Question

The measure of two sides of a triangle are 7 and 11. Determine the range of value for the third side of the triangle

Answer 1

To find the range of possible values for the third side of a triangle, you can use the triangle inequality theorem. This 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.

Given the sides \( a = 7 \), \( b = 11 \), and \( c \) (the third side), the triangle inequalities can be written as follows:

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

Let's apply these inequalities:

1. From \( a + b > c \):
\[
7 + 11 > c \implies 18 > c \implies c < 18
\]

2. From \( a + c > b \):
\[
7 + c > 11 \implies c > 11 - 7 \implies c > 4
\]

3. From \( b + c > a \):
\[
11 + c > 7 \implies c > 7 - 11 \implies c > -4 \quad (\text{This condition is always true since } c > 4)
\]

Combining the two relevant inequalities, we find:
\[
4 < c < 18
\]

Thus, the range of values for the third side \( c \) of the triangle is:
\[
(4, 18)
\]