To determine the possible values of [tex]\( h \)[/tex] in a triangle with side lengths [tex]\( 3x \)[/tex] cm, [tex]\( 7x \)[/tex] cm, and [tex]\( h \)[/tex] cm, we use the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides must be greater than the length of the remaining side. This gives us three inequalities to consider:
1. [tex]\( 3x + 7x > h \)[/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
3. [tex]\( 7x + h > 3x \)[/tex]
Let's solve these inequalities step-by-step.
1. [tex]\( 3x + 7x > h \)[/tex]
[tex]\[
10x > h \quad \text{or} \quad h < 10x
\][/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
[tex]\[
h > 7x - 3x \quad \text{or} \quad h > 4x
\][/tex]
3. [tex]\( 7x + h > 3x \)[/tex]
[tex]\[
h > 3x - 7x \quad \text{or} \quad h > -4x
\][/tex]
However, since [tex]\( h > -4x \)[/tex] is always true for positive lengths, we do not need to consider it. The relevant inequalities are:
[tex]\[
4x < h < 10x
\][/tex]
Thus, the expression describing the possible values of [tex]\( h \)[/tex] in centimeters is:
[tex]\[
4x < h < 10x
\][/tex]
The correct answer is:
[tex]\[
\boxed{4x < h < 10x}
\][/tex]
