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: Apply the property of an acute triangle

For an acute triangle with sides \(a\), \(b\), \(c\) (\(c\) is the longest side), \(a^{2}+b^{2}>c^{2}\). Let \(a = 5\), \(b = 8\), and \(c\) be the unknown side.

Step2: Check each option

• Option A: If \(c = 8\), \(5^{2}+8^{2}=25 + 64=89\), \(8^{2}=64\), \(89>64\). But we want the greatest possible value.
• Option B: If \(c = 9\), \(5^{2}+8^{2}=25 + 64 = 89\), \(9^{2}=81\), \(89>81\).
• Option C: If \(c = 12\), \(5^{2}+8^{2}=25+64 = 89\), \(12^{2}=144\), \(89<144\) (violates the acute - triangle property \(a^{2}+b^{2}>c^{2}\)).
• Option D: If \(c = 13\), \(5^{2}+8^{2}=25 + 64=89\), \(13^{2}=169\), \(89<169\) (violates the acute - triangle property \(a^{2}+b^{2}>c^{2}\)).

Answer:

B. 9 inches