Question

Find the range for the third side of a triangle with the given measures of two sides
7 and 12
the range of the side lengths is between __ and __

Answer 1

To determine the range of the third side \( c \) of a triangle when the other two sides are \( a = 7 \) and \( b = 12 \), we can use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

The triangle inequalities for sides \( a \), \( b \), and \( c \) are as follows:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Using the given side lengths:

1. **From** \( a + b > c \):
\[
7 + 12 > c \implies 19 > c \implies c < 19
\]

2. **From** \( a + c > b \):
\[
7 + c > 12 \implies c > 12 - 7 \implies c > 5
\]

3. **From** \( b + c > a \):
\[
12 + c > 7 \implies c > 7 - 12 \implies c > -5
\]
(This condition is always satisfied since side lengths are positive.)

Combining these inequalities, we find that:
\[
5 < c < 19
\]

Thus, the range for the third side \( c \) is between **5** and **19**. Therefore, the answer is:

The range of the side lengths is between **5** and **19**.