The problem involves determining the type of quadrilateral formed by the points $L(-7, -4)$, $M(-1, -8)$, $N(1, -5)$, and $O(-5, -1)$, using the slopes and lengths of the sides.

Let's break down the information:

1. Slopes of the sides:

   • Slope of $LM = -\frac{2}{3}$
   • Slope of $MN = \frac{3}{2}$
   • Slope of $NO = -\frac{2}{3}$
   • Slope of $OL = \frac{3}{2}$

2. Lengths of the sides:

   • Length of $LM = 2\sqrt{13}$
   • Length of $MN = \sqrt{13}$
   • Length of $NO = 2\sqrt{13}$
   • Length of $OL = \sqrt{13}$

Identifying the quadrilateral:

• The opposite sides $LM$ and $NO$ have equal slopes ($-\frac{2}{3}$) and equal lengths ($2\sqrt{13}$).
• The opposite sides $MN$ and $OL$ also have equal slopes ($\frac{3}{2}$) and equal lengths ($\sqrt{13}$).

When opposite sides are both equal in length and parallel (having the same slope), the quadrilateral is classified as a parallelogram.

Therefore, the quadrilateral $LMNO$ is a parallelogram.

Would you like more detailed steps or clarifications?

Related Questions:

1. How do you calculate the slope between two points?
2. How is the distance formula derived from the Pythagorean theorem?
3. What are the characteristics of different types of quadrilaterals (e.g., rhombus, rectangle, square)?
4. How do parallel lines relate to the slopes of line segments?
5. How do you determine if a quadrilateral is a rectangle or a rhombus?

Tip:

To identify a quadrilateral, always check the slopes and lengths of opposite sides. If both are equal, it's likely a parallelogram!