Question

ramon wants to make an acute triangle with three pieces of wood. so far, he has cut wood lengths of 7 inches and 3 inches. he still needs to cut the longest side. what length must the longest side be in order for the triangle to be acute? exactly $sqrt{58}$ inches greater than $sqrt{58}$ inches but less than 10 inches less than $sqrt{58}$ inches but greater than 7 inches not enough information given

Explanation:

Step1: Recall acute - triangle inequality

For a triangle with side lengths \(a\), \(b\), and \(c\) (\(c\) being the longest side), the triangle is acute if \(a^{2}+b^{2}>c^{2}\). Here \(a = 3\) and \(b = 7\).

Step2: Calculate \(a^{2}+b^{2}\)

\(a^{2}+b^{2}=3^{2}+7^{2}=9 + 49=58\). So \(c^{2}<58\), then \(c<\sqrt{58}\).

Step3: Consider triangle - side relationship

Also, for a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Since \(c\) is the longest side and \(a = 3\), \(b = 7\), \(c>7\) (because if \(c\leq7\), it won't be the longest side).

Answer:

C. less than \(\sqrt{58}\) inches but greater than 7 inches