Select all the correct answers. \( \triangle \mathrm{LMN} \) has vertices \( \mathrm{L}(-2,4), \mathrm{M}(3,2) \), and \( \mathrm{N}(1,-3) \). Which describe the triangle? right obtuse acute isosceles equilateral scalene

To analyze triangle \( \triangle \mathrm{LMN} \), we need to determine the lengths of its sides using the distance formula. 1. Calculate the lengths of sides \( LM \), \( MN \), and \( LN \): - \( LM = \sqrt{(3 - (-2))^2 + (2 - 4)^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} \) - \( MN = \sqrt{(1 - 3)^2 + (-3 - 2)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \) - \( LN = \sqrt{(1 - (-2))^2 + (-3 - 4)^2} = \sqrt{3^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58} \) 2. Since \( LM = MN \), triangle \( \triangle \mathrm{LMN} \) is isosceles. To determine whether it's right, obtuse, or acute: - Check the squared distances: \( LM^2 + MN^2 = 29 + 29 = 58 \) and \( LN^2 = 58 \). - The sum of the squares of the two shorter sides equals the square of the longest side, indicating that the triangle is right. Therefore, the triangle \( \triangle \mathrm{LMN} \) is **isosceles** and **right**.