To determine the possible values for [tex]\( h \)[/tex] in a triangle with side lengths [tex]\(3x \, \text{cm}\)[/tex], [tex]\(7x \, \text{cm}\)[/tex], and [tex]\(h \, \text{cm}\)[/tex], we need to use the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Therefore, for our triangle, we have three inequalities to consider:
1. [tex]\( 3x + 7x > h \)[/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
3. [tex]\( 7x + h > 3x \)[/tex]
Let's analyze these inequalities one by one:
1. [tex]\( 3x + 7x > h \)[/tex]:
[tex]\[ 10x > h \][/tex]
2. [tex]\( 3x + h > 7x \)[/tex]:
[tex]\[ h > 4x \][/tex]
3. [tex]\( 7x + h > 3x \)[/tex]:
[tex]\[ 7x + h > 3x \][/tex]
[tex]\[ h > -4x \][/tex]
Since [tex]\( h \)[/tex] represents a length and must be positive, the inequality [tex]\( h > -4x \)[/tex] will always be true and doesn't restrict our range.
Combining the valid inequalities [tex]\( 10x > h \)[/tex] and [tex]\( h > 4x \)[/tex], we obtain:
[tex]\[ 4x < h < 10x \][/tex]
Thus, the expression that correctly describes the possible values of [tex]\( h \)[/tex] in cm is:
[tex]\[ 4x < h < 10x \][/tex]
So, the correct answer is:
[tex]\[ 4x < h < 10x \][/tex]
