Question

lmnp is a parallelogram. what additional information would prove that lmnp is a rectangle? the length of \\( \overline { l m } \\) is \\( \sqrt { 45 } \\) and the length of \\( \overline { m n } \\) is \\( \sqrt { 5 } \\). the slope of \\( \overline { l p } \\) and \\( \overline { m n } \\) is \\( - 2 \\). \\( \overline { l m } \parallel \overline { p n } \\) \\( \overline { l p } \perp \overline { p n } \\)

Explanation:

Step1: Recall the properties of a rectangle

A rectangle is a parallelogram with one right - angle. In coordinate geometry, if two adjacent sides of a parallelogram are perpendicular (the product of their slopes is - 1), then the parallelogram is a rectangle.

Step2: Analyze each option

• Option 1: Knowing the lengths of two adjacent sides (\(\overline{LM}\) and \(\overline{MN}\)) only gives information about the side - lengths. In a parallelogram, opposite sides are equal. But this does not guarantee a right - angle.
• Option 2: If the slope of \(\overline{LP}\) and \(\overline{MN}\) is - 2. In a parallelogram \(LMNP\), \(\overline{LP}\parallel\overline{MN}\) (property of a parallelogram). Equal slopes confirm parallelism (which we already know from the parallelogram property), but not perpendicularity.
• Option 3: \(\overline{LM}\parallel\overline{PN}\) is a property of a parallelogram (opposite sides of a parallelogram are parallel). This does not add any new information to prove it is a rectangle.
• Option 4: If \(\overline{LP}\perp\overline{PN}\), since \(LMNP\) is a parallelogram (given), and in a parallelogram adjacent sides are \(\overline{LP}\) and \(\overline{PN}\). If two adjacent sides of a parallelogram are perpendicular (the product of their slopes is - 1, which is equivalent to the geometric definition of perpendicular lines in coordinate geometry), then the parallelogram is a rectangle.

Answer:

\(\overline{LP}\perp\overline{PN}\)