The coordinates of the vertices of a quadrilateral are A(-2, 5), B(6, 5), C(4, -3), and D(-4, -3). Use slope to show that ABCD is a parallelogram.
show that slope AB = slope DC
and slope AD = slope BC
I will do the first pair, you do the second
slope AB = (5-5)/6+2) = 0/8 = 0
slope DC = (-3+3)/(-4-4) = 0/-8 = 0
so AB || DC
test if AD || BC
if so, then by definition, it is a parallogram.
To determine whether quadrilateral ABCD is a parallelogram, we need to show that opposite sides are parallel. We can use the concept of slope to do this.
The slope between two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Let's calculate the slopes of the sides:
1. Slope of AB:
Using points A(-2, 5) and B(6, 5):
m_AB = (5 - 5) / (6 - (-2))
= 0 / 8
= 0
2. Slope of BC:
Using points B(6, 5) and C(4, -3):
m_BC = (-3 - 5) / (4 - 6)
= -8 / (-2)
= 4
3. Slope of CD:
Using points C(4, -3) and D(-4, -3):
m_CD = (-3 - (-3)) / (-4 - 4)
= 0 / (-8)
= 0
4. Slope of DA:
Using points D(-4, -3) and A(-2, 5):
m_DA = (5 - (-3)) / (-2 - (-4))
= 8 / 2
= 4
Now, let's check if opposite sides are parallel:
AB || CD if m_AB = m_CD: 0 = 0 (true)
BC || DA if m_BC = m_DA: 4 = 4 (true)
Since the slopes of opposite sides are equal, we can conclude that the quadrilateral ABCD is a parallelogram.