two sides of an acute triangle measure 5 inches and 8 inches. the length of the longest side is unknown…

Explanation:

Step1: Recall the acute - triangle inequality

For an acute triangle with side lengths (a), (b), and (c) ((c) being the longest side), (a^{2}+b^{2}>c^{2}). Here (a = 5) and (b = 8). So (5^{2}+8^{2}>c^{2}), which means (25 + 64>c^{2}), or (c^{2}<89).

Step2: Find the upper - bound of (c)

We know that (c) is a whole number. We find the square - root of 89. (\sqrt{89}\approx9.43). Since (c) is a whole number and (c^{2}<89), the greatest whole - number value of (c) is 9. Also, by the triangle inequality theorem (a + b>c), and (5+8 = 13>9), (8 + 9>5) and (5+9>8).

Answer:

9 inches