quadrilateral abcd has the following vertices a(-4,2) b(-1,4) c(2,4) d(-1,-3) Is it a parallelogram, rhombus or neither?
GRAPH IT !!!
if you are loath to graph it, you must
compare the lengths of AB and CD
compare the slopes of AB and CD
If they are the same, then ABCD is a parallelogram.
If, in addition, the lengths of AB and BC are the same, it is a rhombus.
To determine whether the quadrilateral ABCD is a parallelogram, rhombus, or neither, we need to examine its properties.
A parallelogram is a quadrilateral with opposite sides that are parallel. To check for parallel sides, we need to calculate the slopes of each pair of opposite sides and compare them.
Let's calculate the slopes of the sides of the quadrilateral:
Side AB: (x2 - x1) / (y2 - y1) = (-1 - (-4)) / (4 - 2) = 3 / 2 = 1.5
Side CD: (x2 - x1) / (y2 - y1) = (-1 - 2) / (-3 - 4) = -3 / -7 ≈ 0.429
Side BC: (x2 - x1) / (y2 - y1) = (2 - (-1)) / (4 - 4) = 3 / 0, division by zero is undefined, so we cannot calculate the slope.
Side AD: (x2 - x1) / (y2 - y1) = (-4 - (-1)) / (2 - (-3)) = -3 / 5 ≈ -0.6
Comparing the slopes:
- The slopes of sides AB and CD are different, indicating that they are not parallel.
- The slope of side BC cannot be calculated since it is a vertical line, making it parallel to the y-axis (a slope of infinity).
- The slope of side AD is different from the slope of side BC, indicating that they are not parallel.
Since none of the opposite sides are parallel, quadrilateral ABCD is not a parallelogram.
To determine whether it is a rhombus, we also need to check the lengths of the sides. A rhombus is a quadrilateral with all sides of equal length.
Let's calculate the lengths of the sides:
Side AB: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(-1 - (-4))^2 + (4 - 2)^2] = √[9 + 4] = √13 ≈ 3.61
Side BC: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(2 - (-1))^2 + (4 - 4)^2] = √[9 + 0] = √9 = 3
Side CD: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(-1 - 2)^2 + (-3 - 4)^2] = √[9 + 49] = √58 ≈ 7.62
Side AD: √[(x2 - x1)^2 + (y2 - y1)^2] = √[(-4 - (-1))^2 + (2 - (-3))^2] = √[9 + 25] = √34 ≈ 5.83
The lengths of all four sides are different, so quadrilateral ABCD is not a rhombus either.
In conclusion, based on the information given, quadrilateral ABCD is neither a parallelogram nor a rhombus.