( b = 8 \) inches. We’ll call the length of the third side \( c \).
Using the triangle inequality:
1. **Sum of the two sides greater than the third side**:
- \( a + b > c \)
- \( 6 + 8 > c \)
- \( 14 > c \), which implies \( c < 14 \)
2. **Difference of the two sides less than the third side**:
- \( |a - b| < c \)
- \( |6 - 8| < c \)
- \( 2 < c \), which implies \( c > 2 \)
Summarizing these inequalities, we find:
- From the sum: \( c < 14 \)
- From the difference: \( c > 2 \)
Putting these together, we find that the value of \( c \) must satisfy:
\[ 2 < c < 14 \]
Now, we can evaluate the provided options (1, 2, 4, 10, 12, 14, 15):
- **1 in**: Not valid (less than 2)
- **2 in**: Not valid (not greater than 2)
- **4 in**: Valid (between 2 and 14)
- **10 in**: Valid (between 2 and 14)
- **12 in**: Valid (between 2 and 14)
- **14 in**: Not valid (not less than 14)
- **15 in**: Not valid (greater than 14)
**Conclusion**: The possible lengths for the third side \( c \) are **4 in, 10 in, and 12 in**.
### Final Explanation:
To summarize in terms of the question's placeholders:
1. The **sum** of the two **shorter** side lengths (6 in and 8 in) is **greater** than the length of the **longest** side \( c \).
2. The **difference** of the two **longer** side lengths (8 in - 6 in = 2 in) is **less** than the length of the **third** side \( c \).
1. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. The difference between the lengths of any two sides must be less than the length of the third side.
Let’s denote the two given sides as \( a = 6 \) inches and \( b = 8 \) inches. We’ll call the length of the third side \( c \).
Using the triangle inequality:
1. **Sum of the two sides greater than the third side**:
- \( a + b > c \)
- \( 6 + 8 > c \)
- \( 14 > c \), which implies \( c < 14 \)
2. **Difference of the two sides less than the third side**:
- \( |a - b| < c \)
- \( |6 - 8| < c \)
- \( 2 < c \), which implies \( c > 2 \)
Summarizing these inequalities, we find:
- From the sum: \( c < 14 \)
- From the difference: \( c > 2 \)
Putting these together, we find that the value of \( c \) must satisfy:
\[ 2 < c < 14 \]
Now, we can evaluate the provided options (1, 2, 4, 10, 12, 14, 15):
- **1 in**: Not valid (less than 2)
- **2 in**: Not valid (not greater than 2)
- **4 in**: Valid (between 2 and 14)
- **10 in**: Valid (between 2 and 14)
- **12 in**: Valid (between 2 and 14)
- **14 in**: Not valid (not less than 14)
- **15 in**: Not valid (greater than 14)
**Conclusion**: The possible lengths for the third side \( c \) are **4 in, 10 in, and 12 in**.
### Final Explanation:
To summarize in terms of the question's placeholders:
1. The **sum** of the two **shorter** side lengths (6 in and 8 in) is **greater** than the length of the **longest** side \( c \).
2. The **difference** of the two **longer** side lengths (8 in - 6 in = 2 in) is **less** than the length of the **third** side \( c \).