Question 1116014: The quadrilateral ABCD has vertices A(1,0), B(3,1), C(4,3) and D(2,2).
a) Show that the intervals AC and BD bisect each other.
b) What can be concluded about the type of quadrilateral ABCD?
Answer by greenestamps(13198)   (Show Source):
You can put this solution on YOUR website!
part (a)
Diagonal AC contains the points (1,0) and (4,3); the equation of the line through those two points is y = x-1.
Diagonal BD contains the points (3,1) and (2,2); the equation of the line through those two points is y = -x+4.
The intersection of the two diagonals is the common point on the graphs of y = x-1 and y = -x+4; that point is (2.5,1.5).
The midpoints of AC and BD are also both (2.5,1.5).
The intersection of the two diagonals is the midpoint of both, so the diagonals bisect each other.
part (b)
The fact that the diagonals bisect each other means the quadrilateral is a parallelogram.
In fact, the quadrilateral is a rhombus, because the two diagonals are perpendicular to each other (the product of their slopes is -1).
But we were not asked to show that the diagonals are perpendicular; only knowing that the diagonals bisect each other only tells us that the quadrilateral is a parallelogram.
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