Final answer:
Figure MNOP is best described as a square. This conclusion is based on the parallel opposite sides with equal slopes, the perpendicular consecutive sides, and the equal lengths of all sides.
Step-by-step explanation:
To determine which geometric shape best describes figure MNOP with vertices M(-3, 5), N(0, 9), O(4,6), and P(1, 2), we must calculate the slopes of the sides and their lengths to identify properties like parallel sides and equal lengths.
First, we calculate the slopes of the opposite sides MN and OP:
For MN: Slope = (9 - 5) / (0 + 3)
= 4 / 3
For OP: Slope = (6 - 2) / (4 - 1)
= 4 / 3
Since both slopes are equal, sides MN and OP are parallel.
Next, we calculate the slopes of the other two sides NO and PM:
For NO: Slope = (6 - 9) / (4 - 0)
= -3 / 4
For PM: Slope = (2 - 5) / (1 + 3)
= -3 / 4
Again, these slopes are equal, indicating that NO and PM are also parallel.
Thus, opposite sides are parallel, making this figure a parallelogram. Consecutive sides are perpendicular, indicating a rectangle or a square. To finalize our conclusion, we must calculate the lengths of the sides to check for equal lengths
Using the distance formula, we find:
Length MN = √[(0 + 3)² + (9 - 5)²]
= 5
Length NO = √[(4 - 0)² + (6 - 9)²]
= 5
Length OP = √[(4 - 1)² + (6 - 2)²]
= 5
Length PM = √[(-3 + 3)² + (5 - 2)²]
= 5
All sides are of equal length, which means the figure is not only a parallelogram and a rectangle but is specifically a square.
Therefore, the most precise name for figure MNOP is a square due to the following reasons:
Each set of opposite sides have slopes of 4/3 and -3/4, making the opposite sides parallel.
Consecutive sides are perpendicular to each other.
The lengths of all sides are equal, measuring exactly 5 units.