To determine the possible range of values for the third side, [tex]\( s \)[/tex], of an acute triangle with sides 8 cm and 10 cm, follow these steps:
1. Understand the properties of an acute triangle:
An acute triangle requires all its interior angles to be less than 90 degrees. This means that the square of each side must be less than the sum of the squares of the other two sides.
2. Apply the triangle inequality theorem:
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side.
Therefore:
[tex]\[
s < 8 + 10
\][/tex]
Which implies:
[tex]\[
s < 18
\][/tex]
And,
[tex]\[
s > |8 - 10|
\][/tex]
Which implies:
[tex]\[
s > 2
\][/tex]
3. Confirm with the acute triangle condition:
To ensure that the triangle remains acute, the third side [tex]\( s \)[/tex] must satisfy the following conditions:
[tex]\[
s^2 < 8^2 + 10^2
\][/tex]
This simplifies to:
[tex]\[
s^2 < 64 + 100
\][/tex]
[tex]\[
s^2 < 164
\][/tex]
Taking the square root of both sides:
[tex]\[
s < \sqrt{164} \approx 12.8
\][/tex]
4. Combine all findings:
The above calculations show that the side [tex]\( s \)[/tex] must lie between 2 and 18 while additionally ensuring [tex]\( s < 12.8\)[/tex] to maintain the condition of acuteness.
Given all this information, it is clear that the correct answer is:
[tex]\[ 2 < s < 18 \][/tex]
Thus, the correct representation of the possible range of values for the third side [tex]\( s \)[/tex] in an acute triangle with two sides measuring 8 cm and 10 cm is:
[tex]\[ 2 < s < 18 \][/tex]
