Question

lmnp is a parallelogram.

what additional information would prove that lmnp is a rectangle?

• the length of \\(\overline{lm}\\) is \\(\sqrt{45}\\) and the length of \\(\overline{mn}\\) is \\(\sqrt{5}\\).
• the slope of \\(\overline{lp}\\) and \\(\overline{mn}\\) is \\(-2\\).
• \\(\overline{lm} \parallel \overline{pn}\\)
• \\(\overline{lp} \perp \overline{pn}\\)

Explanation:

Identify the given conditions

Using the Properties of Parallelograms knowledge point

$$ \text{Given: } LMNP \text{ is a parallelogram.} $$

Recall the definition of a rectangle

Using the Properties of Rectangles knowledge point

$$ \text{A rectangle is a parallelogram with four right angles.} $$

Analyze the geometric requirements

To prove a parallelogram is a rectangle, we must show it has at least one right angle (perpendicular consecutive sides).

$$ \overline{LP} \perp \overline{PN} \implies \angle LPN = 90^\circ $$

Evaluate the given multiple-choice options

• Option 1: Side lengths do not guarantee right angles.
• Option 2: Slopes of opposite sides being equal is already true for any parallelogram.
• Option 3: Parallel opposite sides are already true for any parallelogram.
• Option 4: Perpendicular consecutive sides \(\overline{LP} \perp \overline{PN}\) guarantees a right angle.

Answer:

• (A) The length of \(LM\) is \(\sqrt{45}\) and the length of \(MN\) is \(\sqrt{5}\).
• (B) The slope of \(\overline{LP}\) and \(\overline{MN}\) is \(-2\).
• (C) \(\overline{LM} \parallel \overline{PN}\)
• (D) \(\overline{LP} \perp \overline{PN}\) (Correct answer)