Question

the sides of a triangle have lengths 8, 15, and 16. what kind of triangle is it? acute right obtuse

Explanation:

Step1: Recall the Pythagorean - related rules

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}Let \(a = 8\), \(b = 15\), and \(c = 16\).

Step2: Calculate \(a^{2}+b^{2}\) and \(c^{2}\)

Calculate \(a^{2}+b^{2}\): \(a^{2}=8^{2}=64\), \(b^{2}=15^{2}=225\), so \(a^{2}+b^{2}=64 + 225=289\).
Calculate \(c^{2}\): \(c^{2}=16^{2}=256\).

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

Since \(a^{2}+b^{2}=289\) and \(c^{2}=256\), and \(289>256\) (i.e., \(a^{2}+b^{2}>c^{2}\)).

Answer:

acute