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:

A parallelogram is a rectangle if one of its angles is a right angle (i.e., adjacent sides are perpendicular). We analyze each option:

1. Calculating side lengths only confirms it's a parallelogram, not a rectangle.
2. Parallel sides having the same slope is a property of all parallelograms, not unique to rectangles.
3. $\overline{LM} \parallel \overline{PN}$ is already a defining property of parallelograms.
4. If adjacent sides $\overline{LP} \perp \overline{PN}$, this means one angle of the parallelogram is 90°, which proves it is a rectangle.

Answer:

D. $\overline{LP} \perp \overline{PN}$