Question

A triangular garden has its corners at points A(2, 3), B(5, 7), and C(1, 6) on a coordinate plane. If a fourth point D is added to form a parallelogram ABCD, what are the coordinates of point D?

• A. (2,2)
• B. (-4,2)
• C. (-2,2)
• D. (2,4)

Answer 1

The Correct Option is C

In a parallelogram, diagonals bisect each other.
Let D be (x, y).
Midpoint of AC = Midpoint of BD.
Midpoint formula:\(\bigg(\frac{(x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}\bigg)\)
Midpoint of AC =\(\bigg(\frac{(2+1)}{2}, \frac{(3+6)}{2}\bigg) = (\frac{3}{2}, \frac{9}{2})\)
Midpoint of BD =\(\bigg(\frac{(5+x)}{2}, \frac{(7+y)}{2}\bigg)\)
Equating the x-coordinates:
\(\frac{(5+x)}{2} = \frac{3}{2}\)
\(\Rightarrow\)x = -2
Equating the y-coordinates:
\(\frac{(7+y)}{2} = \frac{9}{2}\)
\(\Rightarrow\)y = 2
Therefore, the coordinates of point D are (-2, 2).