which set of numbers can represent the side lengths, in inches, of an acute triangle?\n4, 5, 7\n5, 7, 8\n6…

Explanation:

Step1: Recall the acute - triangle inequality

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

Step2: Check option 1 ((a = 4), (b = 5), (c = 7))

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

Step3: Check option 2 ((a = 5), (b = 7), (c = 8))

Calculate (a^{2}+b^{2}=5^{2}+7^{2}=25 + 49 = 74) and (c^{2}=8^{2}=64). Since (74>64), it could be an acute - triangle.

Step4: Check option 3 ((a = 6), (b = 7), (c = 10))

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

Step5: Check option 4 ((a = 7), (b = 9), (c = 12))

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

Answer:

5, 7, 8