Question

which set of numbers can represent the side lengths, in inches, of an acute triangle? 4, 5, 7 5, 7, 8 6, 7, 10 7, 9, 12

Explanation:

Step1: Recall the acute - triangle condition

For a triangle with side lengths \(a\), \(b\), \(c\) (\(c\) is the longest side), the triangle is acute if \(a^{2}+b^{2}>c^{2}\)

Step2: Check each option

• Option 1: \(a = 4\), \(b = 5\), \(c = 7\)

\(a^{2}+b^{2}=4^{2}+5^{2}=16 + 25=41\), \(c^{2}=7^{2}=49\)
Since \(41<49\), this is not an acute triangle.

• Option 2: \(a = 5\), \(b = 7\), \(c = 8\)

\(a^{2}+b^{2}=5^{2}+7^{2}=25 + 49 = 74\), \(c^{2}=8^{2}=64\)
Since \(74>64\), this is an acute triangle.

• Option 3: \(a = 6\), \(b = 7\), \(c = 10\)

\(a^{2}+b^{2}=6^{2}+7^{2}=36+49 = 85\), \(c^{2}=10^{2}=100\)
Since \(85<100\), this is not an acute triangle.

• Option 4: \(a = 7\), \(b = 9\), \(c = 12\)

\(a^{2}+b^{2}=7^{2}+9^{2}=49 + 81=130\), \(c^{2}=12^{2}=144\)
Since \(130<144\), this is not an acute triangle.

Answer:

\(5,7,8\)