Question

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

Explanation:

Step1: Calculate length of LM

Use distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$\overline{LM} = \sqrt{(3-(-2))^2+(2-4)^2} = \sqrt{5^2+(-2)^2} = \sqrt{25+4} = \sqrt{29}$

Step2: Calculate length of MN

Apply distance formula to M,N
$\overline{MN} = \sqrt{(1-3)^2+(-3-2)^2} = \sqrt{(-2)^2+(-5)^2} = \sqrt{4+25} = \sqrt{29}$

Step3: Calculate length of LN

Apply distance formula to L,N
$\overline{LN} = \sqrt{(1-(-2))^2+(-3-4)^2} = \sqrt{3^2+(-7)^2} = \sqrt{9+49} = \sqrt{58}$

Step4: Classify triangle by sides

Two sides ($\sqrt{29}$) are equal, so it is isosceles.

Step5: Check for right angle

Use Pythagorean theorem: $(\sqrt{29})^2 + (\sqrt{29})^2 = 29+29=58 = (\sqrt{58})^2$
This satisfies $a^2+b^2=c^2$, so it is right.

Answer:

• isosceles
• right