To find the possible range of values for the third side $s$ of an acute triangle given two sides, we can use the triangle inequality theorem and the condition for an acute triangle.

Step 1: Apply the Triangle Inequality Theorem

For any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let the sides of the triangle be $a = 8 \, \text{cm}$, $b = 10 \, \text{cm}$, and $s$ be the unknown side. The inequalities are:

$a + b > s \implies 8 + 10 > s \implies s < 18$ $a + s > b \implies 8 + s > 10 \implies s > 2$ $b + s > a \implies 10 + s > 8 \implies s > -2 \quad (\text{this inequality is already satisfied for any positive } s)$

So from the triangle inequality theorem, we get the range: $2 < s < 18$

Step 2: Ensure the Triangle is Acute

For a triangle to be acute, the square of each side must satisfy the condition: $a^2 + b^2 > s^2$ Applying this for our known sides, $a = 8 \, \text{cm}$ and $b = 10 \, \text{cm}$:

$8^2 + 10^2 > s^2 \implies 64 + 100 > s^2 \implies 164 > s^2 \implies s < \sqrt{164} \approx 12.8$

So for the triangle to remain acute, the third side must also satisfy $s < 12.8$.

Conclusion

Combining both conditions, we get the range: $6 < s < 12.8$ Thus, the correct answer is: $\boxed{6 < s < 12.8}$

Do you have any questions or need more details?

Here are 5 related questions:

1. How does the triangle inequality theorem apply to obtuse triangles?
2. Can you explain how to calculate the area of an acute triangle using Heron's formula?
3. What are the properties of an equilateral triangle in relation to the triangle inequality?
4. How do you determine if a triangle is right, acute, or obtuse based on its side lengths?
5. What is the geometric significance of the circumcenter in an acute triangle?

Tip: Always double-check the conditions (like the triangle inequality) when working with triangle side lengths to avoid impossible cases.