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?
options:
2 < s < 18
6 < s < 12.8
s < 2 or s > 18
s < 6 or s > 12.8

Explanation:

Step1: Recall Triangle Inequality

For any triangle, the sum of two sides must be greater than the third side. So, \(|10 - 8| < s < 10 + 8\), which simplifies to \(2 < s < 18\). But since it's an acute triangle, we need to use the Pythagorean inequality for acute triangles.

Step2: Case 1: \(s\) is the longest side

If \(s\) is the longest side (\(s\geq10\)), then \(s^{2}<8^{2}+10^{2}\). Calculate \(8^{2}+10^{2}=64 + 100 = 164\), so \(s<\sqrt{164}\approx12.8\). Also, from triangle inequality, \(s > 10 - 8 = 2\), but since \(s\geq10\), this gives \(10\leq s<12.8\).

Step3: Case 2: \(10\) is the longest side

If \(10\) is the longest side (\(s<10\)), then \(10^{2}<8^{2}+s^{2}\). Rearranging, \(s^{2}>100 - 64 = 36\), so \(s > 6\). From triangle inequality, \(s>2\) and \(s<10\), so \(6 < s<10\).

Step4: Combine Cases

Combining both cases, for \(s<10\): \(6 < s<10\); for \(s\geq10\): \(10\leq s<12.8\). So overall, \(6 < s<12.8\).

Answer:

\(6 < s < 12.8\) (the second option: \(6 < s < 12.8\))