- Establish the fundamental triangle inequality for sides 8, 10, and \(s\), resulting in \(2 < s < 18\).
- Apply the acute triangle condition \(a^2 + b^2 > c^2\) for each side to find additional constraints: \(s > 6\) and \(s < \sqrt{164}\).
- Find the intersection of these constraints to determine the final range: \(\boxed{6 < s < \sqrt{164} \approx 12.8}\).
Explanation
- Triangle inequality analysis
For any triangle with side lengths \(a\), \(b\), and \(s\), the triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. This gives us three conditions:
- \(a + b > s \implies 8 + 10 > s \implies 18 > s\)
- \(a + s > b \implies 8 + s > 10 \implies s > 2\)
- \(b + s > a \implies 10 + s > 8 \implies s > -2\) (which is trivial for length)
Combining these, we get the base requirement for any triangle with sides 8 and 10: \(2 < s < 18\).
- Acute triangle conditions analysis
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. Given sides \(8\), \(10\), and \(s\):
- For the angle opposite to side \(s\) to be acute: \(s^2 < 8^2 + 10^2 \implies s^2 < 64 + 100 \implies s^2 < 164 \implies s < \sqrt{164} \approx 12.8\)
- For the angle opposite to side \(8\) to be acute: \(8^2 < 10^2 + s^2 \implies 64 < 100 + s^2 \implies s^2 > -36\) (always true)
- For the angle opposite to side \(10\) to be acute: \(10^2 < 8^2 + s^2 \implies 100 < 64 + s^2 \implies s^2 > 36 \implies s > 6\)
Thus, for the triangle to be acute, \(s\) must satisfy both \(s < \sqrt{164}\) and \(s > 6\).
- Determining the range for \(s\)
To find the final range for \(s\), we combine the requirements from both the general triangle inequality and the acute triangle conditions:
- From triangle inequality: \(2 < s < 18\)
- From acute conditions: \(6 < s < \sqrt{164} \approx 12.8\)
Taking the intersection of these two intervals, we find that the more restrictive bounds determine the range: \(6 < s < \sqrt{164}\). Comparing this with the provided choices, it matches the second option.
Examples
In architecture, when designing a support structure like a roof truss, the lengths of the members must adhere to specific geometric constraints to maintain the desired shape. If you have two steel beams of 8 meters and 10 meters and want to join them to form an acute-angled triangle for better weight distribution, this calculation ensures you choose a third beam length that keeps the triangle acute. By ensuring the third beam is between 6 and approximately 12.8 meters, the structural integrity and the intended acute angle design are successfully maintained.