Question

the sides of a triangle have lengths 4, 5, and 7. what kind of triangle is it? acute right obtuse

Explanation:

Step1: Recall the Pythagorean - related inequalities

For a triangle with side lengths \(a\), \(b\), and \(c\) where \(c\) is the longest side: If \(a^{2}+b^{2}=c^{2}\), it's a right - triangle; if \(a^{2}+b^{2}>c^{2}\), it's an acute - triangle; if \(a^{2}+b^{2}

Step2: Identify \(a\), \(b\), and \(c\) and calculate \(a^{2}+b^{2}\) and \(c^{2}\)

Let \(a = 4\), \(b = 5\), and \(c = 7\). Then \(a^{2}=4^{2}=16\), \(b^{2}=5^{2}=25\), and \(c^{2}=7^{2}=49\). Calculate \(a^{2}+b^{2}\): \(a^{2}+b^{2}=16 + 25=41\).

Step3: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(41<49\) (i.e., \(a^{2}+b^{2}

Answer:

obtuse