Answer:
C) Both pairs of opposite sides are parallel because mPN = mLM = (1/2) and mPL = mMN = - 2.
Explanation:
The corners of a rectangle are all right angles. The two sets of sides are parallel. In this example:
PL ║ MN and
LM ║ NP
To prove that the corners are all right angles, we could compre the slopes of the two lines that form each corner. They will be perpendicular if they are 90°.
In this example:
PL ⊥ LM and
MN ⊥ NP
So one approach Sherry may have taken is to compare the slopes of all four lines. To prove the object is a rectangle she could show that:
1. The sets of parallel lines have equal slopes, and
2. The sets of perpendicular lines have slopes that are the negative inverse of each other. (e.g., if a line has slope 5, a perpendicular line will have slope -(1/5))
To prove these points, Sherry probably used a spreadsheet to calculate each line's slopes. See the attached spreadsheet for how she may have set up the calculations. The slopes are the Rise/Run for each line. Rise is the change in y and x is the change in x between the two points.
Note the green cells. These are the slopes. Sherry found that:
a) PL ║ MN and LM ║ NP, since PL and MN both have slopes of -2; and LM and NP both have slopes of -0.5
b) PL ⊥ LM and MN ⊥ NP, since PL and LM and MN and NP both have slopes that are the negative inverse of each other (-2 and -(1/-2) or 0.5)
These are the two conditions Sherry originally established as proof of a rectangle.
Without having checked whether any of the other answer options are viable options, Sherry will likely have seen, and done, enough to have chosen option C as proof that quadrilateral PLMN is a rectangle.