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 triangle inequality and acute triangle conditions

For a triangle with sides \(a\), \(b\), \(c\) (where \(c\) is the longest side), the triangle inequality states \(a + b>c\). For an acute triangle, the Pythagorean inequality for the longest side \(c\) (opposite the largest angle) is \(a^{2}+b^{2}>c^{2}\) (since the angle is acute, the square of the side opposite is less than the sum of the squares of the other two sides). Also, from triangle inequality, \(c < a + b\). Here, \(a = 3\), \(b = 7\), so \(c<3 + 7=10\). For the triangle to be acute, \(3^{2}+7^{2}>c^{2}\), so \(9 + 49>c^{2}\), \(58>c^{2}\), so \(c < \sqrt{58}\approx7.62\). But also, since \(c\) is the longest side, \(c>7\) (because \(7\) is one of the sides and \(c\) must be longer than or equal to the longer of the two given sides, here \(7\) is longer than \(3\), so \(c>7\) to be the longest side).

Step2: Combine the conditions

So the length of the longest side \(c\) must satisfy \(7 < c<\sqrt{58}\) (since \(c\) is the longest side, \(c > 7\), and for acute, \(c<\sqrt{58}\), and from triangle inequality \(c < 10\), but \(\sqrt{58}\approx7.62<10\), so the stricter upper bound is \(\sqrt{58}\)).

Answer:

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