Question

an acute triangle has two sides measuring 8 cm and 10 cm. what is the best representation of the possible range of values for the third side, s?
2 < s < 18
6 < s < 12.8
s < 2 or s > 18
s < 6 or s > 12.8

Explanation:

Step1: Use triangle inequality first

For any triangle, the sum of two sides > third side.
$10+8 > s$, $10+s > 8$, $8+s > 10$
This gives $2 < s < 18$.

Step2: Use acute triangle condition (case1: s is longest side)

If $s \geq 10$, all angles are acute, so $8^2 + 10^2 > s^2$.

$$\begin{align} 64 + 100 &> s^2 \\ 164 &> s^2 \\ s &< \sqrt{164} \approx 12.8 \end{align}$$

Step3: Use acute triangle condition (case2: 10 is longest side)

If $s < 10$, all angles are acute, so $8^2 + s^2 > 10^2$.

$$\begin{align} 64 + s^2 &> 100 \\ s^2 &> 36 \\ s &> 6 \end{align}$$

Step4: Combine the valid ranges

Merge the results from Step2 and Step3.

Answer:

6 < s < 12.8