A parallelogram is a convex quadrilateral with four vertices and two sets of opposite parallel sides. Given 3 of the vertices of a parallelogram at A (0,0) B(2,5) and C(2,0), find all possible coordinates for D.
To find the coordinates of point D, we need to consider the properties of a parallelogram. One of the properties is that opposite sides of a parallelogram are parallel.
Therefore, we can determine the position of point D by applying this property to points A, B, and C.
First, let's consider the line passing through points A and B. We can find its equation using the slope-intercept form, which is:
y - y1 = m(x - x1)
where (x1, y1) are the coordinates of a point on the line and m is the slope of the line.
The slope (m) can be calculated by dividing the difference in y-coordinates by the difference in x-coordinates:
m = (y2 - y1) / (x2 - x1)
Using points A(0,0) and B(2,5), we have:
m = (5 - 0) / (2 - 0) = 5/2
Using the point-slope form, we can write the equation of the line passing through A and B as:
y - 0 = (5/2)(x - 0)
Simplifying the equation gives:
2y = 5x
Next, let's consider the line passing through points C and D. Applying the same approach, the equation of this line is:
y - 0 = m(x - 2)
where m is the slope of the line. Since opposite sides of a parallelogram are parallel, the slope of the line passing through C and D must be the same as the slope of the line passing through A and B.
Therefore, we can write:
y = (5/2)(x - 2)
Now, we need to find the coordinates of point D that satisfy this equation. Let's substitute the coordinates of D as (x, y) into the equation:
y = (5/2)(x - 2)
Since point D is on the line passing through C and D, we can substitute its x-coordinate as 2 to find the corresponding y-coordinate:
y = (5/2)(2 - 2)
Simplifying further:
y = 0
Hence, the possible coordinates for point D are given by (2,0).
In summary, for the given parallelogram with vertices at A(0,0), B(2,5), and C(2,0), the possible coordinates for D are (2,0).