Question

the sides of a triangle have lengths 10, 12, and 15. 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}

Step2: Identify \(a\), \(b\), and \(c\)

Let \(a = 10\), \(b = 12\), and \(c = 15\).

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

\(a^{2}=10^{2}=100\), \(b^{2}=12^{2}=144\), so \(a^{2}+b^{2}=100 + 144=244\). And \(c^{2}=15^{2}=225\).

Step4: Compare \(a^{2}+b^{2}\) and \(c^{2}\)

Since \(a^{2}+b^{2}=244\) and \(c^{2}=225\), and \(244>225\) (i.e., \(a^{2}+b^{2}>c^{2}\)).

Answer:

acute