Question

what is true about triangle lmn? check all that apply. lm ⊥ mn. the triangle is scalene. the triangle is equilateral. the triangle is isosceles. the triangle is a right triangle.

Explanation:

Step1: Check for perpendicular sides

The slope of $\overline{LM}$ is undefined (vertical line $x = - 3$) and the slope of $\overline{MN}$ is $0$ (horizontal line $y=-1$). Since the product of an undefined slope and a zero - slope is undefined in the context of the perpendicular - slope relationship ($m_1\times m_2=- 1$ for non - vertical and non - horizontal perpendicular lines), but geometrically a vertical and a horizontal line are perpendicular. So $\overline{LM}\perp\overline{MN}$.

Step2: Check side lengths

The length of $\overline{LM}=4 - (-1)=5$ (vertical distance, difference in $y$ - coordinates with same $x$ - coordinate). The length of $\overline{MN}=2-(-3)=5$ (horizontal distance, difference in $x$ - coordinates with same $y$ - coordinate). The length of $\overline{LN}=\sqrt{(2 + 3)^2+( - 1 - 4)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}$ using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Since two sides ($\overline{LM}$ and $\overline{MN}$) have the same length, the triangle is isosceles. Also, since it has a right angle (from $\overline{LM}\perp\overline{MN}$), it is a right triangle. It is not scalene (because two sides are equal) and not equilateral (because all three sides are not equal).

Answer:

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