Question

△lmn has vertices l(-2,4), m(3,2), and n(1,-3). which describe the triangle?
equilateral
acute
isosceles
scalene
right
obtuse

Explanation:

Step1: Calculate the lengths of the sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(LM\):
\(x_1=-2,y_1 = 4,x_2=3,y_2=2\)
\(LM=\sqrt{(3+ 2)^2+(2 - 4)^2}=\sqrt{25 + 4}=\sqrt{29}\)

For \(MN\):
\(x_1=3,y_1 = 2,x_2=1,y_2=-3\)
\(MN=\sqrt{(1 - 3)^2+(-3 - 2)^2}=\sqrt{4+25}=\sqrt{29}\)

For \(LN\):
\(x_1=-2,y_1 = 4,x_2=1,y_2=-3\)
\(LN=\sqrt{(1 + 2)^2+(-3 - 4)^2}=\sqrt{9 + 49}=\sqrt{58}\)

Step2: Check the type of triangle

Since \(LM = MN=\sqrt{29}\), by the definition of an isosceles triangle (a triangle with at least two equal - length sides), the triangle is isosceles.
Also, check for right - triangle using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Let \(a = LM=\sqrt{29},b = MN=\sqrt{29},c = LN=\sqrt{58}\)
\((\sqrt{29})^{2}+(\sqrt{29})^{2}=29 + 29=58=(\sqrt{58})^{2}\)

Answer:

isosceles, right