Answer :
To determine the possible range of values for the third side, [tex]\( s \)[/tex], of an acute triangle with sides [tex]\( a = 8 \)[/tex] cm and [tex]\( b = 10 \)[/tex] cm, we need to consider several geometric constraints, particularly those related to the triangle inequality theorem and the conditions specific to an acute triangle.
### Steps for Solution:
1. Lower Bound Calculation:
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. For an acute triangle, the inequality [tex]\( a + b > s \)[/tex] must hold, but the essential constraint is [tex]\( s > |a - b| \)[/tex]:
[tex]\[ s > |8 - 10| = 2 \][/tex]
So, the lower bound for [tex]\( s \)[/tex] is greater than 2.
2. Upper Bound Calculation:
For a triangle to be acute, the square of the longest side must be less than the sum of the squares of the other two sides. Therefore, using:
[tex]\[ s < \sqrt{a^2 + b^2} \][/tex]
Substituting the given values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ s < \sqrt{8^2 + 10^2} = \sqrt{64 + 100} = \sqrt{164} \approx 12.8 \][/tex]
So, the upper bound for [tex]\( s \)[/tex] must be less than approximately 12.8.
3. Conclusion:
By combining both constraints, we derive the final range for [tex]\( s \)[/tex]:
[tex]\[ 2 < s < 12.8 \][/tex]
Thus, the best representation of the possible range of values for the third side [tex]\( s \)[/tex] is:
[tex]\[ 6 < s < 12.8 \][/tex]
Of the given options, the correct one is:
[tex]\[ \boxed{6 < s < 12.8} \][/tex]
### Steps for Solution:
1. Lower Bound Calculation:
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. For an acute triangle, the inequality [tex]\( a + b > s \)[/tex] must hold, but the essential constraint is [tex]\( s > |a - b| \)[/tex]:
[tex]\[ s > |8 - 10| = 2 \][/tex]
So, the lower bound for [tex]\( s \)[/tex] is greater than 2.
2. Upper Bound Calculation:
For a triangle to be acute, the square of the longest side must be less than the sum of the squares of the other two sides. Therefore, using:
[tex]\[ s < \sqrt{a^2 + b^2} \][/tex]
Substituting the given values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ s < \sqrt{8^2 + 10^2} = \sqrt{64 + 100} = \sqrt{164} \approx 12.8 \][/tex]
So, the upper bound for [tex]\( s \)[/tex] must be less than approximately 12.8.
3. Conclusion:
By combining both constraints, we derive the final range for [tex]\( s \)[/tex]:
[tex]\[ 2 < s < 12.8 \][/tex]
Thus, the best representation of the possible range of values for the third side [tex]\( s \)[/tex] is:
[tex]\[ 6 < s < 12.8 \][/tex]
Of the given options, the correct one is:
[tex]\[ \boxed{6 < s < 12.8} \][/tex]
