Answer:
- opposite sides are congruent
- opposite sides are parallel
- adjacent sides are not perpendicular
- the figure is a parallelogram
Explanation:
An easy way to determine the nature of the quadrilateral is to look at the lengths and midpoints of the diagonals. Adding and subtracting the coordinates of one end point from the other can tell what you need to know.
The diagonals of a parallelogram are mutual bisectors. The diagonals of a rhombus are perpendicular. The diagonals of a rectangle are congruent.
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diagonal AC
The difference of end points of diagonal AC is ...
C -A = (7, 2) -(-6, 1) = (13, 1)
The sum of the end points of diagonal AC is ...
C +A = (7, 2) +(-6, 1) = (1, 3)
diagonal BD
The difference of end points of diagonal BD is ...
B -D = (0, 7) -(1, -4) = (-1, 11)
The sum of end points of diagonal BD is ...
B +D = (0, 7) +(1, -4) = (1, 3)
comparison of diagonals
midpoints
The two diagonals have the same end point sum, (1, 3), meaning their midpoints are coincident. The figure is a parallelogram.
lengths
The lengths of the diagonals will be the same if the sum of squares of the coordinates of the end-point differences are the same.
Σ(C -A)² = 13² +1² = 170
Σ(B -D)² = (-1)² +11² = 122 ≠ 170 . . . . . not a rectangle
perpendicularity
The diagonals will be perpendicular if the "dot product" of the end-point differences is zero.
x1·x2 +y1·y2 = 13(-1) +1(11) = -2 ≠ 0 . . . . not a rhombus
The most specific name for this quadrilateral is parallelogram.
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Opposite sides of a parallelogram are congruent and parallel.
Adjacent sides of a rectangle are perpendicular. This is not a rectangle, so adjacent sides are not perpendicular.
A parallelogram that is not a rectangle or rhombus is described most specifically as a parallelogram.