To find the possible range of values for the third side [tex]\( s \)[/tex] of an acute triangle with sides measuring 8 cm and 10 cm, we will use two key properties:
1. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
2. Acute Triangle Condition: Each angle in an acute triangle must be less than 90 degrees, which implies that the square of the length of the longest side must be less than the sum of the squares of the lengths of the other two sides.
Let's derive the range step-by-step:
### 1. Triangle Inequality Theorem
For a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] (where [tex]\( a = 8 \)[/tex] cm, [tex]\( b = 10 \)[/tex] cm, and [tex]\( c = s \)[/tex]):
- [tex]\( a + b > c \)[/tex]
- [tex]\( a + c > b \)[/tex]
- [tex]\( b + c > a \)[/tex]
Given [tex]\( a = 8 \)[/tex] and [tex]\( b = 10 \)[/tex]:
- [tex]\( 8 + 10 > s \Rightarrow s < 18 \)[/tex]
- [tex]\( 8 + s > 10 \Rightarrow s > 2 \)[/tex]
- [tex]\( 10 + s > 8 \)[/tex]
The third condition [tex]\( 10 + s > 8 \)[/tex] is always true for any [tex]\( s > 0 \)[/tex], so we do not need to explicitly consider it. Thus, based on the Triangle Inequality Theorem, we have:
[tex]\[ 2 < s < 18 \][/tex]
### 2. Acute Triangle Condition
For the triangle to be acute, the square of the length of any side must be less than the sum of the squares of the lengths of the other two sides:
- [tex]\( s^2 < a^2 + b^2 \Rightarrow s^2 < 8^2 + 10^2 \Rightarrow s^2 < 64 + 100 \Rightarrow s^2 < 164 \Rightarrow s < \sqrt{164} \)[/tex]
Calculating [tex]\( \sqrt{164} \)[/tex]:
[tex]\[ s < \sqrt{164} \approx 12.8 \][/tex]
### Combining Both Conditions
To satisfy both the Triangle Inequality Theorem and acute triangle condition:
[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]\[ \boxed{6 < s < 12.8} \][/tex]
