Which set of numbers can represent the side lengths, in inches, of an acute triangle?
To determine if a set of side lengths represents an acute triangle, we must first verify that they form a valid triangle using the triangle inequality. If the side lengths are \(a\), \(b\), and \(c\) with \(c\) being the longest side, the triangle inequality requires \(a + b > c\).
Once the triangle inequality is satisfied, we classify the triangle based on the relationship between the squares of its sides. A triangle is acute if the square of the longest side is less than the sum of the squares of the other two sides:
\[a^2 + b^2 > c^2\]
If \(a^2 + b^2 = c^2\), the triangle is a right triangle.
If \(a^2 + b^2 < c^2\), the triangle is an obtuse triangle.
We analyze each given set of side lengths \((a, b, c)\) by checking the triangle inequality \(a + b > c\) and then comparing \(a^2 + b^2\) with \(c^2\).
After testing each set of side lengths, we found that the set \(5, 7, 8\) is the only one that satisfies the condition \(a^2 + b^2 > c^2\), confirming it is an acute triangle.
The final answer is \(\boxed{5, 7, 8}\).
Understanding the properties of triangles is crucial in fields like architecture and structural engineering. For example, when building a roof truss, engineers must ensure the angles are correct to maintain structural stability. By checking whether a triangle is acute, right, or obtuse using side lengths, designers can ensure the geometry supports the intended loads efficiently. For instance, testing if \(5^2 + 7^2 > 8^2\) (which is \(74 > 64\)) confirms the structural integrity of this particular configuration.