Question

which best explains whether a triangle with side lengths 2 in., 5 in., and 4 in. is an acute triangle? the triangle is acute because 2 + 4 > 5. the triangle is acute because 2² + 5² > 4². the triangle is not acute because 2² < 4² + 5². the triangle is not acute because 2² + 4² < 5².

Explanation:

Step1: Recall the acute - triangle inequality

For a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), the triangle is acute if \(a^{2}+b^{2}>c^{2}\), right - angled if \(a^{2}+b^{2}=c^{2}\), and obtuse if \(a^{2}+b^{2}In the triangle with side lengths \(a = 2\), \(b = 4\), and \(c = 5\) (since \(5\) is the longest side).

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

Calculate \(a^{2}+b^{2}\): \(2^{2}+4^{2}=4 + 16=20\).
Calculate \(c^{2}\): \(5^{2}=25\).
Since \(2^{2}+4^{2}=20<25 = 5^{2}\), the triangle is not acute.

Answer:

The triangle is not acute because \(2^{2}+4^{2}<5^{2}\).