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: Triangle inequality

For any triangle with sides \(a = 8\), \(b = 10\), and \(c=s\), the triangle - inequality theorem states that \(|a - b|\lt s\lt a + b\). So \(|8 - 10|\lt s\lt8 + 10\), which simplifies to \(2\lt s\lt18\). But since it's an acute triangle, we use the Pythagorean - related inequalities.
Let \(a = 8\), \(b = 10\), \(c=s\).
If \(c\) is the longest side (\(c\geq10\)), then \(a^{2}+b^{2}>c^{2}\). Substituting \(a = 8\) and \(b = 10\), we get \(8^{2}+10^{2}>s^{2}\), \(64 + 100>s^{2}\), \(s^{2}<164\), \(s<\sqrt{164}\approx12.8\).
If \(b = 10\) is the longest side (\(s\leq10\)), then \(a^{2}+s^{2}>b^{2}\). Substituting \(a = 8\) and \(b = 10\), we get \(8^{2}+s^{2}>10^{2}\), \(s^{2}>100 - 64\), \(s^{2}>36\), \(s>6\).

Answer:

\(6\lt s\lt12.8\)