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 theorem

For any triangle with sides \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Here \(a = 8\) and \(b=10\), so \(|10 - 8|\lt s\lt10 + 8\), which simplifies to \(2\lt s\lt18\). But since it's an acute - triangle, we use the Pythagorean inequality.

Step2: Acute - triangle Pythagorean inequalities

Case 1: Let \(s\) be the longest side (\(s\geq10\)). By the Pythagorean inequality for acute triangles \(a^{2}+b^{2}>c^{2}\). Substituting \(a = 8\), \(b = 10\), and \(c=s\), we get \(8^{2}+10^{2}>s^{2}\), i.e., \(s^{2}<64 + 100=164\), so \(s<\sqrt{164}\approx12.8\).
Case 2: Let \(10\) be the longest side (\(s<10\)). Then \(s^{2}+8^{2}>10^{2}\), \(s^{2}>100 - 64 = 36\), so \(s>6\).

Combining the two cases from the acute - triangle condition with the triangle - inequality, we get \(6\lt s\lt12.8\).

Answer:

\(6\lt s\lt12.8\) (the second option)