Final answer:
The correct answers are options A and B. To prove LMNP is a parallelogram, show the midpoints of LN and MP are the same (Option A) and that LM equals NP and MN equals LP (Option C). These methods confirm the properties of parallelograms in coordinate geometry.
Step-by-step explanation:
To show that the quadrilateral LMNP on the coordinate plane is a parallelogram, you can follow several methods:
- A. Show that the midpoints of line segments LN and MP are the same. This demonstrates that LN and MP are bisected at their respective midpoints, which implies that LN and MP are parallel and equal in length, two properties of opposite sides in a parallelogram.
- B. Showing that LN = MP also demonstrates that opposite sides of the quadrilateral are congruent, suggesting it could be a parallelogram. However, this alone is not conclusive without additional information about the angles or parallelism of the sides.
- C. Demonstrating that LM = NP and MN = LP proves that both pairs of opposite sides are congruent, which is a defining property of a parallelogram.
- D. Showing that the product of the slopes of line segments LN and MP equals -1 reveals that these segments are perpendicular to each other, which is not a property of a parallelogram. Therefore, this choice does not prove that LMNP is a parallelogram.
Of the given options, A and C are correct. They align with the properties of a parallelogram where opposite sides are equal in length and bisected at their midpoints. The slope calculation pertains to the concept of perpendicular lines, not parallelism. Selecting appropriate strategies to confirm the properties of a parallelogram is vital in coordinate geometry.