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 rule for acute triangles

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

Step2: Check option 4, 5, 7

Longest side \(c = 7\). Calculate \(4^{2}+5^{2}=16 + 25=41\), \(7^{2}=49\). Since \(41<49\), this is an obtuse triangle.

Step3: Check option 5, 7, 8

Longest side \(c = 8\). Calculate \(5^{2}+7^{2}=25 + 49 = 74\), \(8^{2}=64\). Since \(74>64\), this is an acute triangle. Let's check other options to be sure.

Step4: Check option 6, 7, 10

Longest side \(c = 10\). Calculate \(6^{2}+7^{2}=36+49 = 85\), \(10^{2}=100\). Since \(85<100\), this is an obtuse triangle.

Step5: Check option 7, 9, 12

Longest side \(c = 12\). Calculate \(7^{2}+9^{2}=49 + 81=130\), \(12^{2}=144\). Since \(130<144\), this is an obtuse triangle.

Answer:

B. 5, 7, 8