Which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.? acute, because 10^2+12^2gt 15^2 acute, because 12^2+15^2gt 10^2 obtuse, because 10^2+12^2gt 15^2 obtuse, because 12^2+15^2gt 10^2

To classify the triangle based on its side lengths, we can use the Pythagorean Theorem and its converse. For a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the longest side):<br /><br />- If \(a^2 + b^2 = c^2\), the triangle is a right triangle.<br />- If \(a^2 + b^2 > c^2\), the triangle is an acute triangle.<br />- If \(a^2 + b^2 < c^2\), the triangle is an obtuse triangle.<br /><br />Given the side lengths 10 in., 12 in., and 15 in., we identify 15 in. as the longest side (\(c\)).<br /><br />Calculate \(10^2 + 12^2\) and compare it to \(15^2\):<br /><br />\[ <br />10^2 + 12^2 = 100 + 144 = 244 <br />\]<br /><br />\[ <br />15^2 = 225 <br />\]<br /><br />Since \(244 > 225\), we have:<br /><br />\[ <br />10^2 + 12^2 > 15^2 <br />\]<br /><br />This indicates that the triangle is an acute triangle.<br /><br />Therefore, the correct classification is:<br /><br />**acute, because \(10^2 + 12^2 > 15^2\)**.