Answer:
Parallelogram
Explanation:
You want the name of the figure defined by vertices I(2, -3), J(0, 0), K(4, 6) and L(6, 3).
Parallelogram
A quick check will tell if the figure is a parallelogram. The diagonals will have the same midpoint.
(I +K)/2 = (J +L)/2
I +K = J +L . . . . . . . . multiply by 2 to make the math easier
(2, -3) +(4, 6) = (0, 0) +(6, 3) . . . . . substitute the given points
(6, 3) = (6, 3) . . . . . . . . . . . . . . the sums are equal
The figure is a parallelogram.
Diagonals
A parallelogram can be a rectangle, square, or rhombus. We can determine if it is a square or rectangle by looking at the lengths of the diagonals. First, we need to find the differences of their endpoints.
K -I = (4, 6) -(2, -3) = (4 -2, 6 +3) = (2, 9)
L -J = (6, 3) -(0, 0) = (6, 3)
The lengths will be the root of the sum of the squares. We can compare the lengths by comparing the sum of the squares (without taking the root):
2² +9² = 4 +81 = 85
6² +3² = 36 +9 = 45
The diagonals are different lengths, so the figure is not a rectangle.
Right angles
We can make another check to see if the figure is a rhombus. In a rhombus, the diagonals are perpendicular. The "dot product" of the differences we just found will be zero in that case.
(2, 9)•(6, 3) = (2)(6) +(9)(3) = 12 +27 = 39 ≠ 0
The diagonals are not perpendicular, so the parallelogram is not a rhombus.
The most specific name of the figure is Parallelogram.
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Additional comment
A lot is answered by graphing the points. The slopes and lengths of the lines can be found by counting grid squares between points horizontally and vertically. It is easy to see the diagonals cross at their midpoints, and that they are different lengths.
You can figure segment lengths and slopes for showing opposite sides are the same lengths and parallel, but we find the diagonal midpoint computation to be much easier for showing the figure is a parallelogram.
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