Question

lmnp is a parallelogram. what additional information would prove that lmnp is a rectangle? the length of lm is $sqrt{45}$ and the length of mn is $sqrt{5}$. the slope of lp and mn is -2. lm || pn lp ⊥ pn

Explanation:

Step1: Recall the properties of a rectangle

A rectangle is a parallelogram with one right - angle. In a coordinate - plane, if the product of the slopes of two adjacent sides of a parallelogram is - 1, then the angle between them is a right - angle.

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, not about the angles. So, it cannot prove the parallelogram is a rectangle.
• Option 2: The slope of $\overline{LP}$ and $\overline{MN}$ being equal only shows that these two sides are parallel (which is already true for a parallelogram), not that there is a right - angle.
• Option 3: $\overline{LM}\parallel\overline{PN}$ is a property of a parallelogram, not a property that distinguishes a rectangle from a general parallelogram.
• Option 4: If $\overline{LP}\perp\overline{PN}$, then there is a right - angle at $\angle LPN$. Since LMNP is a parallelogram, if one angle is a right - angle, then all angles are right - angles (opposite angles of a parallelogram are equal and adjacent angles are supplementary), and the parallelogram is a rectangle.

Answer:

The additional information that would prove that LMNP is a rectangle is $\overline{LP}\perp\overline{PN}$. So the answer is $\overline{LP}\perp\overline{PN}$.