To determine the range of possible values for the third side of a triangle where the other two sides are 10 cm and 16 cm, you need to use the triangle inequality theorem. This theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
For this specific triangle:
1. Let [tex]\( a = 10 \)[/tex], [tex]\( b = 16 \)[/tex], and [tex]\( c \)[/tex] be the unknown side.
2. We apply the inequalities as follows:
- [tex]\( 10 + 16 > c \implies 26 > c \)[/tex]
- [tex]\( 10 + c > 16 \implies c > 6 \)[/tex]
- [tex]\( 16 + c > 10 \implies this is always true since \( c \)[/tex] is positive.
Combining the first two conditions, we get:
[tex]\[ 6 < c < 26 \][/tex]
Therefore, the length of the third side lies between 6 cm and 26 cm.
The correct answer is:
[tex]\[ 6 < x < 26 \][/tex]
