The quadrilateral with vertices at (4,0), (8,0), (4,-4), and (8,-4) is a rhombus. All four sides have equal lengths, and based on the given information, it cannot be conclusively identified as a rectangle or square.
The correct answer is C. Rhombus.
To determine the type of quadrilateral formed by the vertices (4,0), (8,0), (4,-4), and (8,-4), we can use the Distance Formula to calculate the lengths of the sides and examine the relationships between these lengths.
The Distance Formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Calculating the distances between adjacent vertices:
Distance between (4,0) and (8,0):
d1 = sqrt((8 - 4)^2 + (0 - 0)^2) = 4
Distance between (8,0) and (8,-4):
d2 = sqrt((8 - 8)^2 + (0 - (-4))^2) = 4
Distance between (8,-4) and (4,-4):
d3 = sqrt((4 - 8)^2 + ((-4) - (-4))^2) = 4
Distance between (4,-4) and (4,0):
d4 = sqrt((4 - 4)^2 + ((-4) - 0)^2) = 4
All four sides have equal lengths (4), which implies that the quadrilateral is a rhombus. A rhombus is a quadrilateral with all sides of equal length.
Now, to further classify the rhombus, we consider the angles between adjacent sides. If the angles are right angles, the rhombus is also a rectangle. If all angles are right angles and all sides are of equal length, then it is a square.
In this case, without additional information about the angles, we can conclude that the quadrilateral is a rhombus (Option C) based on the equal side lengths.