To determine the possible values of [tex]\( h \)[/tex] for a triangle with side lengths [tex]\( 3x \, \text{cm} \)[/tex], [tex]\( 7x \, \text{cm} \)[/tex], and [tex]\( h \, \text{cm} \)[/tex], 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 remaining side.
We need to check the three possible combinations:
1. [tex]\( (3x + 7x) > h \)[/tex]
2. [tex]\( (3x + h) > 7x \)[/tex]
3. [tex]\( (7x + h) > 3x \)[/tex]
Let’s simplify each inequality individually:
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]
Since [tex]\( h \)[/tex] and [tex]\( x \)[/tex] both represent positive lengths, [tex]\( h > -4x \)[/tex] is always true and does not provide additional constraints.
Combining the simplified inequalities from (1) and (2), we get:
[tex]\[
4x < h < 10x
\][/tex]
Therefore, the correct expression that describes the possible values of [tex]\( h \)[/tex], in cm, is:
[tex]\[
4x < h < 10x
\][/tex]
