Question

which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.?
acute, because
10² + 12² > 15²
obtuse, because
12² + 15² > 10²
acute, because
12² + 15² > 10²
obtuse, because
10² + 12² > 15²

Explanation:

Step1: Recall the triangle inequality for angles

For a triangle with side lengths \(a\), \(b\), and \(c\) (where \(c\) is the longest side), we use the following:

• If \(a^{2}+b^{2}>c^{2}\), the triangle is acute.
• If \(a^{2}+b^{2}=c^{2}\), the triangle is right - angled.
• If \(a^{2}+b^{2}

In the given triangle, the side lengths are \(10\) in, \(12\) in, and \(15\) in. The longest side is \(c = 15\) in, and \(a=10\) in, \(b = 12\) in.

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

Calculate \(a^{2}+b^{2}\):
\(a^{2}+b^{2}=10^{2}+12^{2}=100 + 144=244\)

Calculate \(c^{2}\):
\(c^{2}=15^{2}=225\)

Since \(244>225\), i.e., \(10^{2}+12^{2}>15^{2}\), the triangle is acute.

Answer:

acute, because \(10^{2}+12^{2}>15^{2}\)