Consider the line segments AB and BC graphed on the coordinate plane shown. What will be the coordinates of point D if ABCD forms a parallelogram?
A graph is shown in the xy-plane. The values on the x-axis ranges from negative 8 to 8 in increments of 2 and the values on the y-axis ranges from negative 8 to 8 in increments of 2. Three points Upper A, Upper B, and Upper C are marked on the graph with coordinates (negative 4, negative 3), (negative 4, 7), and (3, 4) respectively. A line is formed by joining the points Upper A and Upper B and a line is formed by joining the points Upper B and Upper C.
A.
(3,−3)
B.
(3,0)
C.
(3,−4)
D.
(3,−6)
Since AB and BC are two sides of a parallelogram, their slopes must be the same:
slope of AB = (7 - (-3))/(-4 - (-4)) = 5
slope of BC = (4 - 7)/(3 - (-4)) = -3/7
Since these slopes are not equal, ABCD cannot be a parallelogram. Therefore, the question is invalid and there is no answer.
To find the coordinates of point D, we need to determine the coordinates of point A, B, and C, and then use their properties to find point D.
Given:
Point A: (-4, -3)
Point B: (-4, 7)
Point C: (3, 4)
We know that opposite sides of a parallelogram are parallel and equal in length. So, the line segment AB is parallel to the line segment CD, and the line segment BC is parallel to the line segment AD.
Since AB is parallel to CD, the slope of AB should be equal to the slope of CD.
The slope of AB:
m1 = (change in y) / (change in x)
m1 = (7 - (-3)) / (-4 - (-4))
m1 = 10 / 0
m1 is undefined
Since the slope of AB is undefined, we know that the line segment CD should also have an undefined slope.
The only option that has an undefined slope is option B: (3,0).
Thus, the coordinates of point D, if ABCD forms a parallelogram, would be (3,0).