Question
triangle with side lengths
a=?
b=6
c=7
a=?
b=6
c=7
Answers
Answered by
GPT-4o mini
To find the length of side \( a \) in a triangle where the other two sides \( b \) and \( c \) are given as 6 and 7, we can use the triangle inequality theorem. The triangle inequality states that, for any triangle with sides of lengths \( a \), \( b \), and \( c \):
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting \( b = 6 \) and \( c = 7 \):
1. \( a + 6 > 7 \) → \( a > 1 \)
2. \( a + 7 > 6 \) → \( a > -1 \) (this condition is always satisfied for positive side lengths)
3. \( 6 + 7 > a \) → \( 13 > a \) or \( a < 13 \)
From these inequalities, we can conclude:
- \( 1 < a < 13 \)
This means that the length of side \( a \) must be greater than 1 and less than 13 for the side lengths to form a valid triangle. You can choose any value for \( a \) within this range. For example, \( a \) could be 2, 3, 4, ..., up to 12.
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting \( b = 6 \) and \( c = 7 \):
1. \( a + 6 > 7 \) → \( a > 1 \)
2. \( a + 7 > 6 \) → \( a > -1 \) (this condition is always satisfied for positive side lengths)
3. \( 6 + 7 > a \) → \( 13 > a \) or \( a < 13 \)
From these inequalities, we can conclude:
- \( 1 < a < 13 \)
This means that the length of side \( a \) must be greater than 1 and less than 13 for the side lengths to form a valid triangle. You can choose any value for \( a \) within this range. For example, \( a \) could be 2, 3, 4, ..., up to 12.