Figures in the Coordinate Plane LMNP is a parallelogram. What additional information would prove that LMNP is a rectangle? The length of LM is
and the length of MN is
. The slope of LP and MN is -2. LM \parallel PN LP \perp P
The Correct Answer and Explanation is:
To determine which additional information would prove that LMNP is a rectangle, let’s first review what makes a parallelogram a rectangle:
A parallelogram becomes a rectangle when:
- All angles are right angles, or
- Adjacent sides are perpendicular, or
- Diagonals are equal and bisect each other at right angles, or
- One angle is a right angle (which implies all others are right angles due to parallel opposite sides).
Now, in the given image, LMNP is already a parallelogram, so:
- Opposite sides are parallel.
- Opposite sides are equal in length.
To prove that this parallelogram is a rectangle, we need evidence of right angles.
The correct answer:
“LP ⊥ PN” (LP is perpendicular to PN)
Explanation
In coordinate geometry, a rectangle is a special type of parallelogram where adjacent sides meet at right angles (90°). The given figure, LMNP, is a parallelogram, which means opposite sides are parallel and equal in length, but this alone does not confirm the shape is a rectangle.
To confirm LMNP is a rectangle, we need to prove that at least one angle is a right angle. In coordinate geometry, this can be done by verifying the perpendicularity of adjacent sides using slopes. If the product of the slopes of two adjacent sides is -1, they are perpendicular.
Among the answer choices:
- Length of sides only tells us about the size, not the shape.
- Same slope for LP and MN only shows they are parallel, not perpendicular.
- LM ∥ PN again confirms parallelism, not perpendicularity.
However, “LP ⊥ PN” (LP is perpendicular to PN) tells us that two adjacent sides form a right angle, which is exactly what’s needed to show that the parallelogram is a rectangle. If one interior angle is a right angle in a parallelogram, then all interior angles must be right angles, confirming the shape is a rectangle.
Therefore, the additional information that would prove LMNP is a rectangle is:
✅ LP ⊥ PN (correct answer)
