To determine the possible range of values for the third side [tex]\( s \)[/tex] of an acute triangle with given sides 8 cm and 10 cm, we need to consider two conditions:
1. The Triangle Inequality Theorem: For any three sides to form a triangle, the sum of any two sides must be greater than the third side.
2. The condition for an acute triangle: The square of the third side must be less than the sum of the squares of the other two sides.
Given the two sides of the triangle as 8 cm and 10 cm, let's consider these conditions one by one.
### 1. Triangle Inequality Theorem
For [tex]\( s \)[/tex] to form a triangle with sides 8 cm and 10 cm:
- [tex]\( 8 + 10 > s \)[/tex]
- [tex]\( 8 + s > 10 \)[/tex]
- [tex]\( 10 + s > 8 \)[/tex]
These inequalities can be simplified to:
- [tex]\( s < 18 \)[/tex]
- [tex]\( s > 2 \)[/tex]
- This already indicates that [tex]\( 2 < s < 18 \)[/tex].
### 2. Acute Triangle Condition
For a triangle to be acute, the square of each side must be less than the sum of the squares of the other two sides.
- [tex]\( s^2 < 8^2 + 10^2 \)[/tex]
- [tex]\( s^2 < 64 + 100 \)[/tex]
- [tex]\( s^2 < 164 \)[/tex]
- [tex]\( s < \sqrt{164} \)[/tex]
- [tex]\( s < 12.8 \)[/tex]
Combining these two conditions:
- From the Triangle Inequality, we have [tex]\( 2 < s < 18 \)[/tex].
- From the acute triangle condition, we also have [tex]\( s < 12.8 \)[/tex].
Taking the more restrictive upper bound, we get [tex]\( 2 < s < 12.8 \)[/tex].
So the best representation of the possible range of values for the third side [tex]\( s \)[/tex] is:
[tex]\[ 2 < s < 12.8 \][/tex]
