To determine the possible range of values for the third side [tex]\( s \)[/tex] of an acute triangle given two sides measuring 8 cm and 10 cm, we need to follow these steps:
1. Triangle Inequality Theorem:
This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's denote the three sides of the triangle as [tex]\( a = 8 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = s \)[/tex] (where [tex]\( s \)[/tex] is the unknown third side).
According to the Triangle Inequality Theorem, we have:
[tex]\[
\begin{align*}
a + b > c & \implies 8 + 10 > s \implies 18 > s \implies s < 18, \\
a + c > b & \implies 8 + s > 10 \implies s > 2, \\
b + c > a & \implies 10 + s > 8 \implies s > -2.
\end{align*}
\][/tex]
However, [tex]\( s > -2 \)[/tex] is redundant since [tex]\( s > 2 \)[/tex] and [tex]\( s \)[/tex] must be a positive value.
Therefore, combining these inequalities, we get:
[tex]\[
2 < s < 18.
\][/tex]
2. Acuteness Condition:
For the triangle to be acute, the square of each side must be less than the sum of the squares of the other two sides. That means:
[tex]\[
s^2 < a^2 + b^2.
\][/tex]
Specifically:
[tex]\[
s^2 < 8^2 + 10^2 = 64 + 100 = 164 \implies s < \sqrt{164} \approx 12.8.
\][/tex]
Additionally, for the triangle to specifically remain acute:
[tex]\[
\begin{align*}
8^2 & < 10^2 + s^2 \implies 64 < 100 + s^2 \implies -36 < s^2 \implies (always true). \\
10^2 & < 8^2 + s^2 \implies 100 < 64 + s^2 \implies 36 < s^2 \implies s > 6.
\end{align*}
\][/tex]
Combining our findings, the more restrictive range due to the acuteness condition is:
[tex]\[
6 < s < 12.8.
\][/tex]
Given this thorough investigation, the best representation of the possible range of values for the third side [tex]\( s \)[/tex] in an acute triangle is [tex]\( 6 < s < 12.8 \)[/tex]. Thus, the correct answer is:
[tex]\[ 6 < s < 12.8 \][/tex]
