Question

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

Explanation:

Step1: Calculate the lengths of the sides

Use 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 \(LM=\sqrt{29}\), \(MN=\sqrt{29}\), \(LN=\sqrt{58}\), \(LM = MN\), so it is isosceles.
• Also, check if it is right - angled:

\(LM^{2}+MN^{2}=29+29 = 58\), \(LN^{2}=58\)
By Pythagoras theorem \(a^{2}+b^{2}=c^{2}\) (where \(a = LM\), \(b = MN\), \(c = LN\)), so it is right - angled.

Answer:

right, isosceles