State whether the following statement is true or false. Justify your answer.
The points $A(-1, 0)$,$B(3, 1)$,$C(2, 2)$,and $D(-2, 1)$ are the vertices of a parallelogram.
The points $A(-1, 0)$,$B(3, 1)$,$C(2, 2)$,and $D(-2, 1)$ are the vertices of a parallelogram.
(A) The statement is true.
$A$ quadrilateral is a parallelogram if its diagonals bisect each other,which means they share the same midpoint.
$1$. Midpoint of diagonal $AC$:
Midpoint $= (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) = (\frac{-1 + 2}{2}, \frac{0 + 2}{2}) = (\frac{1}{2}, 1)$.
$2$. Midpoint of diagonal $BD$:
Midpoint $= (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) = (\frac{3 - 2}{2}, \frac{1 + 1}{2}) = (\frac{1}{2}, 1)$.
Since the midpoints of both diagonals $AC$ and $BD$ are the same $(\frac{1}{2}, 1)$,the diagonals bisect each other. Therefore,the points $A, B, C,$ and $D$ form a parallelogram.
$A$ quadrilateral is a parallelogram if its diagonals bisect each other,which means they share the same midpoint.
$1$. Midpoint of diagonal $AC$:
Midpoint $= (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) = (\frac{-1 + 2}{2}, \frac{0 + 2}{2}) = (\frac{1}{2}, 1)$.
$2$. Midpoint of diagonal $BD$:
Midpoint $= (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) = (\frac{3 - 2}{2}, \frac{1 + 1}{2}) = (\frac{1}{2}, 1)$.
Since the midpoints of both diagonals $AC$ and $BD$ are the same $(\frac{1}{2}, 1)$,the diagonals bisect each other. Therefore,the points $A, B, C,$ and $D$ form a parallelogram.
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