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An acute triangle has two sides measuring 8 cm and 10 cm. What is the best representation of the possible range of values for the third side, [tex]\( s \)[/tex]?

A. [tex]\(2 \ \textless \ s \ \textless \ 18\)[/tex]
B. [tex]\(6 \ \textless \ s \ \textless \ 12.8\)[/tex]
C. [tex]\(s \ \textless \ 2\)[/tex] or [tex]\(s \ \textgreater \ 18\)[/tex]
D. [tex]\(s \ \textless \ 6\)[/tex] or [tex]\(s \ \textgreater \ 12.8\)[/tex]

Sagot :

GinnyAnswer
To determine the possible range of values for the third side, [tex]\( s \)[/tex], of an acute triangle given the two sides are 8 cm and 10 cm, we need to use the triangle inequality theorem. The triangle inequality theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Specifically, for a triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(s\)[/tex]:

1. [tex]\( a + b > s \)[/tex]
2. [tex]\( a + s > b \)[/tex]
3. [tex]\( b + s > a \)[/tex]

We are given two sides: [tex]\(a = 8\)[/tex] cm and [tex]\(b = 10\)[/tex] cm.

First, let's consider the inequalities involving the given sides and the third side [tex]\( s \)[/tex]:

1. [tex]\( 8 + 10 > s \)[/tex]
2. [tex]\( 8 + s > 10 \)[/tex]
3. [tex]\( 10 + s > 8 \)[/tex]

We can simplify these inequalities one by one:

1. [tex]\( 18 > s \)[/tex], which can be written as [tex]\( s < 18 \)[/tex]
2. [tex]\( 8 + s > 10 \)[/tex] simplifies to [tex]\( s > 2 \)[/tex]
3. [tex]\( 10 + s > 8 \)[/tex] simplifies to [tex]\( s > -2 \)[/tex], but since [tex]\( s \)[/tex] must be a positive length, this inequality does not provide new information

Combining these inequalities, we find that [tex]\( s \)[/tex] must satisfy both:

[tex]\[ s < 18 \][/tex]
[tex]\[ s > 2 \][/tex]

Therefore, the best representation of the possible range of values for the third side, [tex]\( s \)[/tex], is:

[tex]\[ 2 < s < 18 \][/tex]

Thus, the correct option is:
[tex]\[ 2 < s < 18 \][/tex]
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