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A triangle has side lengths measuring [tex]$3x$[/tex] cm, [tex]$7x$[/tex] cm, and [tex][tex]$h$[/tex][/tex] cm. Which expression describes the possible values of [tex]$h$[/tex], in cm?

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]



Answer :

GinnyAnswer
To solve for the range of possible values for [tex]\( h \)[/tex] using the triangle inequality theorem, we need to analyze the side lengths given:

1. Side 1: [tex]\( 3x \)[/tex] cm
2. Side 2: [tex]\( 7x \)[/tex] cm
3. Side 3: [tex]\( h \)[/tex] cm

The triangle inequality 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]

Let’s apply these inequalities to our triangle:

1. [tex]\( 3x + 7x > h \)[/tex]
[tex]\[ 10x > h \][/tex]
[tex]\[ h < 10x \][/tex]

2. [tex]\( 3x + h > 7x \)[/tex]
[tex]\[ h > 7x - 3x \][/tex]
[tex]\[ h > 4x \][/tex]

3. [tex]\( 7x + h > 3x \)[/tex]
[tex]\[ h > 3x - 7x \][/tex]
[tex]\[ h > -4x \][/tex]

Since [tex]\( h > -4x \)[/tex] will always be true as long as [tex]\( h > 4x \)[/tex], it isn't a restrictive condition in this context.

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

Therefore, the expression that describes the possible values of [tex]\( h \)[/tex] in cm is:

[tex]\[ 4x < h < 10x \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{4x < h < 10x} \][/tex]

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