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?
o exactly \\( \sqrt { 58 } \\) inches
greater than \\( \sqrt { 58 } \\) inches but less than 10 inches
o less than \\( \sqrt { 58 } \\) inches but greater than 7 inches
o not enough information given

Explanation:

Step1: Triangle - side length relationship

For a triangle with side lengths \(a\), \(b\), \(c\) (\(c\) is the longest side), the triangle - inequality theorem states that \(a + b>c\). Here \(a = 3\), \(b = 7\), so \(3 + 7>c\), i.e., \(c<10\).

Step2: Acute - triangle condition

For an acute triangle with side lengths \(a\), \(b\), \(c\) (\(c\) is the longest side), the Pythagorean - like condition \(a^{2}+b^{2}>c^{2}\) must hold. Substitute \(a = 3\) and \(b = 7\) into \(a^{2}+b^{2}\): \(3^{2}+7^{2}=9 + 49=58\). So \(c^{2}<58\), then \(c<\sqrt{58}\). Also, since \(c\) is the longest side and \(7>3\), \(c>7\).

Answer:

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