To determine the possible range of values for the third side [tex]\( s \)[/tex] of an acute triangle with the given sides [tex]\( 8 \)[/tex] cm and [tex]\( 10 \)[/tex] cm, we can use the properties of triangles and specifically, the properties of an acute triangle.
In any triangle, the length of one side must be less than the sum of the other two sides and more than the absolute value of their difference.
For an acute triangle, all angles should be less than 90 degrees. This adds an additional requirement that the length of any side must be less than the sum of the lengths of the other two sides but greater than the difference of the lengths of the other two sides.
Given:
- Side 1: [tex]\( 8 \)[/tex] cm
- Side 2: [tex]\( 10 \)[/tex] cm
- Third side: [tex]\( s \)[/tex]
We need to find the range for [tex]\( s \)[/tex].
1. Applying the Triangle Inequality Theorem:
According to the triangle inequality theorem:
[tex]\[ |8 - 10| < s < 8 + 10 \][/tex]
This translates to:
[tex]\[ 2 < s < 18 \][/tex]
2. Ensuring the triangle is acute:
For the triangle to be acute with sides [tex]\( 8 \)[/tex] cm, [tex]\( 10 \)[/tex] cm, and [tex]\( s \)[/tex], each angle should be less than 90 degrees. There is an additional requirement to ensure that the largest angle is opposite the largest side.
The properties to check are:
[tex]\[ s^2 < 8^2 + 10^2 \][/tex]
[tex]\[ 8^2 < 10^2 + s^2 \][/tex]
[tex]\[ 10^2 < 8^2 + s^2 \][/tex]
From these conditions:
[tex]\[ s^2 < 64 + 100 \implies s^2 < 164 \implies s < \sqrt{164} \approx 12.8 \][/tex]
Thus, the range established by the acute triangle condition is combined with the general triangle condition:
[tex]\[ 6 < s < 12.8 \][/tex]
Therefore, the correct representation of the possible range of values for [tex]\( s \)[/tex] is:
[tex]\[ 6 < s < 12.8 \][/tex]
Hence, the correct answer is:
[tex]\[ 6 < s < 12.8 \][/tex]
