True or False. A quadrilateral with these following vertices is a parallelogram.
(-4,-3),(2,-3),(4,-6),(-4,-6)
Calling the points ABCD,
slope of AB = 0
slope of CD = 0 so AB || CD
slope of BC = -3/2
AD is vertical
so BD not || to AD
no parallelogram - it is, however, a trapezoid.
To determine if a quadrilateral is a parallelogram, we need to check if it satisfies the properties of a parallelogram.
A parallelogram is a quadrilateral with two pairs of parallel sides. One way to verify if a quadrilateral is a parallelogram is by comparing the slopes of its sides.
1. Find the slopes of the sides:
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁).
For the given quadrilateral with vertices (-4,-3), (2,-3), (4,-6), and (-4,-6), let's find the slopes of the sides:
Side AB: A(-4,-3), B(2,-3)
slope AB = (-3 - (-3)) / (2 - (-4)) = 0 / 6 = 0
Side BC: B(2,-3), C(4,-6)
slope BC = (-6 - (-3)) / (4 - 2) = (-3) / 2 = -1.5
Side CD: C(4,-6), D(-4,-6)
slope CD = (-6 - (-6)) / (-4 - 4) = 0 / (-8) = 0
Side DA: D(-4,-6), A(-4,-3)
slope DA = (-3 - (-6)) / (-4 - (-4)) = 3 / 0 (undefined slope)
2. Check the conditions:
If the slopes of opposite sides are equal, and the slopes of adjacent sides are different, then the quadrilateral is a parallelogram.
In this case, slope AB = slope CD = 0, and slope BC ≠ slope DA. Therefore, the given quadrilateral satisfies the conditions for a parallelogram.
Conclusion:
True. The quadrilateral with vertices (-4,-3), (2,-3), (4,-6), and (-4,-6) is a parallelogram.