Question 6 of 10

A triangle has sides measuring 8 inches and 12 inches. If [tex]$x$[/tex] represents the length in inches of the third side, which inequality gives the range of possible values for [tex]$x$[/tex]?

A. [tex]$8 \leq x \leq 12$[/tex]
B. [tex][tex]$8 \ \textless \ x \ \textless \ 12$[/tex][/tex]
C. [tex]$4 \leq x \leq 20$[/tex]
D. [tex]$4 \ \textless \ x \ \textless \ 20$[/tex]



Answer :

Let's analyze the problem using the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's denote the sides of the triangle as [tex]\( a = 8 \)[/tex] inches, [tex]\( b = 12 \)[/tex] inches, and [tex]\( c = x \)[/tex] inches (where [tex]\( x \)[/tex] is the unknown side length).

We need to satisfy the following three inequalities based on the triangle inequality theorem:

1. The sum of the first two sides must be greater than the third side:
[tex]\[ a + b > x \][/tex]
[tex]\[ 8 + 12 > x \][/tex]
[tex]\[ 20 > x \][/tex]
which simplifies to:
[tex]\[ x < 20 \][/tex]

2. The sum of [tex]\( a \)[/tex] and [tex]\( c \)[/tex] must be greater than [tex]\( b \)[/tex]:
[tex]\[ a + x > b \][/tex]
[tex]\[ 8 + x > 12 \][/tex]
[tex]\[ x > 4 \][/tex]

3. The sum of [tex]\( b \)[/tex] and [tex]\( c \)[/tex] must be greater than [tex]\( a \)[/tex]:
[tex]\[ b + x > a \][/tex]
[tex]\[ 12 + x > 8 \][/tex]
This simplifies to:
[tex]\[ x > -4 \][/tex]
However, since side length must always be positive, this condition is inherently satisfied for any positive [tex]\( x \)[/tex].

Combining the valid inequalities from the above analysis, we get:
[tex]\[ 4 < x < 20 \][/tex]

Thus, the range of possible values for [tex]\( x \)[/tex] is:
[tex]\[ 4 < x < 20 \][/tex]

This corresponds to the answer choice:

D. [tex]\( 4 < x < 20 \)[/tex]