Question

two sides of an acute triangle measure 5 inches and 8 inches. the length of the longest side is unknown. what is the greatest possible whole - number length of the unknown side?
8 inches
9 inches
12 inches
13 inches

Explanation:

Step1: Recall triangle inequality and acute triangle condition

For a triangle with sides \(a\), \(b\), \(c\) (where \(c\) is the longest side), the triangle inequality states \(a + b>c\), and for an acute triangle, \(a^{2}+b^{2}>c^{2}\) (when \(c\) is the longest side). Let the unknown side be \(x\), and assume \(x\) is the longest side (so \(x\geq8\) and \(x\geq5\)).

Step2: Apply the acute triangle inequality

We have \(a = 5\), \(b = 8\), \(c=x\). The inequality is \(5^{2}+8^{2}>x^{2}\), so \(25 + 64>x^{2}\), \(89>x^{2}\), so \(x<\sqrt{89}\approx9.43\). Also, from triangle inequality, \(5 + 8>x\), so \(x < 13\). But since we want the greatest whole - number \(x\) (and \(x\) is the longest side, so \(x\geq8\)), and \(x<9.43\), the greatest whole number less than \(9.43\) is \(9\). We also check the other case where \(8\) is the longest side: then \(x\leq8\), and the greatest whole number in that case would be \(8\), but we want the greatest possible, so we consider \(x\) as the longest side.

Answer:

B. 9 inches