Question

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

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 $M(3,2)$:
\[

$$\begin{align*} LM&=\sqrt{(3 + 2)^2+(2 - 4)^2}\\ &=\sqrt{25+4}\\ &=\sqrt{29} \end{align*}$$

\]
For side $MN$ with $M(3,2)$ and $N(1,-3)$:
\[

$$\begin{align*} MN&=\sqrt{(1 - 3)^2+(-3 - 2)^2}\\ &=\sqrt{4 + 25}\\ &=\sqrt{29} \end{align*}$$

\]
For side $LN$ with $L(-2,4)$ and $N(1,-3)$:
\[

$$\begin{align*} LN&=\sqrt{(1 + 2)^2+(-3 - 4)^2}\\ &=\sqrt{9 + 49}\\ &=\sqrt{58} \end{align*}$$

\]

Step2: Determine triangle type by side - lengths and angles

Since $LM = MN=\sqrt{29}$ and $LN=\sqrt{58}$, the triangle is isosceles.
To check the angle type, we can use the Pythagorean theorem's converse.
\[LM^{2}+MN^{2}=29 + 29=58=LN^{2}\]
So, $\angle M = 90^{\circ}$, and the triangle is right - angled.

Answer:

isosceles, right