Final answer:
To classify triangle LMN, one must calculate the lengths of its sides using the distance formula, then assess the relationships between the sides to determine its type (right, isosceles, obtuse, acute, equilateral, or scalene).
Step-by-step explanation:
The question involves determining the type of triangle given three vertices: I(-2,4), M(3,2), and N(1,-3). To classify the triangle as right, isosceles, obtuse, acute, equilateral, or scalene, we analyze the lengths of its sides, which can be calculated using the distance formula. Then, we can determine the type of angles the triangle has (right, obtuse, or acute) by applying the Pythagorean theorem or by comparing the squared lengths of the sides.
First, calculate the distance between each pair of vertices to get the lengths of sides IM, IN, and MN:
- IM = √((3 - (-2))^2 + (2 - 4)^2)
- IN = √((1 - (-2))^2 + (-3 - 4)^2)
- MN = √((3 - 1)^2 + (2 - (-3))^2)
Once we have the lengths, we can check if any two sides are equal to determine if it's an isosceles triangle. If all three sides are different, it's scalene. If the square of the longest side is equal to the sum of the squares of the other two sides, it's a right triangle. If the square of the longest side is greater than the sum of the squares of the other two sides, it's an obtuse triangle. If it's less, then the triangle is acute. Lastly, if all three sides are equal, the triangle is equilateral.