Question

what is true about triangle lmn? check all that apply. \\(\overline{lm} \perp \overline{mn}\\). the triangle is scalene. the triangle is equilateral. the triangle is isosceles. the triangle is a right triangle.

Explanation:

Step1: Identify coordinates of vertices

Coordinates: $L(-3, 4)$, $M(-3, -1)$, $N(2, -1)$

Step2: Calculate side lengths

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:

• $LM$: $\sqrt{(-3+3)^2+(-1-4)^2}=\sqrt{0+25}=5$
• $MN$: $\sqrt{(2+3)^2+(-1+1)^2}=\sqrt{25+0}=5$
• $LN$: $\sqrt{(2+3)^2+(-1-4)^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}$

Step3: Check perpendicularity

Slope of $LM$: $\frac{-1-4}{-3+3}$ (undefined, vertical line)
Slope of $MN$: $\frac{-1+1}{2+3}=0$ (horizontal line)
Vertical and horizontal lines are perpendicular, so $\overline{LM} \perp \overline{MN}$.

Step4: Classify triangle

Two sides ($LM=MN=5$) are equal, so it is isosceles.
It has a right angle ($\angle M$), so it is a right triangle.

Answer:

$\overline{LM} \perp \overline{MN}$.
The triangle is isosceles.
The triangle is a right triangle.