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 √58 inches
o greater than √58 inches but less than 10 inches
o less than √58 inches but greater than 7 inches
o not enough information given

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 \(a^{2}+b^{2}>c^{2}\). Here \(a = 3\) and \(b = 7\), so \(a^{2}+b^{2}=3^{2}+7^{2}=9 + 49=58\).

Step2: Determine the range of the longest - side length

Let the length of the longest side be \(c\). For the triangle to be acute, \(c^{2}7\) (since \(c\) is the longest side). So the length of the longest side must be less than \(\sqrt{58}\) inches but greater than 7 inches.

Answer:

less than $\sqrt{58}$ inches but greater than 7 inches