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A triangle has side lengths measuring [tex]\(3x \, \text{cm}\)[/tex], [tex]\(7x \, \text{cm}\)[/tex], and [tex]\(h \, \text{cm}\)[/tex]. Which expression describes the possible values of [tex]\(h\)[/tex] in [tex]\(\text{cm}\)[/tex]?

A. [tex]\(4x \ \textless \ h \ \textless \ 10x\)[/tex]
B. [tex]\(10x \ \textless \ h \ \textless \ 4x\)[/tex]
C. [tex]\(h = 4x\)[/tex]
D. [tex]\(h = 10x\)[/tex]

Sagot :

To determine the possible values of [tex]\( h \)[/tex] in terms of [tex]\( x \)[/tex], given the side lengths of a triangle as [tex]\( 3x \)[/tex] cm, [tex]\( 7x \)[/tex] cm, and [tex]\( h \)[/tex] cm, we need to apply 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. We need to consider these three inequalities:

1. [tex]\( 3x + 7x > h \)[/tex]
2. [tex]\( 3x + h > 7x \)[/tex]
3. [tex]\( 7x + h > 3x \)[/tex]

Let's break these down one by one:

### Step 1: Simplify the inequalities

1. First Inequality:
[tex]\[ 3x + 7x > h \][/tex]
[tex]\[ 10x > h \][/tex]
This simplifies to:
[tex]\[ h < 10x \][/tex]

2. Second Inequality:
[tex]\[ 3x + h > 7x \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ h > 4x \][/tex]

3. Third Inequality:
[tex]\[ 7x + h > 3x \][/tex]
Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 7x + h - 3x > 0 \][/tex]
[tex]\[ 4x + h > 0 \][/tex]
This simplifies to:
[tex]\[ h > -4x \][/tex]
However, since [tex]\( h \)[/tex] represents a side length of a triangle and must be positive, this inequality is always true and doesn't provide any new information.

### Step 2: Combine the valid inequalities

From the simplified inequalities, we have:
[tex]\[ h < 10x \][/tex]
and
[tex]\[ h > 4x \][/tex]

Combining these two, we get:
[tex]\[ 4x < h < 10x \][/tex]

### Conclusion

The expression that describes the possible values of [tex]\( h \)[/tex] is [tex]\( 4x < h < 10x \)[/tex]. Therefore, the correct answer is:
[tex]\[ 4x < h < 10x \][/tex]