Answer:
rectangle
Step-by-step explanation:
You want to know the name of the polygon with vertices D(3, 1), E(1, 5), F(9, 9), and G(11, 5).
Diagonal length
It can be informative to look at the diagonal vectors DF and EG.
DF = F -D = (9, 9) -(3, 1) = (6, 8)
EG = G -E = (11, 5) -(1, 5) = (10, 0)
The lengths of these are found by the Pythagorean theorem:
DF = √(6²+8²) = √(36 +64) = √100 = 10
EG = √(10²+0²) = 10
The diagonals are the same length.
Diagonal midpoints
The midpoints of the diagonals are ...
midpoint DF = (D +F)/2 = ((9, 9) +(3, 1))/2 = (12, 10)/2 = (6, 5)
midpoint EG = (E +G)/2 = ((11, 5) +(1, 5))/2 = (12, 10)/2 = (6, 5)
The midpoints of the diagonals are the same point. This means the figure is a parallelogram.
A quadrilateral that has congruent diagonals that bisect each other is a rectangle.
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Additional comment
Looking at the graph, we see the slopes of the line segments are 1/2 and -2, opposite reciprocals. This means they are perpendicular, another indication the figure is a rectangle.
The diagonal vectors do not have opposite reciprocal slope, so this is not a rhombus or square or kite.
