Question

the sides of a triangle have lengths 27, 45, and 48. 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 \(a = 27\), \(b = 45\), and \(c = 48\).

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

\(a^{2}=27^{2}=729\), \(b^{2}=45^{2}=2025\), so \(a^{2}+b^{2}=729 + 2025=2754\). And \(c^{2}=48^{2}=2304\).

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

Since \(a^{2}+b^{2}=2754\) and \(c^{2}=2304\), and \(2754>2304\) (i.e., \(a^{2}+b^{2}>c^{2}\)).

Answer:

acute