Question
List the range of possible measures for the third side in a triangle given two sides have lengths 6 and 19.
Answers
GPT-4o mini
To determine the possible range of lengths for the third side of a triangle when two sides have lengths \( a = 6 \) and \( b = 19 \), we can use the triangle inequality theorem. The theorem states that for any triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the length of the third side), the following inequalities must hold:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting in the values we have:
1. From \( a + b > c \):
\[
6 + 19 > c \implies 25 > c \implies c < 25
\]
2. From \( a + c > b \):
\[
6 + c > 19 \implies c > 19 - 6 \implies c > 13
\]
3. From \( b + c > a \):
\[
19 + c > 6 \text{ (this inequality will always hold true since } c > 0\text{)}
\]
Combining the results from the first two inequalities, we find that \( c \) must satisfy:
\[
13 < c < 25
\]
Thus, the range of possible measures for the third side of the triangle is:
\[
(13, 25)
\]
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)
Substituting in the values we have:
1. From \( a + b > c \):
\[
6 + 19 > c \implies 25 > c \implies c < 25
\]
2. From \( a + c > b \):
\[
6 + c > 19 \implies c > 19 - 6 \implies c > 13
\]
3. From \( b + c > a \):
\[
19 + c > 6 \text{ (this inequality will always hold true since } c > 0\text{)}
\]
Combining the results from the first two inequalities, we find that \( c \) must satisfy:
\[
13 < c < 25
\]
Thus, the range of possible measures for the third side of the triangle is:
\[
(13, 25)
\]