Parallelogram Diagonals and Midpoints

Quick answer: The diagonals of a parallelogram bisect each other. For consecutive vertices A, B, C, D, equal diagonal midpoints imply A + C = B + D coordinate-wise, so D = A + C − B.

Worked example

Find D if A(1, 2), B(5, 3), C(7, 7) are consecutive vertices.

1. x_D = 1 + 7 − 5 = 3.
2. y_D = 2 + 7 − 3 = 6.
3. Thus D = (3, 6).
4. Midpoint of AC is ((1 + 7)/2, (2 + 7)/2) = (4, 4.5).

Check the result

Midpoint of BD is ((5 + 3)/2, (3 + 6)/2) = (4, 4.5), so the diagonals bisect.

Common mistake to avoid

The vertex order matters. If three points are given without their order, different fourth vertices may be possible.

Try it yourself

1. For A(0,0), B(4,0), C(6,3), find D. 2. Do parallelogram diagonals always have equal lengths?

Answers and reasoning

1. D = (2,3). 2. No; they bisect each other, but equal lengths require an additional condition, as in a rectangle.

Need help with your own attempt? Ask a study question. AI answers require an account and paid credits. Check important steps with your textbook or teacher.