Question

the sides of a triangle have lengths 5, 6, and 7. what kind of triangle is it? acute right obtuse

Explanation:

Step1: Recall the Pythagorean - related rule

For a triangle with side lengths \(a\), \(b\), and \(c\) (\(c\) 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

Let \(a = 5\), \(b = 6\), and \(c = 7\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\). \(a^{2}=5^{2}=25\), \(b^{2}=6^{2}=36\), so \(a^{2}+b^{2}=25 + 36=61\). And \(c^{2}=7^{2}=49\).

Step3: Compare the values

Since \(a^{2}+b^{2}=61\) and \(c^{2}=49\), and \(61>49\) (i.e., \(a^{2}+b^{2}>c^{2}\)).

Answer:

acute