# A(-2,2) and C(4,-1)are opposite vertices of a parallelogram ABCD whose sides are parallel to the lines x=0 and 3y=x, a)Find the coordinates of B and D

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Given the points A(-2, 2) and c(4,-1) are opposite vertices's of a parallelogram.

Then the equations of the lines for the parallelogram can be obtained .

A(-2,2) is on both lines, so it satisfies both equations.

==> y-2 = m (x+2)

Now the slope for one side is the same slope for the line x= 0

==> Then, one side of the parallelogram is parallel to the y-axis.

==> Then, the equation of the line is

x = -2.............(1)

The other side of the parallelogram is parallel to 3y=x ==> Then the slope is (1/3)

==> y-2 = (1/3)(x+2)

==> y= (1/3)x + 2/3 + 2

==> y= (1/3)x +8/3 ............(2)

Then, we have the equation of the lines of the sides of the parallelogram that intersects at the point (-2,2)

Now we will find the other sides.

==> We will use the point B(4,-1)

==> The first equation of the line that is parallel to the y-axis.

==> Then the equation is x = 4 ..........(3)

==> The other side has the following equation.

==> y +1 = (1/3) (x-4)

==> y= (1/3)x - 4/3 -1

==> y= (1/3)x -7/3 ..............(4)

Now the remaining points are the intersection points between the lines (1) with (4) and the lines (2) with (3).

Intersection point between equation (1) and (4):

==> x= -2 and y= (1/3)x - 7/3

==> y= (-2/3 - 7/3 = -9/3 = -3

==> Then, one of the points is ( -2, -3)

Intersection between lines (2) and (3).

==> x= 4 and y= (1/3)x +8/3 ==> y= 4/3 + 8/3 = 12/3 = 4

Then, the point is (4,4)

**==> Then, the points B and D are (4,4) and (-2,-3).**