Question

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

Explanation:

Step1: Calculate side - lengths using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $LM$ with $L(-2,4)$ and $M(3,2)$:
$LM=\sqrt{(3 + 2)^2+(2 - 4)^2}=\sqrt{25 + 4}=\sqrt{29}$
For side $MN$ with $M(3,2)$ and $N(1,-3)$:
$MN=\sqrt{(1 - 3)^2+(-3 - 2)^2}=\sqrt{4 + 25}=\sqrt{29}$
For side $LN$ with $L(-2,4)$ and $N(1,-3)$:
$LN=\sqrt{(1 + 2)^2+(-3 - 4)^2}=\sqrt{9 + 49}=\sqrt{58}$

Step2: Analyze the type of triangle based on side - lengths

Since $LM = MN=\sqrt{29}$ and $LN=\sqrt{58}$, the triangle is isosceles.
Also, check the angle - type using the Pythagorean theorem. $(LM)^2+(MN)^2=29 + 29=58=(LN)^2$, so the triangle is right - angled.

Answer:

B. isosceles, F. right