Question

which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.?

• acute, because (10^2 + 12^2 > 15^2)
• acute, because (12^2 + 15^2 > 10^2)
• obtuse, because (10^2 + 12^2 > 15^2)
• obtuse, because (12^2 + 15^2 > 10^2)

Explanation:

Step1: Recall the triangle inequality for angles

For a triangle with side lengths \(a\), \(b\), \(c\) (where \(c\) is the longest side), we use the Pythagorean theorem extension:

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

Step2: Identify the longest side and apply the formula

Here, the side lengths are \(10\) in, \(12\) in, and \(15\) in. The longest side \(c = 15\) in. The other two sides \(a = 10\) in and \(b = 12\) in.

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

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

Now compare \(a^{2}+b^{2}\) and \(c^{2}\):
Since \(244>225\) (i.e., \(10^{2}+12^{2}>15^{2}\)), the triangle is acute.

Answer:

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